Springboard Unit 1 - Equations, Inequalities and Functions

Unit 1Planning the Unit I n this unit, students model real-world situations by using one- and two-variable equations. They study inverse functions, composite functions, and piecewisedefined functions, perform operations on functions, and solve systems of equations and inequalities. Vocabulary Development The key terms for this unit can be found on the Unit Opener page. These terms are divided into Academic Vocabulary and Math Terms. Academic Vocabulary includes terms that have additional meaning outside of math. These terms are listed separately to help students transition from their current understanding of a term to its meaning as a mathematics term. To help students learn new vocabulary: Have students discuss meaning and use graphic organizers to record their understanding of new words. Remind students to place their graphic organizers in their math notebooks and revisit their notes as their understanding of vocabulary grows. As needed, pronounce new words and place pronunciation guides and definitions on the class Word Wall. Embedded Assessments © 2015 College Board. All rights reserved. Embedded Assessments allow students to do the following: Demonstrate their understanding of new concepts. Integrate previous and new knowledge by solving real-world problems presented in new settings. They also provide formative information to help you adjust instruction to meet your students’ learning needs. Prior to beginning instruction, have students unpack the first Embedded Assessment in the unit to identify the skills and knowledge necessary for successful completion of that assessment. Help students create a visual display of the unpacked assessment and post it in your class. As students learn new knowledge and skills, remind them that they will be expected to apply that knowledge to the assessment. After students complete each Embedded Assessment, turn to the next one in the unit and repeat the process of unpacking that assessment with students. AP / College Readiness Unit 1 continues to prepare students for Advanced Placement courses by: Modeling real-world situations using one- and two-variable equations. Increasing student ability to work with a wide variety of functions. Unpacking the Embedded Assessments The following are the key skills and knowledge students will need to know for each assessment. Embedded Assessment 1 Equations, Inequalities, and Systems, Gaming Systems Systems of equations Systems of inequalities Absolute value equations Embedded Assessment 2 Piecewise-Defined, Composite, and Inverse Functions, Currency Conversion Piecewise-defined functions Composition of functions Inverse functions Unit 1 • Equations, Inequalities, Functions 1a Planning the Unit continued Suggested Pacing The following table provides suggestions for pacing using a 45-minute class period. Space is left for you to write your own pacing guidelines based on your experiences in using the materials. 45-Minute Period Unit Overview/Getting Ready 1 Activity 1 3 Activity 2 2 Activity 3 4 Embedded Assessment 1 1 Activity 4 3 Activity 5 3 Activity 6 2 Embedded Assessment 2 1 Total 45-Minute Periods 20 Your Comments on Pacing Additional Resources Unit Practice (additional problems for each activity) Getting Ready Practice (additional lessons and practice problems for the prerequisite skills) Mini-Lessons (instructional support for concepts related to lesson content) 1b SpringBoard® Mathematics Algebra 2 © 2015 College Board. All rights reserved. Additional resources that you may find helpful for your instruction include the following, which may be found in the Teacher Resources at SpringBoard Digital. 001-002_SB_AG2_SE_U01_UO.indd Page 1 24/04/14 11:19 PM user-g-w-728 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ ... Unit Overview Equations, Inequalities, Functions ESSENTIAL QUESTIONS Unit Overview In this unit, you will model real-world situations by using one- and two-variable linear equations. You will extend your knowledge of linear relationships through the study of inverse functions, composite functions, piecewise-defined functions, operations on functions, and systems of linear equations and inequalities. Key Terms As you study this unit, add these and other terms to your math notebook. Include in your notes your prior knowledge of each word, as well as your experiences in using the word in different mathematical examples. If needed, ask for help in pronouncing new words and add information on pronunciation to your math notebook. It is important that you learn new terms and use them correctly in your class discussions and in your problem solutions. © 2015 College Board. All rights reserved. © 2015 College Board. All rights reserved. Academic Vocabulary • interpret • compare • contrast Math Terms • • • • • • • • • • • • absolute value equation absolute value inequality constraints consistent inconsistent independent dependent ordered triple Gaussian elimination matrix dimensions of a matrix square matrix 1 • feasible • confirm • prove • • • • • • • • • • • • As students encounter new terms in this unit, help them to choose an appropriate graphic organizer for their word study. As they complete a graphic organizer, have them place it in their math notebooks and revisit as needed as they gain additional knowledge about each word or concept. Read the essential questions with students and ask them to share possible answers. As students complete the unit, revisit the essential questions to help them adjust their initial answers as needed. How are composite and inverse functions useful in problem solving? Unpacking Embedded Assessments EMBEDDED ASSESSMENTS This unit has two embedded assessments, following Activities 3 and 6. They will give you an opportunity to demonstrate your understanding of equations, inequalities, and functions. Embedded Assessment 1: multiplicative identity matrix multiplicative inverse matrix matrix equation coefficient matrix variable matrix constant matrix piecewise-defined function step function parent function composition composite function inverse function Key Terms Essential Questions How are linear equations and systems of equations and inequalities used to model and solve real-world problems? Equations, Inequalities, and Systems Ask students to read the unit overview and mark the text to identify key phrases that indicate what they will learn in this unit. p. 55 Embedded Assessment 2: Prior to beginning the first activity in this unit, turn to Embedded Assessment 1 and have students unpack the assessment by identifying the skills and knowledge they will need to complete the assessment successfully. Guide students through a close reading of the assessment, and use a graphic organizer or other means to capture their identification of the skills and knowledge. Repeat the process for each Embedded Assessment in the unit. Piecewise-Defined, Composite, and Inverse Functions p. 99 1 Developing Math Language As this unit progresses, help students make the transition from general words they may already know (the Academic Vocabulary) to the meanings of those words in mathematics. You may want students to work in pairs or small groups to facilitate discussion and to build confidence and fluency as they internalize new language. Ask students to discuss new academic and mathematics terms as they are introduced, identifying meaning as well as pronunciation and common usage. Remind students to use their math notebooks to record their understanding of new terms and concepts. As needed, pronounce new terms clearly and monitor students’ use of words in their discussions to ensure that they are using terms correctly. Encourage students to practice fluency with new words as they gain greater understanding of mathematical and other terms. 1 001-002_SB_AG2_SE_U01_UO.indd Page 2 05/12/13 9:54 AM user-s068a 111 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ ... UNIT 1 Getting Ready UNIT 1 Getting Ready Use some or all of these exercises for formative evaluation of students’ readiness for Unit 1 topics. Write your answers on notebook paper. Show your work. Prerequisite Skills • Evaluating functions (Item 1) HSF-IF.A.2 • Finding slope and intercepts (Item 2) HSF-IF.B.4 • Graphing linear equations (Item 3) HSF-IF.C.7a • Writing linear equations (Items 4–5) HSF-IF.B.4 • Finding additive and multiplicative inverses (Item 6) 7.NS.A.1b • Solving linear and literal equations (Items 7, 9, 10) HSA-REI.B.3, HSA-REI.D.10 • Understanding absolute value (Item 8) 6.NS.C.7 • Finding domain and range (Item 11) HSF-IF.B.5 • Identifying lines of symmetry (Item 12) 4.G.A.3 1. Given f(x) = x2 − 4x + 5, find each value. a. f(2) b. f(−6) 2. Find the slope and y-intercept. a. y = 3x − 4 b. 4x − 5y = 15 3. Graph each equation. a. 2x + 3y = 12 b. x = 7 4. Write an equation for each line. a. line with slope 3 and y-intercept −2 b. line passing through (2, 5) and (−4, 1) 10 output −1 c. −3 −5 y 4 y 2 8 –6 –4 –2 6 2 –2 4 6 2 4 x –4 4 2 2 4 6 8 10 d. x y 4 2 –4 –6 –8 –6 –4 –2 –8 –2 –10 –4 7. Solve 3(x + 2) + 4 = 5x + 7. 8. What is the absolute value of 2 and of −2? Explain your response. 8 8 x –6 6. Using the whole number 5, define the additive inverse and the multiplicative inverse. y 6 –8 12. How many lines of symmetry exist in the figure shown in Item 11c? 9. Solve the equation for x. 3x + y =2 z x=7 2x + 3y = 12 4 2 –10 –8 –6 –4 –2 –2 2 4 6 8 10 x –4 –6 –8 –10 4. a. y = 3x − 2 b. 2x − 3y = −11 5. 3x + 4y = 24 6. Sample answer: The additive inverse of 5 is −5 because 5 + (−5) = 0 and (−5) + 5 = 0. The multiplicative inverse of 5 is 1 because 5 5 1 = 1 and 1 5 = 1. 5 5 7. x = 3 2 8. 2; Absolute value is the distance from 0 on a number line, so it cannot be a negative number. The absolute value of both 2 and −2 is 2. 2z − y 9. x = 3 10. C. Sample explanation: When −1 is substituted for x and −2 is substituted for y in the equation, you get 6(−1) − 5(−2) = −6 + 10 = 4. So, (−1, −2) is a solution. () 2 () 2 SpringBoard® Mathematics Algebra 2, Unit 1 • Equations, Inequalities, Functions 11. a. domain and range: all real numbers b. domain: 3, 7, 11; range: −1, −3, −5 c. domain: −5 ≤ x ≤ 5; range: −1 ≤ y ≤ 1 d. domain and range: all real numbers 12. There are two lines of symmetry, the x- and y-axes. SpringBoard® Mathematics Algebra 2, Unit 1 • Equations, Inequalities, Functions Getting Ready Practice For students who may need additional instruction on one or more of the prerequisite skills for this unit, Getting Ready practice pages are available in the Teacher Resources at SpringBoard Digital. These practice pages include worked-out examples as well as multiple opportunities for students to apply concepts learned. © 2015 College Board. All rights reserved. –2 © 2015 College Board. All rights reserved. –10 –8 –6 –4 –2 1. a. 1 b. 65 2. a. slope: 3, y-intercept: −4 b. slope: 4 , y-intercept: −3 5 3. a–b. 6 11. Find the domain and range of each relation. a. y = 2x + 1 b. input 3 7 11 5. Write the equation of the line below. Answer Key 10 10. Which point is a solution to the equation 6x − 5y = 4? Justify your choice. A. (1, 2) B. (1, −2) C. (−1, −2) D. (−1, 2) 003-016_SB_AG2_SE_U01_A01.indd Page 3 25/02/15 7:33 AM s-059 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ ... ACTIVITY Creating Equations ACTIVITY 1 Directed One to Two Lesson 1-1 One-Variable Equations Learning Targets: • Create an equation in one variable from a real-world context. • Solve an equation in one variable. Activity Standards Focus In Activity 1, students write and solve linear equations in one variable, including multistep equations and equations with variables on both sides. They also write equations in two variables and show solutions to those equations on a coordinate plane. Finally, they write, solve, and graph absolute value equations and inequalities. Throughout this activity, emphasize the importance of performing the same operation on both sides of an equation or inequality in an effort to keep the equation or inequality balanced. My Notes SUGGESTED LEARNING STRATEGIES: Shared Reading, Activating Prior Knowledge, Create Representations, Identify a Subtask, ThinkPair-Share, Close Reading A new water park called Sapphire Island is about to have its official grand opening. The staff is putting up signs to provide information to customers before the park opens to the general public. As you read the following scenario, mark the text to identify key information and parts of sentences that help you make meaning from the text. The Penguin, one of the park’s tube rides, has two water slides that share a single line of riders. The table presents information about the number of riders and tubes that can use each slide. Lesson 1-1 PLAN Penguin Water Slides Slide Number Tube Size Tube Release Time 1 2 riders every 0.75 min 2 4 riders every 1.25 min © 2015 College Board. All rights reserved. © 2015 College Board. All rights reserved. Jaabir places a sign in the waiting line for the Penguin. When a rider reaches the sign, there will be approximately 100 people in front of him or her waiting for either slide. The sign states, “From this point, your wait time is minutes.” Jaabir needs to determine the number of approximately minutes to write on the sign. Work with a partner or with your group on Items 1–7. 1. Let the variable r represent the number of riders taking slide 1. Write an algebraic expression for the number of tubes this many riders will need, assuming each tube is full. 1 Pacing: 1 class period Chunking the Lesson #1–2 #3–5 #6–7 #8 #9 #10–11 Check Your Understanding Lesson Practice MATH TIP TEACH An algebraic expression includes at least one variable. It may also include numbers and operations, such as addition, subtraction, multiplication, and division. It does not include an equal sign. Ask students to translate each phrase to an equation. r 2 Bell-Ringer Activity 1. Six more than twice a number c is 24. [6 + 2c = 24] 2. One-third of a number y is 45.  1 y = 45   3  3. Seven less than the product of a number and 10 is 50. [10n − 7 = 50] Discuss with students the methods they used to translate the sentences into equations. 2. Next, write an expression for the time in minutes it will take r riders to go down slide 1. ( 2r )0.75 Developing Math Language 3. Assuming that r riders take slide 1 and that there are 100 riders in all, write an expression for the number of riders who will take slide 2. 100 − r Activity 1 • Creating Equations 3 Common Core State Standards for Activity 1 HSA-CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear functions. HSA-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. This lesson refers to both expressions and equations. An expression can be made up of numbers, variables, constants, arithmetic operation symbols, and grouping symbols. If an expression is algebraic, then it must contain at least one variable. Equal signs are never a part of an expression. An equation is a mathematical sentence that contains an equal sign, showing that two expressions are equivalent to each other. 1–2 Activating Prior Knowledge, Visualization, Create Representations If students are struggling with Item 2, give them a hint to multiply the number of tubes needed for r riders by the time needed for each tube to go down the slide. This expression will combine the result from Item 1 with the information that is found in the table. Activity 1 • Creating Equations 3 003-016_SB_AG2_SE_U01_A01.indd Page 4 05/12/13 2:50 AM s-059 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ ... ACTIVITY 1 Continued Lesson 1-1 One-Variable Equations ACTIVITY 1 Show English language learners various pictures of water slides. Explain that riders climb up stairs or take an elevator to the top of the slide and then slide down the slide into a pool of water below. Ask students to share if there is any type of ride or activity similar to this in their native countries. My Notes 100 − r 4 5. Write an expression for the time in minutes needed for the riders taking slide 2 to go down the slide. 3–5 Activating Prior Knowledge, Simplify the Problem If students are struggling with Item 3, then share this simpler but similar problem: Suppose Jaabir is aware that of the 100 riders, 46 plan on riding slide 1. How many of those riders are planning on riding slide 2? How did you find this? [by using subtraction] Now the only difference is that Jaabir does not know the exact number of riders for slide 1; however, the variable r represents slide 1 riders. Use the same method you used to find slide 2 riders—subtraction. The only difference is you do not know the exact number of each. It may be beneficial for some students to review the properties of equality using numeric examples, rather than only the algebraic definitions. You can use the following examples to demonstrate the properties numerically: • Addition Property of Equality: Start with 8 = 8; add 4 to both sides. 8 + 4 = 8 + 4, or 12 = 12✓ (1004− r )1.25 6. Since Jaabir wants to know how long it takes for 100 riders to complete the ride when both slides are in use, the total time for the riders taking slide 1 should equal the total time for the riders taking slide 2. Write an equation that sets your expression from Item 2 (the time for the slide 1 riders) equal to your expression from Item 5 (the time for the slide 2 riders). ( 2r )0.75 = (1004− r )1.25 MATH TIP 7. Reason abstractly and quantitatively. Solve your equation from Item 6. Describe each step to justify your solution. These properties of real numbers can help you solve equations. Addition Property of Equality If a = b, then a + c = b + c. Subtraction Property of Equality If a = b, then a − c = b − c. Multiplication Property of Equality If a = b, then ca = cb. Sample work: ( 2r )0.75 = (1004− r )1.25 0.375r = (100 − r)0.3125 Simplify each side. 0.375r = 31.25 − 0.3125r Distributive Property 0.6875r = 31.25 r ≈ 45.5 Add 0.3125r to each side. Divide each side by 0.6875. Division Property of Equality If a = b and c ≠ 0, then a = b . c c Distributive Property a(b + c) = ab + ac • Subtraction Property of Equality: Start with −4 = − 4; subtract 2 from both sides. −4 − 2 = −4 − 2, or −6 = − 6✓ • Multiplication Property of Equality: Start with 16 = 16; multiply both sides by −3. 16(−3) = 16(−3), or −48 = −48✓ • Division Property of Equality: Start with −45 = −45; divide both sides by −9. −45 = −45 , −9 −9 or 5 = 5✓ 4 4 SpringBoard® Mathematics Algebra 2, Unit 1 • Equations, Inequalities, Functions 6–7 Discussion Groups, Sharing and Responding, Think-Pair-Share After students complete Item 6, they can discuss the process of solving this problem in Item 7. If there are students within a group having trouble getting from one step to the next in the solution, other students may provide an explanation. Additionally, these steps are samples, so if any students approach the problem with a different strategy (for example, they multiply through first by 100 to eliminate some of the decimals and/or they multiply both sides of the equation by 4 in order to eliminate fractions), they could share this with their peers and demonstrate that there is more than one way to arrive at the correct solution. SpringBoard® Mathematics Algebra 2, Unit 1 • Equations, Inequalities, Functions © 2015 College Board. All rights reserved. Differentiating Instruction 4. Using the expression you wrote in Item 3, write an expression for the number of tubes the riders taking slide 2 will need, assuming each tube is full. © 2015 College Board. All rights reserved. ELL Support continued 25 MATH TIP When you evaluate an algebraic expression. a. It is not certain that there will be exactly 100 people in front of the sign. ( 2r )0. For example. Furthermore. Construct an Argument. does not divide evenly by the 1.indd Page 5 15/04/14 10:20 AM user-g-w-728 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . Also. The term evaluate will surface again in later chapters. the answer in Item 9.75 and 1. you replace each variable in the expression by a given value and simplify the result. Activity 1 • Creating Equations 5 This lesson contains the vocabulary term evaluate. given that r represents a number of riders (people). when students will be asked to evaluate the value of a function with a given input value. © 2015 College Board. Make sense of problems. then why? Ask students to reflect on whether their results or solutions to a problem make sense. Construct an Argument Have students work with a partner or in a small group to interpret the solution of r = 45. your wait is approximately 17 8 Discussion Groups. All rights reserved. © 2015 College Board. My Notes Because the value of r represents a number of riders. b. 10. 17. you substitute values for the variables and then simplify the expression. there will be approximately 100 people waiting in front of him or her. a. Recall that when a rider reaches the sign. When you evaluate an expression. What number should Jaabir write to complete the statement on the sign? From this point. Consider the meaning of the solution from Item 7.003-016_SB_AG2_SE_U01_A01. for several reasons. students read that it will take about the same amount of time for the remaining riders to go down slide 2.75 minutes). Explain why you should or should not round the value of r to the nearest whole number. So. Jaabir should place 17 minutes on the board as the best approximation.. the “common multiples” of 0. Have them explore whether this solution actually makes sense. Discussion Groups. Since the result from Item 9b is 17 minutes. All rights reserved. 9 Critique Reasoning. you should round r to the nearest whole number. Activity 1 • Creating Equations 5 . 10–11 Discussion Groups. 17 minutes The rest of the 100 riders will go through slide 2 in about the same amount of time. and it doesn’t make sense to have a fraction of a rider.5..75 = ( 462 )0. Debriefing Prior to Item 10.25-minute timing cycle of slide 2 (neither does 17 minutes.25 minutes closest to 17 are 15 and 18. your answer to Item 9b gives an estimate of the number of minutes it will take all 100 riders to go down the Penguin slides. There are different factors that could affect this number. Use the expression you wrote in Item 2 to determine how long it will take the number of riders from Item 8b to go through slide 1. whether people stand close together or farther apart could impact the total number. Is this a situation where rounding to the nearest whole number is necessary? If so.25 minutes. minutes. Construct an Argument It is important that students understand that the number of minutes written on the sign is an estimate. This problem depends upon approximations and estimations. Evaluate the expression for the appropriate value of r. How many people out of the 100 riders will take slide 1? Developing Math Language 46 riders 9. the size (age and weight) of the people would have an impact on the number of people in line. How many minutes will it take the riders to go through slide 1? Round to the nearest minute. b.75 = 17. ACTIVITY 1 Continued Lesson 1-1 One-Variable Equations ACTIVITY 1 continued 8. 16. to 5 p. 12. along with numbers and operations. 12. have them write out word equations. or the approximate number of slide 2 riders) and multiply it by 1. minutes. you state the meaning of the solution in the context of the problem or real-world situation. When full. Solve your equation from Item 14. expression. h ≈ 2. Functions © 2015 College Board. and interpret the solution. See the Activity Practice for additional problems for this lesson. 15. 16. 10–11 (continued) Another way to check the answer would be to take 54 (which is the value of 100 − r. All rights reserved. Unit 1 • Equations. 17. When students answer Item 12. Students may note that the number of gallons added to the pool plus the 20. An expression can include variables. Describe how you could check that your answer to Item 10 is reasonable.. Sample answer: I could substitute the value of r into the equation and check that it makes the equation true. depending on the answer to Item 17 SpringBoard® Mathematics Algebra 2. Some students may write and solve an equation to find the answer. Explain the relationships among the terms variable. ASSESS Students’ answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the lesson concepts and their ability to apply their learning. 14. Other students may reason that the wait time for 250 riders will be 2. Check Your Understanding Check Your Understanding Debrief students’ answers to these items to ensure that they understand concepts related to interpreting the solution to an equation. The park has 8 ticket booths. you may also want to have them explain how they arrived at their answer. 43 min. Use this information for Items 16–18.1. the water park may not necessarily be filling every spot in each tube. ACTIVITY 1 Continued Lesson 1-1 One-Variable Equations ACTIVITY 1 continued My Notes 11. All rights reserved. there will be approximately 250 people in front of him or her. ACADEMIC VOCABULARY When you interpret a solution.000 gallons. Other students will note that the park technically needs more than 13 ticket sellers and will give an answer of 14. . Functions 15. An equation is a statement that two expressions are equal. Alert students that even though each tube on slide 1 will hold 2 passengers and each tube on slide 2 will hold 4 passengers. Explanations will vary. ticket sellers work 30 hours per week.000 gallons of water and is being filled at a rate of 130 gallons per minute. Sapphire Island is open 7 days a week. I could also evaluate the expression from Item 5 to check that I get the same time for slide 2 as for slide 1.1 is sufficiently close to 13 that 13 ticket sellers will be enough for the park. Suppose that Jaabir needs to place a second sign in the waiting line for the Penguin slides. 30t = 49(8) or equivalent. Students may note that the minimum number of ticket sellers times 30 hours per week equals the total number of hours worked by the ticket sellers per week. Answers 13. and lunch breaks.003-016_SB_AG2_SE_U01_A01. 130(60h) + 20. ADAPT Check students’ answers to the Lesson Practice to ensure that they understand basic concepts related to writing and solving equations as well as interpreting solutions of equations.000 gallons of water. and interpret the solution.000 gallons already in the pool must equal 43. and each booth has a ticket seller from 10:00 a. Write an equation that can be used to find t. LESSON 1-1 PRACTICE Use this information for Items 14–15.000. 18. vacation. The park plans to hire 20 percent more than the minimum number of ticket sellers needed in order to account for sickness. The solution shows that it will take about 3 more hours to fill the pool..m. 18. From this point. Unit 1 • Equations.25. They may also note that the minutes needed to fill the pool equals the number of hours h times 60. How many ticket sellers should the park hire? Explain. Explanations may vary.m. the minimum number of ticket sellers the park needs.875 ≈ 17 minutes. Students may also note that the total number of hours worked by the ticket sellers each week is equal to the number of hours the park is open times the number of booths.000 = 43. 16 or 17.5 times the wait time for 100 riders. Some students may reason that 13. 54 × 1. Solve your equation from Item 16. Inequalities. t ≈ 13. or 49(8). The pool currently holds 20. Explain the steps you used to write your equation. Explanations may vary. 6 6 SpringBoard® Mathematics Algebra 2. one of the pools at Sapphire Island will hold 43. On average. Model with mathematics. and equation. What number should Jaabir write to complete the statement on this sign? Explain how you determined your answer.9. Inequalities. Sample answer: A variable is a letter or symbol that represents an unknown value or values. Write an equation that can be used to find h. If students are having difficulty creating equations that model the situation.indd Page 6 05/12/13 2:50 AM s-059 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . You may assign the problems here or use them as a culmination for the activity. 17. LESSON 1-1 PRACTICE 14. Explain the steps you used to write your equation.25 = 16. your wait is approximately © 2015 College Board. When a rider reaches this sign. 13. the number of hours it will take to fill the pool from its current level. Debriefing.. Role Play. © 2015 College Board. Maria works at the rental booth and is preparing materials so that visitors and employees will understand the pricing of the tubes. 4x − 7 = 2y [4x − 2y = 7] 2. and C are integers and A is nonnegative. Next. 3–7 Activating Prior Knowledge. write the terms Dependent variable and Independent variable. As you work in groups on Items 1–7. Look for a Pattern. drawing from what they remember from their previous math courses. Lesson 1-2 MATH TIP Recall that in a relationship between two variables. Encourage them to find a pattern from one number to the next. All rights reserved. Maria started making a table that relates the number of hours a tube is rented to the cost of renting the tube. For example. • Graph two-variable equations. The time in hours is the independent variable because this value determines the cost of renting the tube. make notes about what you want to say. After visiting various groups. Chunking the Activity. Listen carefully to other group members as they describe their ideas. What does the independent variable x represent in this situation? Explain. have the students share their patterns and any other findings with the class as a whole. PLAN My Notes Pacing: 1 class period Chunking the Lesson #1–2 #3–7 #8–13 #14 Check Your Understanding Lesson Practice SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge. ACTIVITY 1 Continued Lesson 1-2 Two-Variable Equations ACTIVITY 1 continued Learning Targets: equations in two variables to represent relationships between • Create quantities. Sample answer: Add $2 to the cost of renting a tube for 5 hours: $15 + $2 = $17. Use the information above to help you complete the table. 3. going downward in both columns of the table.003-016_SB_AG2_SE_U01_A01. that employees can use to calculate the cost of renting a tube for any number of hours. Tube Rentals Hours Rented Cost ($) 1 7 2 9 3 11 4 13 5 15 1. Explain how a customer could use the pattern in the table to determine the cost of renting a tube for 6 hours. circulate around the room and have students tell you the rental fee for some arbitrary number of hours that is not already in the table. Look for a Pattern. Beneath the Know column. Want to Know. Summarizing. 5y = x − 3 [x − 5y = 3] DISCUSSION GROUP TIP If you need help in describing your ideas during group discussions. 12 = y − 8x [8x − y = −12] 3. Discuss your understanding of the problems and ask peers or your teacher to clarify any areas that are not clear. Reason abstractly. All rights reserved. The number of hours the tube is rented. 2. While they are working on this..indd Page 7 25/02/15 7:34 AM s-059 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . dependent in the real world. x and y. Identify a Subtask At Sapphire Island. Construct a KWL Chart by writing Know. TEACH Bell-Ringer Activity Have students write each equation in the form Ax + By = C. Create Representations. and Learn as column headings in one row across the board. review the above problem scenario carefully and explore together the information provided and how to use it to create potential solutions. and ask for clarification of meaning for any words routinely used by group members. and ask students to define them in their own words. 1. Maria wants to write an equation in two variables. Activity 1 • Creating Equations 7 . Interactive Word Wall. Renting a tube costs a flat fee of $5 plus an additional $2 per hour. Activity 1 • Creating Equations 7 1–2 Activating Prior Knowledge. B. © 2015 College Board. Think-Pair-Share. Paraphrasing. visitors can rent inner tubes to use in several of the park’s rides and pools. students are dependents of their parents. where A. the value of the independent variable determines the value of the dependent variable. Group Presentation Have students work in pairs or small groups to complete the second column of the table. KWL Chart Discuss with students what it means to be independent vs. At this screen. © 2015 College Board. as follows: Press y= to enter the equation 5 + 2x. Functions SpringBoard® Mathematics Algebra 2. (2. write how the items in the first two columns can be tied together to answer Items 6 and 7. . For additional technology resources. ACTIVITY 1 Continued Lesson 1-2 Two-Variable Equations ACTIVITY 1 continued My Notes The cost in dollars of renting the tube. x. 15) SpringBoard® Mathematics Algebra 2. (4. students can type in any number of hours at the X= prompt. Unit 1 • Equations. Press 2nd WINDOW to access TBLSET. The cost is the dependent variable because this value depends on the number of hours the tube is rented. 7. y = 5 + 2x Technology Tip Students can use the table function on a graphing calculator to find how much to charge a customer. 3–7 (continued) Beneath the Want to Know column. What does the dependent variable y represent in this situation? Explain. 8. How can you tell whether the equation you wrote in Item 5 correctly models the situation? Sample answer: Substitute values for x into the equation and check whether they give the correct values for y by comparing the results to the table on the previous page. This value is the amount to charge the customer in dollars. Explain how an employee could use the equation to determine how much to charge a customer. MATH TIP Before you can graph the equation. Tbl = 1. Functions © 2015 College Board. 9). List five ordered pairs that lie on the graph of the relationship between x and y. 8 4. 7). visit SpringBoard Digital. Write an equation that models the situation. One way to do this is by using pairs of corresponding values from the table on the previous page. the employee should solve the equation for y. write how the dependent and independent variables are represented in this situation. All rights reserved. Inequalities. Sample answer: The employee should substitute the number of hours the customer rented the tube for x in the equation. Set the Tblstart = 0. Sample answer: (1. you need to determine the coordinates of several points that lie on its graph. 8 Maria also thinks it would be useful to make a graph of the equation that relates the time in hours a tube is rented and the cost in dollars of renting a tube. 6. Unit 1 • Equations.003-016_SB_AG2_SE_U01_A01. as well as a linear equation that can be written to model the situation (refer to Bell-Ringer Activity if needed). Construct viable arguments.” Then press 2nd GRAPH to access the table.indd Page 8 05/12/13 2:50 AM s-059 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . 5. Next.. 11). 13). This is a great way to check their work and practice using technology. y. The corresponding charge will appear next to it in the y1 column.. Beneath the Learn column. (5. and Indpnt: to “Ask. Inequalities. All rights reserved. (3. based upon the number of hours. You can also choose several values of x and substitute them into the equation to determine the corresponding values of y. © 2015 College Board. Debriefing Have students predict what they think the graph will look like based upon the previous items leading up to this set of items. Sample explanation: The graph has a constant rate of change and therefore models a linear equation. Item 12 refers to the y-intercept.003-016_SB_AG2_SE_U01_A01. 2.. What is the y-intercept of the graph? Describe what the y-intercept represents in this situation. B. Use the grid below to complete parts a and b. Write an appropriate title for the graph based on the real-world situation. MATH TIP The variable x represents time and the variable y represents cost. 5. A common error students make is not connecting.and y-axes. where A. and that every point on the line is a solution. In this particular case. which can be plotted on a coordinate plane. Based on the graph. Also write appropriate titles for the x. The slope of a line is the ratio of the change in y to the change in x between any two points. It would not make sense for either of these variables to be negative. The axes should be labeled according to what the x. the flat fee in dollars for renting a tube 13. b. this is a linear equation. Explain why the graph is only the first quadrant. Reason quantitatively. MATH TIP 11. 8–13 Predict and Confirm. explain why there is no x-intercept (at least for this situation). © 2015 College Board. and C are integers and A is nonnegative. Recall that a linear equation is an equation whose graph is a line. a. explain how you know whether the equation that models this situation is or is not a linear equation. My Notes Check students’ answers.. What is the slope of the graph? Describe what the slope represents in this situation. A linear equation can be written in standard form Ax + By = C. The easiest way to get these ordered pairs is from the table created in Item 1.indd Page 9 13/01/15 11:18 AM ehi-6 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . the graph is in the first quadrant because you cannot rent for a negative number of hours. When graphed. Remind students that a solution of an equation in two variables is an ordered pair of numbers. x 10. 12. the hourly cost in dollars of renting a tube The y-intercept of a graph is the y-coordinate of a point where the graph intersects the y-axis. Tube Rentals Cost ($) y 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Universal Access 1 2 3 4 5 Number of Hours Rented 6 Yes. or drawing a line through. the ordered pairs should form a straight line. Graph the ordered pairs you listed in Item 8. All rights reserved. Activity 1 • Creating Equations 9 Activity 1 • Creating Equations 9 . ACTIVITY 1 Continued Lesson 1-2 Two-Variable Equations ACTIVITY 1 continued 9. Then connect the points with a line or a smooth curve. Emphasize that there are infinite solutions to equations in two variables. All rights reserved.and y-values represent— hours and cost. respectively. the points. Graph the equation.000 80 19. Sample explanation: The equation cannot be written in the form Ax + By = C.000 gallons of water. The directions for the chemical say to add 16 fluid ounces per 10. 21.000 16 20.. and 50. The cost of renting a tube increases as the number of hours it is rented increases. m. See the Activity Practice for additional problems for this lesson. Construct viable arguments.000. . Amount of Chemical to Add Gallons of Water Fluid Ounces of Chemical 10. is −8. 30. What quantity and units would be represented on each axis? Check Your Understanding Debrief students’ answers to these items to ensure that they understand concepts related to two-variable linear equations. Write a linear equation in two variables that models the situation. SpringBoard® Mathematics Algebra 2.000. ACTIVITY 1 Continued Lesson 1-2 Two-Variable Equations ACTIVITY 1 continued My Notes DISCUSSION GROUP TIP Share your description with your group members and list any details you may not have considered before. Ask students to share their ideas and have a classroom discussion as to why their scenarios are or are not plausible. 20. Answers LESSON 1-2 PRACTICE 15. The equation models the cost y in dollars of purchasing x tickets to the water park with a coupon. Be sure to include titles and use an appropriate scale on each axis. in thousands of gallons. 10 10 SpringBoard® Mathematics Algebra 2. is 40.000 gallons of water. Tickets to the park normally cost $40 each. What are the slope and y-intercept of the graph? What do they represent in the situation? 22. Functions © 2015 College Board. No. not a line. If students are having difficulty graphing the equations. Work with your group. Use nonverbal cues such as raising your hand to ask for clarification of others’ ideas. © 2015 College Board. Unit 1 • Equations. 19. Highlight that the slope of the equation. use synonyms or request assistance from group members to help you convey your ideas. be sure to use appropriate vocabulary. Write the equation y = 40x − 8 right below.000. 14 Marking the Text. 16. 17. Some of the water features at Sapphire Island are periodically treated with a chemical that prevents algae growth. If you do not know the exact words to describe your ideas. Sample answer: The water park mails out coupons for $8 off the total cost of a ticket purchase. Make a table that shows how much of the chemical to add for water features that hold 10. Inequalities. Functions LESSON 1-2 PRACTICE 18. 16. Quantities and units will vary depending on the scenario the student wrote for Item 14. Use this information for Items 18–22. y represents the number of fluid ounces of the chemical to add to a water feature. for which this equation could be modeled. All rights reserved. y increases as x increases. Explain why the slope of the line you graphed in Item 9 is positive.000 48 40. ADAPT Check students’ answers to the Lesson Practice to ensure that they understand basic concepts related to writing and graphing equations in two variables.000 64 50. Then use the values of x and y to write ordered pairs. Inequalities. Unit 1 • Equations. Explain how you would graph the equation from Item 14. Discussion Groups. both real-world and mathematical. 18. 14. Have students spend a few minutes collaborating in small groups to create a plausible scenario. and x represents the amount of water the feature holds.indd Page 10 25/02/15 7:35 AM s-059 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . So. Debriefing.003-016_SB_AG2_SE_U01_A01. 20. review the process of creating a table of values and identifying slope and intercepts.000. which indicates a positive slope. graph the ordered pairs and draw a line through them. Group Presentation Write the slope-intercept form of a line (y = mx + b) on the board. Describe a plausible scenario related to the water park that could be modeled by this equation: y = 40x − 8. The graph of the equation is a curve. ASSESS Students’ answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the lesson concepts and their ability to apply their learning. Refer to the Word Wall and any notes you may have made to help you choose words for your description. Be sure to stress this is a negative value this time.000 gallons of water. related to the water park. Sample answer: Substitute several values of x into the equation to find the corresponding values of y. 40. Finally. Check Your Understanding 15.000 32 30.. Sample answer: y = 1. Tell what each variable in the equation represents. 17. Is the equation y = −2x + x2 a linear equation? Explain how you know.6x. An employee adds 160 fluid ounces of the chemical to a feature that holds 120. In your description. All rights reserved. Did the employee add the correct amount? Explain. You may assign the problems here or use them as a culmination for the activity. b. Highlight that the y-intercept. 11 Activity 1 • Creating Equations 11 .6. All rights reserved. Add 5 to both 2|x − 1| = 6 sides and then divide by 2. It should be emphasized that when solving absolute value equations. Since −(ax + b) = c is equivalent to ax + b = −c. |x + 2| + 3 = 1 no solution LESSON 1-3 PRACTICE (continued) 20. |x|= 6 4 0 Example C Example B Have students evaluate the following: To graph the solutions. Amount of Chemical (fl oz) © 2015 College Board. MATH TIP Recall that the geometric interpretation of |x| is the distance from the number x to 0 on a number line. |x − 1| = 3 Step 2: Write and solve two equations x − 1 = 3 or x − 1 = −3 x = 4 or x = −2 using the definition of absolute value. Substitute 4 and −2 for x in the original equation. x = 1.. Pacing: 1 class period Chunking the Lesson SUGGESTED LEARNING STRATEGIES: Marking the Text. –5 –4 –3 –2 –1 #1 Lesson Practice MATH TERMS 2|−2 − 1| − 5 = 1 2|−3| − 5 = 1 2(3) − 5 = 1 6−5=1 –2 Example A Check Your Understanding Check to see if both solutions satisfy the original equation. and graph absolute value equations. Step 1: Isolate the absolute value 2|x − 1| − 5 = 1 expression. No. x = −3 x=3 Bell-Ringer Activity Students should recall that an absolute value of a number is its distance from zero on a number line. plot points at 4 and −2 on a number line. All rights reserved. the goal is to isolate the variable. Interactive Word Wall Point out the Math Tip to reinforce why two solutions exist. the slope represents the number of fluid ounces of the chemical to add to 1000 gallons of water. Try These A x = 5. Self Revision/Peer Revision You can use the definition of absolute value to solve absolute value equations algebraically. the absolute value equation |ax + b| = c is equivalent to ax + b = −c or ax + b = c. Graph the solutions on a number line. |x + 1| − 4 = −2 c. the y-intercept shows that no chemical should be added when a feature contains no water. This mathematical understanding is necessary for students to be able to check their results. as there are two numbers that have a specific distance from zero on a number line. Close Reading. x = 4 and x = −2 Solution: There are two solutions: Lesson 1-3 Activity 1 • Creating Equations 21. © 2015 College Board. Interactive Word Wall. Remind students that solutions to an equation make the equation a true statement. |x − 2| = 3 b. Just like when solving algebraic equations without absolute value bars. Create Representations. If |x| = 5. Work through the solutions to the equation algebraically.indd Page 11 05/12/13 2:50 AM s-059 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ .  ax + b if ax + b ≥ 0 then the equation |ax + b| = c is equivalent to −(ax + b) = c or (ax + b) = c. Sample explanation: Substituting 120 for x in the equation and solving for y shows that the employee should have added 192 fluid ounces of the chemical. solve. Solve 2|x − 1| − 5 = 1.. Example A 2|4 − 1| − 5 = 1 2|3| − 5 = 1 2(3) − 5 = 1 6−5=1 An absolute value equation is an equation involving the absolute value of a variable expression.003-016_SB_AG2_SE_U01_A01. |6| [6] 2. 1 2 3 4 5 d. The slope is 1. Quickwrite. In this case. 22. Graph the solutions on a number line. a. isolate the absolute value bars because they contain the variable. The y-intercept is 0. students must think of two cases. ACTIVITY 1 Continued Lesson 1-3 Absolute Value Equations and Inequalities ACTIVITY 1 continued PLAN My Notes Learning Targets: • Write. Since −(ax + b) if ax + b < 0 ax + b =  . then x = −5 or x = 5 because those two values are both 5 units away from 0 on a number line. • Solve and graph absolute value inequalities. Developing Math Language Solve each absolute value equation. Identify a Subtask. Think-Pair-Share. x = −1 #2 3. Amount of Chemical to Add y 96 80 64 48 32 16 10 20 30 40 50 Amount of Water (1000 gal) x 1. [x = 6 or x = −6] An absolute value equation is an equation involving an absolute value of an expression containing a variable. |−6| [6] Then have students solve the following equation. |x − 3| + 4 = 4 TEACH Example A Marking the Text. Unit 1 • Equations. Have students look back at Try These A. This means that the coffee could cost as little as $7 per pound or as much as $9 per pound. and an extreme value is a minimum value if it is the smallest possible amount (least value). ACTIVITY 1 Continued Lesson 1-3 Absolute Value Equations and Inequalities ACTIVITY 1 continued My Notes There are possibly two. Let t represent the temperature extremes of the wave pool in degrees Fahrenheit. Now have students work in small groups to examine and solve Example B by implementing an absolute value equation. p = 7. and c are real numbers? © 2015 College Board. Quickwrite When solving absolute value equations. Reviewing the definition of the absolute value function as a piecewise-defined function with two rules may enable students to see the reason why two equations are necessary. b. Justify the reasonableness of your answer to part b. Use the definition of absolute value to solve for t. and interpret the solutions. However. b.2. ask them to take the problem a step further and graph its solution on a number line. Sample answer: 7. less than or greater than. How many solutions are possible for an absolute value equation having the form |ax + b| = c. An extreme value is a maximum value if it is the largest possible amount (greatest value). and so on.) Step 1: Write an absolute value equation to represent the situation. Values can vary upward or downward. 1 Identify a Subtask. |t − 82| = 4. You may wish to present a simpler example such as: The average cost of a pound of coffee is $8. There are different ways of thinking about how values can vary. Example B The temperature of the wave pool at Sapphire Island can vary up to 4.indd Page 12 05/12/13 2:50 AM s-059 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ .5. in a positive direction or a negative direction..2. Example B Marking the Text.5°F. and the least possible temperature is 77.003-016_SB_AG2_SE_U01_A01. The greatest allowed pH is 7.3. or zero solutions to an absolute value equation having this form. p represents the extreme pH values of the water. Solve your equation.8. 12 SpringBoard® Mathematics Algebra 2.2 is 0. regardless of how you think of it. Functions © 2015 College Board.5°F from the target temperature of 82°F. Critique Reasoning.8. All rights reserved.5| = 0. and the least allowed pH is 7. Simplify the Problem. Use this information for parts a–c.5°F.5°F. |t − 82| = 4.3 more than 7. All rights reserved. where a. Write an absolute value equation that can be used to find the extreme pH values of the water on the Seal Slide.5 or t = 77. the cost sometimes varies by $1. 1. students may not see the purpose in creating two equations. Group Presentation Start with emphasizing the word vary in the Example. You know that the distance from t to 82°F on a thermometer is 4.5 MATH TIP p = 7. 7. the importance comes in realizing that there are two different directions.3 less than 7. Functions SpringBoard® Mathematics Algebra 2. Have volunteers construct a graph of the two piecewise-defined functions used to write each equation and then discuss how the solution set is represented by the graph. Unit 1 • Equations.8 is 0. Reason quantitatively. one.5 or t − 82 = −4. Have groups present their findings to the class. Additionally. . Write and solve an absolute value equation to find the temperature extremes of the wave pool. Inequalities. (The temperature extremes are the least and greatest possible temperatures. |p − 7.. explain that the word varies in mathematics means changes. Both of these temperatures are 4.3 from the target pH of 7.5°F from the target temperature of 82°F.5 t − 82 = 4. discussing what it means when something varies. Be sure to explain what the variable represents. The pH of the water on the Seal Slide can vary up to 0.5 Solution: The greatest possible temperature of the wave pool is 86. For those students for whom English is a second language.5.5 t = 86. 12 Step 2: Try These B ELL Support Also address the word extremes as it pertains to mathematics.5. Reason abstractly. Inequalities. This distance can be modeled with the absolute value expression |t − 82|. However. parts c and d. a. The pH of water is a measure of its acidity. c. |2x + 3| + 1 > 6 Step 1: Isolate the absolute |2x + 3| + 1 > 6 value expression. then a < b . |3x − 1| + 5 < 7 |3x − 1| < 2 −2 < 3x − 1 < 2 Try These C Solve and graph each absolute value inequality. ACTIVITY 1 Continued Lesson 1-3 Absolute Value Equations and Inequalities ACTIVITY 1 continued My Notes Solving absolute value inequalities algebraically is similar to solving absolute value equations. Step 3: Solve the inequality. are known as disjunctions and are written as A < −b or A > b. The solution can be 2 represented on a number line and written as x < −4 or x > 1. d. ≤. which can also be written as −c < ax + b < c. This also holds true for |A| ≥ b. then ca < cb. • |ax + b| < c. Use graphs on a number line of the solutions of simple equations and inequalities and absolute value equations and inequalities to show how these are all related. Multiplying the first inequality by −1. Subtraction Property of Inequality If a > b. 2 x + 7 − 4 < 1 −6 < x < −1 1 For example. c c If a > b and c < 0. Step 2: Write the compound inequality. then a + c > b + c. |x| > 5 means the value of the variable x is more than 5 units away from the origin (zero) on a number line. Debriefing Before addressing Example C. x − 2 > 3 x > 5 or x < −1 5 –1 c. ≥. or ≠. 5x − 2 + 1 ≥ 4 x ≥ 1 or x ≤ − 1 5 –1 5 –6 See graph A. Inequalities with |A| < b. ≥. where b is a positive number. c > 0. MATH TIP Developing Math Language –1 TEACHER to TEACHER Activity 1 • Creating Equations A –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 B 13 Another method for solving inequalities relies on the geometric definition of absolute value |x − a| as the distance from x to a. x > 1 or x < −4 Solution: 0 –4 b. Multiplication Property of Inequality If a > b and c > 0. 0 –4 Example C Simplify the Problem. By the definition of absolute value. Activity 1 • Creating Equations 13 . >. After they have some time to work through parts a and b. Point out that they should proceed to solve these just as they would an algebraic equation. b. then ca > cb.. © 2015 College Board. See graph B. except in two parts. Example C Solve each inequality. discuss the solutions with the whole class. |3x − 1| + 5 < 7 Step 1: Isolate the absolute value expression. All rights reserved. The solution is x < − 5 or x > 5. then a > b . |ax + b| > c.indd Page 13 05/12/13 2:51 AM s-059 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . • |ax + b| > c. a. If a > b and c < 0. Students can apply these generalizations to Example C. This also holds true for |A| ≤ b. 2x + 3 > 5 or 2x + 3 < −5 Step 3: Solve each inequality. It still involves an absolute value expression that contains a variable. gives these statements: MATH TERMS An absolute value inequality is an inequality involving the absolute value of a variable expression. The properties also apply to inequalities that include <. as shown above. then a − c > b − c. except that the equal sign is now an inequality symbol: <. c > 0. |2x + 3| > 5 Step 2: Write two inequalities. 1 Addition Property of Inequality If a > b. are known as conjunctions and are written as −b < A < b. Graph the solutions on a number line. or as −b < A and A < b. is equivalent to ax + b < c or ax + b > −c. x + 2 − 3 ≤ −1 −4 ≤ x ≤ 0 a. c c 1 –1 3 An absolute value inequality is basically the same as an absolute value equation. or ≤. this means the value of the variable x is less than 5 units away from the origin (zero) on a number line. Division Property of Inequality If a > b and c > 0. is equivalent to ax + b < −c or ax + b > c. All rights reserved. For example: |x| < 5. just like before. the solution set is all values of x whose distance from − 3 is greater 2 than 5 . −1 < x <1 3 Solution: © 2015 College Board. discuss the following: Inequalities with |A| > b. where c > 0. The solution is −5 < x < 5. These properties of real numbers can help you solve inequalities. where b is a positive number. is equivalent to −(ax + b) > c or ax + b > c. Here’s how you can solve the inequality in the example: |2x + 3| + 1 > 6 |2x + 3| > 5 2 x − −3 > 5 2 x − −3 > 5 2 2 Thus. and then using a similar method for |ax + b| < c..003-016_SB_AG2_SE_U01_A01. |5x − 6| ≥ 9 is equivalent to 5x − 6 ≥ 9 or 5x − 6 ≤ −9. |2x + 9| − 10 = 5 9. |3x − 10| − 5 ≥ −1 14. 6. Critique the reasoning of others. Inequalities. x ≤ 2 or x ≥ 14 3 0 2 4 6 –4 –2 0 2 4 –2 14. the inequality would be impossible or trivial because absolute value is the distance from zero on a number line. review the process of rewriting an absolute value equation or inequality as two equations or inequalities.. The greatest flow rate is 730 gallons per minute and the least flow rate is 550 gallons per minute. She did not isolate the absolute value expression first. Why is the condition c > 0 necessary for |ax + b| < c to have a solution? If c = 0 or if c < 0 (c is negative). |x − 6| = 5 7. x = 11 x = 19 . 3. Thus. you describe ways in which they are alike and ways in which they are different. Unit 1 • Equations. Explain your steps. a. 0 14 SpringBoard® Mathematics Algebra 2. f = 730. Debriefing Use the investigation regarding the restriction c > 0 as an opportunity to discuss the need to identify impossible situations involving inequalities. ACTIVITY 1 Continued Lesson 1-3 Absolute Value Equations and Inequalities ACTIVITY 1 continued My Notes 2. Graph the solutions on a number line. Functions 6 8 © 2015 College Board. Unit 1 • Equations. LESSON 1-3 PRACTICE Solve each absolute value equation. Check Your Understanding Debrief students’ answers to these items to ensure that they understand concepts related to absolute value equations. Inequalities.. c. Model with mathematics. Sample answer: A linear equation of the form ax + b = c has 1 solution. Sample explanation: First. 9.003-016_SB_AG2_SE_U01_A01. All rights reserved. You may assign the problems here or use them as a culmination for the activity. Paige incorrectly solved an absolute value equation as shown below. |4x + 3| − 9 < 5 Students’ answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the lesson concepts and their ability to apply their learning. The equation |x + 5| = −4 has no solutions because the absolute value of an expression cannot be negative. Have groups of students present their solutions to Item 4. Explain how to write the inequality |5x − 6| ≥ 9 without using an absolute value expression. |3x − 7| = 12 8. 11. 14 x = 1. How could Paige have determined that her solutions are incorrect? c. 1. Compare and contrast a linear equation having the form ax + b = c with an absolute value equation having the form |ax + b| = c. Write and solve an absolute value equation to find the extreme values of the flow rate on the Otter River Run. |x − 7| > 1 12. b. where c ≥ 0. Make sense of problems. 10. −2 |x + 5| = 8 −2(x + 5) = 8 or −2(x + 5) = −8 −2x − 10 = 8 or −2x − 10 = −8 x = −9 or x = −1 a. −2 ≤ x ≤ 7 –4 –2 0 2 4 13. 4. Self Revision/Peer Revision. |5x − 3| + 12 = 4 10. Solve each absolute value inequality. 5. ADAPT Check students’ answers to the Lesson Practice to ensure that they understand basic concepts related to writing and solving absolute value equations and inequalities and graphing the solutions of absolute value equations and inequalities. Sample answer: The inequality has the form |ax − b| ≥ c. .indd Page 14 15/04/14 10:26 AM user-g-w-728 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . isolate the absolute value expression: |x + 5| = −4. Solve the equation correctly. or 2 solutions depending on the value of c. See the Activity Practice for additional problems for this lesson. 8. 3. If students are still having difficulty. but an absolute value equation of the form |ax + b| = c may have 0. Answers Check Your Understanding © 2015 College Board. What mistake did Paige make? b. so it can be written as ax − b ≥ c or ax − b ≤ −c. x < 6 or x > 8 6. ASSESS 11. ACADEMIC VOCABULARY When you compare and contrast two topics. All rights reserved. 4. |2x − 5| ≤ 9 13. 2 Quickwrite. where f represents the extreme flow rates in gallons per minute. 7. x = −12 no solution | f − 640| = 90. x = − 5 3 3 x = 3. No solution. Sample answer: She could have substituted the values for x into the original equation to check whether they satisfy the equation. − 17 < x < 11 4 4 8 SpringBoard® Mathematics Algebra 2. f = 550. 5. Functions LESSON 1-3 PRACTICE 2 4 6 12. The flow rate on the Otter River Run can vary up to 90 gallons per minute from the target flow rate of 640 gallons per minute. It takes the team 4 minutes to reach the roof. Sample answer: The graph makes it easy to see that as the trip length increases. B 18.98 + 0. The independent variable x represents the length of the trip in miles. p ≈ 28. Be sure to include a title for the graph and for each axis. including sales tax of 8. 2. y = 8000x + 6000(8 − x) or equivalent 64 Yearly Expenses ($1000) © 2015 College Board. and 20 is less than 24.50 3 9. she is renting a booth at a craft fair for $60..25 percent. The y-intercept is 3. 16.50 for taking a taxi.m. 9.. Explain what the independent variable and the dependent variable represent.5 + 2x. Graph the equation. Sample explanation: The cost of the original sweater with tax is $28.7 miles. 11. ACTIVITY 1 Continued Creating Equations One to Two ACTIVITY 1 continued Jerome bought a sweater that was on sale for 20 percent off. At 2:42 p. Use this information for Items 4–6. The dependent variable y represents the cost of the trip in dollars.98) = $31. the number of minutes it will take the helicopter to reach the hospital? A.m. Solve the equation. 60 8 6 4 56 2 52 1 48 2 3 4 x 5 Trip Length (mi) 44 1 2 3 4 5 6 7 8 Number of Lions x 13.m.0825($28. The original cost of the sweater is $28. A medical rescue helicopter is flying at an average speed of 172 miles per hour toward its base hospital. 7. 3:03 p. Which equation can be used to determine m. This Saturday. An emergency team needs to be on the roof of the hospital 3 minutes before the helicopter arrives. Sample explanation: The helicopter will reach the hospital 28 minutes after 2:42 p. 3.003-016_SB_AG2_SE_U01_A01. the cost also increases. $10. 4. 4. © 2015 College Board. Jerome paid $25.8. will she meet her profit goal? Explain why or why not. $6. 172 m = 80 60 12. the helicopter is 80 miles from the hospital. y = 3. 10. Sample explanation: The graph of the equation is a line.0825(0. Activity 1 • Creating Equations 15 . Lesson 1-1 7. 3.m. 172 = 80 m 60m 5. Write an equation that can be used to find the number of purses Susan must sell to make a profit of $250 at the fair. and interpret the solution.00 charge per mile for taking a taxi..50 11. where p is the original price in dollars 8.80p + 0. Show your work. 15. 13.50 5 13.37. 9.90 (not including tip) 17. The slope is 2. p ≈ 23. The purses cost her $12 each to make. ( ) 14. where p is the number of purses sold 2. Write an equation in two variables that models this situation. or at 3:10 p. 3.10 or equivalent. Make a table that shows what it would cost to take a taxi for trips of 1. Yes. How much it will cost Shelley to take a taxi to the café? 10. 4.80p) = 25. 12. Shelley uses her phone to determine that the distance from her apartment to Blue Café is 3. How much money did Jerome save by buying the sweater on sale? Explain how you determined your answer.50 2 7. and interpret the solution. Sample explanation: Susan must sell at least 24 purses to meet her profit goal. Solve the equation. It will take the helicopter about 28 minutes to reach the hospital. 1. ACTIVITY 1 PRACTICE Write your answers on notebook paper.37 − $25. Is the equation that models this situation a linear equation? Explain why or why not. 6.5. C. 8.27. m ≈ 27. Solve the equation. It represents the initial fee of $3. and interpret the solution. The difference between the original cost of the sweater with tax and the sale price of the sweater with tax is $31. 0. Use this information for Items 10–16. Describe one advantage of the graph compared to the equation.50 4 11. Taxi Fare Trip Length (mi) Cost ($) 1 5. Susan makes and sells purses. B 5. At what time should the team start moving to the roof to meet the helicopter? Explain your reasoning.9.indd Page 15 05/12/13 2:51 AM s-059 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . ACTIVITY PRACTICE 1.10 for the sweater.27. Write an equation that can be used to find the original price of the sweater. or at 3:03 p.10 = $6.00 per mile. Use this information for Items 7–9. What are the slope and y-intercept of the graph? What do they represent in the situation? 6. Use this information for Items 1–3. ( ) 15. 172 60 = 80 D.. The team needs to start moving 7 minutes before the helicopter arrives. All rights reserved. If Susan sells 20 purses at the fair.. Susan must sell 24 purses to make a profit of $250. Taxi Fare y 14 19.98. No.m. and she sells them for $25.50 plus $2. It represents the $2. 172(60m) = 80 B. 16. 25p − 12p − 60 = 250 or equivalent.98. and 5 miles. All rights reserved. A taxi company charges an initial fee of $3. Lesson 1-2 2. y Yearly Expenses of Activity 1 • Creating Equations Large-Cat Exhibit 12 15 10 Cost ($) 14. 5x + 40. y represents the monthly cost of the cell phone. 25. Graph the equation. no solutions 23. This point represents a yearly cost of $58. ACTIVITY 1 Continued 2 –6 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 25.000. which means that x cannot be negative. The least possible monthly cost of the cell phone is $40. |7x + 1| − 7 = 3 e. Unit 1 • Equations. Thus.1. 21.. the only points on the line that are solutions are those with an x-coordinate that is a whole number no greater than 8. so the greatest value of x is 8. What are the units of the slope of the linear equation? Explain. Check students’ graphs. Also. y = 2 x + 6 B. a. 16 SpringBoard® Mathematics Algebra 2. x = 9. All rights reserved. The greatest acceptable center thickness is 5. Which number line shows the solutions of the inequality 2|x − 1| ≥ 4? 20.6 or equivalent. −17 < x < 7 b. x = −14 c. and expenses for a leopard will be about $6000 per year.50 per day. 24. e. –2 –1 0 1 2 3 4 B. y = 45.. The thermometer indicates that Zachary’s temperature is 101.5x + 40 represents the monthly cost y in dollars of Lesley’s cell phone. Functions © 2015 College Board.3°F. SpringBoard® Mathematics Algebra 2. Graph the equation.150 mm. where t represents the extreme acceptable values for the center thickness of the lens. 2 ≤ x ≤ 2 5 d. so I let the y-axis go up to 48 to include all of the y-values as x increases from 0 to 10. What are the units on each axis of your graph? d.850 mm.150. A.1°F and 102. x = 9 . |5x + 2| ≥ 13 c. Inequalities. Solve each absolute value equation. which means that y cannot be negative. Graph the solutions on a number line.6°F. A. Write a different plausible scenario—not related to cell phone costs—that could be modeled using the equation y = 0. x ≥ 11 or x ≤ −3 5 c. Inequalities. Use this information for Items 18–21. |10x − 12| − 9 ≤ −1 d. a. How did you determine the range of values to show on each axis of your graph? c./103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . y = 4 − 1 3 x C. |−2x + 5| + 6 ≥ 4 27. y-axis: dollars d. either. 22. $58. The equation y = 0. so I put a break in the y-axis between 0 and 38 so that the graph would not need to be so tall. –2 –1 0 1 2 3 4 MATHEMATICAL PRACTICES Attend to Precision 28. Dollars per minute over 750. No. 26. 18.850. If the center thickness of a lens varies more than 0. |t − 101. b. so x must be a whole number. the yearly expenses for the eight cats in the exhibit when x of the cats are lions. 21. |x − 7| + 3 > 8 e.3. Monthly Cost ($) 46 44 16 42 40 38 2 4 6 8 10 x Talk Minutes over 750 ADDITIONAL PRACTICE If students need more practice on the concepts in this activity. Write and solve an absolute value equation to find the extreme possible values of Zachary’s actual temperature. Write an equation that can be used to find y. 19. Write and solve an absolute value equation to find the extreme acceptable values for the center thickness of the lens. x-axis: minutes over 750. |2x| − 3 = −5 23.000 millimeters. Zachary’s actual temperature is between 101. so the units of the slope are units of y per units of x. x = −4 A zoo is building a new large-cat exhibit. –2 –1 0 1 2 3 4 C. t = 102. see the Teacher Resources at SpringBoard Digital for additional practice problems.150| = 5. |t − 0. no solutions 27. Solve the equation |2x + 4| − 1 = 7. Then graph the solutions on a number line. where x is the number of talk minutes over 750 that Lesley uses. t = 101. a. t = 5. Lesley’s Cell Phone y Creating Equations One to Two ACTIVITY 1 continued 17.7| = 0. 24. Part of the space will be used for lions and part for leopards. a. Be sure to include a title for the graph and for each axis. 3x + 2y = 8 D. a. x > 12 or x < 2 e. C 28.150 millimeter from the target thickness of 5. t = 4. x = 2. Sample explanation: The slope is the ratio of the change in y to the change in x between any two points on the line. 003-016_SB_AG2_SE_U01_A01. Sample answer: The equation represents the amount of money Gary saves if he started with $40 and then saved $0. x = 10 d. Functions 28. –2 –1 0 1 2 3 4 D. 20.indd Page 16 13/01/15 11:27 AM ehi-6 . no more than 8 lions can be in the exhibit. When x = 10. Answers may vary. Choose the equation that is not linear. both real-world and mathematical. Lesson 1-3 22. x = −2 b. Solve each absolute value inequality. Unit 1 • Equations. What would the yearly expenses be if five of the cats in the exhibit are lions and the rest are leopards? Explain how you found your answer. |x − 10| − 11 = 12 − 23 d.7°F.000 when 5 of the large cats in the exhibit are lions. Sample answer: x represents a number of talk minutes. a. x = 5. The exhibit will house eight large cats in all. |2x + 5| = 23 c. and the least acceptable center thickness is 4.000 or equivalent. |x + 5| < 12 b. where t represents the extreme possible values of Zachary’s actual temperature in °F. e. c. Expenses for a lion will be about $8000 per year. All rights reserved. 26. |2x − 3| = 7 b. Be sure to use appropriate vocabulary. x = − 11 7 7 e. Are all points on the line you graphed solutions in this situation? Explain. the lens cannot be used. Sample explanation: The graph includes the point (5. A thermometer is accurate to within 0. x represents a number of lions. b. 58). x = −6 © 2015 College Board. Investigative Activity Standards Focus Learning Targets: © 2015 College Board.indd Page 17 05/12/13 1:45 PM user-s068a 123 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ .. How much money will Roy have available to spend on performances. During his trip to New York City. After surfing the web. and incidentals. Look for a Pattern. All rights reserved. SUGGESTED LEARNING STRATEGIES: Shared Reading. a ticket for a performance in New York City costs $100. and any other expenses that might arise after paying for his hotel and taxis? Show your work. meals.017-028_SB_AG2_SE_U01_A02. 1 Debriefing This item is designed as an entry-level question. students represent constraints using equations and/or inequalities. 2 Bell-Ringer Activity Have students write a function for each situation. The prize was a five-day trip to New York City. ACTIVITY Graphing to Find Solutions ACTIVITY 2 Choices Lesson 2-1 Graphing Two-Variable Equations In Activity 2. Create Representations. Activity 2 • Graphing to Find Solutions 17 . Reason quantitatively. Throughout this activity. 1.. Roy wants to spend only his winnings from the contest. and the trip includes staying five nights. • Graph equations on coordinate axes with labels and scales. Once he has done this. He wants to focus on two of his favorite pastimes: attending theater or musical performances and dining in restaurants. • A taxi between New York City and LaGuardia Airport will cost $45 each way. Activity 2 • Graphing to Find Solutions 17 Common Core State Standards for Activity 2 HSA-CED. descent of hot air balloon: 5 ft/min initial height of balloon: 250 ft [H(m) = 250 − 5m] 3. and by systems of equations and/or inequalities.2 Create equations in two or more variables to represent relationships between quantities. To prepare for his trip. Roy gathered this information. cost of plumbing repairs: $35/hr initial fee for repair: $50 [C(h) = 50 + 35h] 2. Roy determines the following facts: • On average. HSA-CED. including a round-trip airplane ticket and $3000 in cash. It will be used throughout the activity. Roy can begin to make plans to enjoy the city with his remaining prize funds. • He will spend on average $40 per meal. The money will pay the cost of a hotel room. #8–10 TEACH $1360 Sample answer: 3000 − 5(310) − 2(45) = 1360 © 2015 College Board. and interpret solutions as viable or nonviable options in a modeling context. number of students: 18 per bus other students: 135 [S(b) = 135 + 18b] Discuss with students the method they used to write the functions and the definitions of the variables they chose. Group Presentation.A. PLAN Pacing: 1 class period Chunking the Lesson Roy must set aside the cash required to pay for his hotel room and for taxi service to and from the airport. Activating Prior Knowledge Roy recently won a trivia contest.A. All rights reserved. graph equations on coordinate axes with labels and scales. meals. entertainment. Graphic Organizer. emphasize the process of writing equations and inequalities from verbal descriptions and generating solutions once the constraints are graphed on the coordinate plane. Lesson 2-1 • A hotel room in New York City costs $310 per night. Marking the Text. My Notes equations in two variables to represent relationships between • Write quantities. They graph these constraints on a coordinate plane. #1 #2–4 #5–6 #7 Check Your Understanding Lesson Practice 1.3 Represent constraints by equations or inequalities. so a debriefing is important to make sure that all students have the correct answer as they progress through the activity. Then they use their graphs to determine solutions to a system of equations or system of inequalities. Quickwrite Ensure that students know the difference between discrete and continuous data. 2. Group Presentation. and 13 tickets. My Notes 2–4 Create Representations. Unit 1 • Equations. slope-intercept form.indd Page 18 15/04/14 10:37 AM user-g-w-728 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . Model with mathematics. All rights reserved. make sure they relate the fact that the amount of money decreasing represents a negative rate of change.. It may be possible to work through these items quickly without using the mini-lessons on this page and the next. a mini-lesson is available to provide practice. only a whole number of tickets can be purchased..017-028_SB_AG2_SE_U01_A02. Look for a Pattern. 10. Functions MINI-LESSON: Slope-Intercept Form of the Equation of a Line If students need additional help with finding slope or finding the equation of a line in slope-intercept form. Sample answer: There is a linear pattern in both the graph and the table such that the amount of money available decreases by $100 for each ticket purchased. See the Teacher Resources at SpringBoard Digital for a student page for this mini-lesson. and use the graph to review how to find the slope of a linear function. take the time to have discussions about rate. Functions © 2015 College Board. M(t) = 1360 − 100t SpringBoard® Mathematics Algebra 2. subtract $100 from the amount available. and point-slope form as needed. Roy wants to know how the purchase of each ticket affects his available money. 18 15 Tickets 3. so the graph is individual points rather than a line. 5 Create Representations. slope. If students need more support. © 2015 College Board. Write a function M(t) that represents the amount of money that Roy has left after purchasing t tickets. and 560 as their last three entries in the table. Tickets (t) Money Available (M) 0 1360 1 1260 2 1160 3 1060 4 960 5 860 8 560 10 360 13 60 M 1400 1200 1000 800 600 400 200 5 10 t Sample explanation: The rate of change is −100. 660. Explain how you determined the values for 8. Money Available ($) Differentiating Instruction continued . Have students look at successive differences in the table. All rights reserved. Plot the points on the grid. What patterns do you notice? When students look for the pattern in Item 3. Inequalities. These students may not realize that the numbers of tickets in the last three rows of the table are no longer increasing by 1 each time. A function is a relationship between two quantities in which each input has exactly one output. In this case. Review slope as a constant rate of change for linear functions. SpringBoard® Mathematics Algebra 2. Watch for students that have 760. Unit 1 • Equations. ACTIVITY 2 Continued Lesson 2-1 Graphing Two-Variable Equations ACTIVITY 2 Items 2–10 are designed to review how to work with linear equations and activate students’ prior knowledge. Fill in the table below. Inequalities. MATH TIP 18 5. 4. Debriefing Ask students to share their answers. so for each additional ticket purchased. y-intercept. Activating Prior Knowledge. 9. Use mathematical terminology to explain what −100 and 1360 each represent in your function in Item 5. given that m represents the number of meals that Roy can buy? Explain. 7. including units? −40 MATH TIP $ .017-028_SB_AG2_SE_U01_A02. D(m) CONNECT TO TECHNOLOGY 1400 You can also graph the function by using a graphing calculator. Quickwrite. Make sense of problems. CONNECT TO TECHNOLOGY For additional technology resources. No. Activity 2 • Graphing to Find Solutions 19 MINI-LESSON: Point-Slope Form of the Equation of a Line If students need additional help with writing an equation using point-slope form. Write a function D(m) that represents the amount of money Roy has left after purchasing m number of meals. Money Left ($) 1200 1000 800 6 Think-Pair-Share. Activity 2 • Graphing to Find Solutions 19 . When entering the equation. visit SpringBoard Digital. 600 8–10 Look for a Pattern. Quickwrite Ensure students make the connection between the constant term and the y-intercept of a line in slope-intercept form. 7 Create Representations Students will replicate what they did in Items 2–6 without the scaffolding. © 2015 College Board. use x for the independent variable and y for the dependent variable. and the constant 1360 is the y-intercept of the line. Are all the values for m on your graph valid in this situation. D(m) = 1360 − 40m b. Debriefing Review the concept of domain and clarify the idea of the contextual domain.. Discuss why negative values have no meaning in the situation. −100. Sample explanation: The coefficient of t. 400 200 20 40 60 m Meals 8. All rights reserved. The rate of change of a function is the ratio of the amount of change in the dependent variable to the amount of change in the independent variable. a mini-lesson is available to provide practice. All rights reserved. or −$40 per meal meal 10. What kind of function is D(m)? © 2015 College Board.indd Page 19 15/04/14 10:38 AM user-g-w-728 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . Sample explanation: The contextual domain is restricted to positive integers less than 35 because the number of meals will be a positive number and the amount of money left cannot be negative. See the Teacher Resources at SpringBoard Digital for a student page for this mini-lesson. It is linear. Use this time to help individual students who need more support in the review process. What is the rate of change for D(m). Roy wonders how his meal costs will affect his spending money. is the slope. ACTIVITY 2 Continued Lesson 2-1 Graphing Two-Variable Equations ACTIVITY 2 continued My Notes 6. Interactive Word Wall. Graph your function on the grid. a.. See the Activity Practice for additional problems for this lesson. What do the x.. In addition. Reason quantitatively.320 13. Finally.200 + 2d SpringBoard® Mathematics Algebra 2.. 15. 14. The rate of change of a linear function is the same as the slope.240 13.000 for f(d) and then solve for d.000 frequent flyer miles? Explain how you determined your answer. All rights reserved. All rights reserved. and he will earn 2832 more miles from his round-trip flight to New York City. Inequalities. have them practice writing word expressions and translating the word expressions into algebraic expressions. How many dollars will Roy need to charge on his credit card to have a total of 15. ACTIVITY 2 Continued Lesson 2-1 Graphing Two-Variable Equations ACTIVITY 2 Debrief students’ answers to these items to ensure that they understand concepts related to interpreting the rate of change and intercepts of an equation. Roy already has 10. y = 3 x − 4 2 8 6 4 2 –10 –8 –6 –4 –2 –2 2 4 6 8 10 x –4 –6 ADAPT Check students’ answers to the Lesson Practice to ensure that they understand basic concepts related to writing and graphing linear equations. You may assign the problems here or use them as a culmination for the activity. f(d) = 13. Graph the function f (x ) = 3 − 1 (x − 2) . If you know the coordinates of two points on the graph of a linear function. Then graph the two points and draw a line through them to determine the y-intercept of the line. −3) and has a slope of 5. If students are continuing to having difficulty writing equations to model a given situation. 16. 20 20 SpringBoard® Mathematics Algebra 2. he earns 2 frequent flyer miles for each dollar he charges on his credit card. Sample answer: Use the coordinates of the two points to find the slope of the line. The x-intercept of 34 represents the greatest number of meals Roy can buy before running out of money. Unit 1 • Equations.360 13. Model with mathematics. LESSON 2-1 PRACTICE 15. Write the equation of a function f(d) that represents the total number of frequent flyer miles Roy will have after his trip if he charges d dollars on his credit card. 17. 20. $900. Unit 1 • Equations. Functions 16. 12. y = 5x + 7 10 17. where m is the slope and b is the y-intercept. Sample explanation: Substitute 15. Check Your Understanding d Amount Charged to Credit Card ($) 20. Write the ratio of the change in the dependent variable to the change in the independent variable. explain how to write the equation of a line when you are given the coordinates of two points on the line. .280 13. Graph the function. y 10 6 6 4 4 2 2 2 4 6 8 10 x –10 –8 –6 –4 –2 –2 4 6 –4 –4 –6 –6 –8 –8 –10 –10 2 18. as well as how to interpret rate of change and intercepts of equations. continued My Notes Check Your Understanding Answers 11. 14. 13. Write the equation of the line that passes through the point (−2. Graph the equation. Inequalities. Functions 8 10 x Roy’s Frequent Flyer Miles f(d) 8 8 –10 –8 –6 –4 –2 –2 19. 2 Use the following information for Items 18–20. 18. The y-intercept of 1360 represents the amount of money in dollars Roy will have if he does not buy any meals. 19. using appropriate scales on the axes. write the equation in slope-intercept form. Using your answers to Items 12 and 13.and y-intercepts of your graph in Item 7 represent? 12.200 20 40 60 80 100 © 2015 College Board. 2 Graph the equation. 11.017-028_SB_AG2_SE_U01_A02. What is the relationship between the rate of change of a linear function and the slope of its graph? MATH TIP The slope-intercept form of a line is y = mx + b.400 13. ASSESS Students’ answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the lesson concepts and their ability to apply their learning. LESSON 2-1 PRACTICE 10 y © 2015 College Board. y Frequent Flyer Miles –8 –10 13. how can you determine the function’s rate of change? 13.368 frequent flyer miles.indd Page 20 05/12/13 1:45 PM user-s068a 123 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . Write the equation of the line with y-intercept −4 and a slope of 3 . you lose. MATH TERMS A linear inequality is an inequality that can be written in one of these forms. The statement in Item 2b assumes that Roy will buy some number of meals greater than zero. 4–5 Think-Pair-Share. When using algebraic expressions to find solution sets within an applied setting. students must always interpret the solution set for reasonableness within the context of the applied setting. 55). they may be confused by m as a variable for the number of meals that Roy may eat. and meals must be greater than zero. Ask and answer questions clearly to aid comprehension and to ensure understanding of all group members’ ideas. 60). Interactive Word Wall.6 Because you cannot purchase part of a ticket. Create Representations. you tie] 2 Think-Pair-Share This item returns to the domain constraints of the problem. 1. ACTIVITY 2 Continued Lesson 2-2 Graphing Systems of Inequalities ACTIVITY 2 continued PLAN My Notes Learning Targets: • Represent constraints by equations or inequalities. assume that Roy has no additional expenses.5 11 890 2. where A and B are not both equal to 0: Ax + By < C. If graphed in the coordinate plane. For all the ordered pairs (t. some students may give t ≤ 9. Work Backward. height) on the board. 9 tickets. m) that are feasible options. Discuss how those constraints affect the feasible options for the conditions given in Item 1. Show your work. determine the total number of tickets that he could purchase in five days. Pacing: 1 class period Chunking the Lesson #1 #2 #3 #4–5 #6 Check Your Understanding #11 #12–13 #14–15 Check Your Understanding Lesson Practice SUGGESTED LEARNING STRATEGIES: Think-Pair-Share. Debriefing For this item. • Use a graph to determine solutions of a system of inequalities. © 2015 College Board. All rights reserved.6 as the answer. so only coordinate pairs that are integers will be solutions. at most 9 tickets can be purchased.. © 2015 College Board. Ax + By ≤ C. If Roy buys exactly two meals each day. Danielle(10.. 48). Discussion Groups In Item 4. Write a linear inequality that represents all ordered pairs (t. 1 Interactive Word Wall Introduce students to the word feasible as it applies to the solution sets of equations and inequalities. Candice(14. m) that are feasible options for Roy. As needed. all ordered pairs would fall either in the first quadrant or on the positive m-axis. Debriefing. Roy will eat three meals per day. pose the following to students: You and a friend are playing a game of Tic-Tac-Toe. From their perspective. Work Backward. Tickets (t) Meals (m) Total Cost Is it feasible? Rationale 6 16 1240 Yes 1240 ≤ 1360 8 14 1360 Yes 1360 = 1360 10 12 1480 No 1480 > 1360 No You cannot buy half a ticket. Have students discuss which constraint prevents students from riding the ride. Although this is a correct solution to the inequality. Lesson 2-2 #7 TEACH ACADEMIC VOCABULARY The term feasible means that something is possible in a given situation. Sample explanation: Tickets must be greater than or equal to zero. Activity 2 • Graphing to Find Solutions 21 Bell-Ringer Activity Write the statement “You must be at least 13 years old and at least 54 inches tall to ride this ride. What are the feasible results of the game? [you win. or 15 meals. Roy’s spending money depends on both the number of tickets t and the number of meals m.” on the board. Sample explanation: Both meals and tickets are integral values. Sample answer: 100t + 40(10) ≤ 1360 100t ≤ 960 t ≤ 9. Ask students which of the following people could ride the ride: Anna(11. 3. it does not answer the question asked. Therefore. a.indd Page 21 25/02/15 7:38 AM s-059 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . 41). Ax + By > C. or Ax + By ≥ C. Construct viable arguments. 3 Create Representations. To further illustrate the word feasible. Make notes as you listen to group members to help you remember the meaning of new words and how they are used to describe mathematical concepts. All rights reserved. be sure to use mathematical terms and academic vocabulary precisely. 100t + 40m ≤ 1360 for t ≥ 0 and m > 0 4. b. Activating Prior Knowledge Work with your group on Items 1 through 5. DISCUSSION GROUP TIPS As you share your ideas. 4. Close Reading.017-028_SB_AG2_SE_U01_A02. during his entire stay in New York City. refer to the Glossary to review translations of key terms. explain why each statement below must be true. All coordinates in the ordered pairs are integer values. Write the general ordered pair (age. Determine whether each option is feasible for Roy and provide a rationale in the table below. Incorporate your understanding into group discussions to confirm your knowledge and use of key mathematical language. Discussion Groups. 58). Ben(13. Activity 2 • Graphing to Find Solutions 21 . Students may incorrectly assume that everyone eats three meals per day. which requires an integer answer. Ed(15. ELL Support Support students whose first language is not English by pairing them with more-fluent speakers for Item 6. If students need additional help.. the question asks for a specific number of meals. Show your work. 22 SpringBoard® Mathematics Algebra 2. If Roy buys exactly one ticket each day. To see what the feasible options are. Unit 1 • Equations. ACTIVITY 2 Continued Lesson 2-2 Graphing Systems of Inequalities ACTIVITY 2 continued My Notes 21 meals. Although this is a solution to the inequality. the focus is on one inequality. 22 SpringBoard® Mathematics Algebra 2. you can use a visual display of the values on a graph. . Inequalities. students will graph all the constraint inequalities on one grid.017-028_SB_AG2_SE_U01_A02. First. a. Quickwrite Review how to graph a linear inequality. Functions © 2015 College Board. Activating Prior Knowledge. Later. find the maximum number of meals that he could eat in the five days. at most 21 meals can be purchased. Finally. 0) satisfies the inequality. This makes it easier to determine whether or not students have any misunderstandings about the procedure for graphing linear inequalities. 4–5 (continued) Likewise. See the Teacher Resources at SpringBoard Digital for a student page for this mini-lesson. Unit 1 • Equations. a mini-lesson is available to provide practice. assign Mini-Lesson: Graphing Linear Inequalities. Functions MINI-LESSON: Graphing Linear Inequalities If students need additional help with graphing linear inequalities. All rights reserved. 6. MATH TIP 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 2 4 6 8 10 12 14 16 t Number of Tickets b. Which half-plane is shaded? How did you decide? The lower half-plane is shaded. Recall how to graph linear inequalities. For now. some students may give m ≤ 21. Attend to precision. shade the half-plane that contains the solution set.indd Page 22 10/02/14 8:00 AM user-g-w-728 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . All rights reserved. Pairs can practice their listening and speaking skills as they take turns describing the process of graphing a linear inequality. Encourage students to use precise mathematical language in their discussions. 5. Then choose a test point not on the line to determine which half-plane contains the set of solutions to the inequality. Inequalities. which also must be an integer. Sample answer: 100(5) + 40m ≤ 1360 40m ≤ 860 m ≤ 21. m Number of Meals 6 Create Representations. Graph your inequality from Item 3 on the grid below. in Item 5.. in Item 8.5 Because you cannot purchase part of a meal. What is the boundary line of the graph? 100t + 40m = 1360 c. © 2015 College Board.5 as the answer. graph the corresponding linear equation. Sample explanation: The test point (0. Activity 2 • Graphing to Find Solutions Technology Tip If students are using TI-Nspire technology. Now press GRAPH .5x TECHNOLOGY TIP To enter an equation in a graphing calculator. ACTIVITY 2 Continued Lesson 2-2 Graphing Systems of Inequalities ACTIVITY 2 continued My Notes d. Discuss that t is replaced with x because t (the number of tickets) is the independent variable and m (the number of meals) depends on the number of tickets. an Xmax of 16. a Ymin of 0.. © 2015 College Board. TECHNOLOGY TIP To graph an inequality that includes ≥ or ≤. the graph displayed on the calculator will not completely match. Describe the graph. Item 4 Item 5 (9. d. you may or may not be able to see the boundary line. All rights reserved.5 X (use the green letter key for x). 100t + 40m ≤ 1360 → 100x + 40y ≤ 1360 40y ≤ 1360 − 100x y ≤ 34 − 2. Debriefing Part a provides an opportunity to review the concept of independent and dependent variables. 10) (5. visit SpringBoard Digital. b. but students should be able to relate one to the other. Are both those points in the shaded region of your graph? Explain. Enter this inequality into your graphing calculator. Press ENTER until the symbol to the left of Y1 changes to  . Step 2: Change the equal sign to a less than sign by using the CLEAR key followed by the < key located in the leftmost column of white keys. you would use the symbol or . students can adjust the viewing window by pressing WINDOW and entering an Xmin of 0. Write your response for each item as points in the form (t. an Xscl of 1. Students should discuss why replacing t with y and m with x does not work. What does this symbol indicate about the graph? Sample answer: This symbol indicates that the half-plane below the boundary line will be shaded. Yes. a Ymax of 36. m). Now follow these steps to graph the inequality on a graphing calculator. Sample answer: The graph shows a line with negative slope. Replace t with x. You need to indicate whether the half-plane above or below the boundary line will be shaded. Then press GRAPH again. For additional technology resources. Step 4: Adjust the viewing window as needed to view the graph.. All rights reserved. © 2015 College Board. a. Use the left arrow key to move the cursor to the far left of the equation you entered. and a Yscl of 1. Sample explanation: Both are near but below the boundary line. with shading below and to the left of the line. Step 3: Enter the function f1(x) as 34 − 2. 7 Activating Prior Knowledge. Check students’ graphs. 21) e. start with Y= . Interactive Word Wall. c. 23 Activity 2 • Graphing to Find Solutions 23 . provide the following directions for how to graph the inequality given in Item 7: Step 1: Choose Graphs&Geometry from the home screen. Use appropriate tools strategically. 7.indd Page 23 23/01/14 7:28 PM ehi-6 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . Press WINDOW and adjust the viewing window so that it matches the graph from Item 6. and replace m with y. Then solve the inequality for y.017-028_SB_AG2_SE_U01_A02. Depending on your window settings. Note that due to the height of the graph on the student page. In part c. Then choose several values of x and substitute them into the equation to find the corresponding values of y. ask students to use a number line to identify numbers that satisfy both phrases and then numbers that satisfy just one phrase but not the other. Use a dashed boundary line for inequalities that include > or <. m ≤ 2t. Inequalities. Solutions below the boundary line represent solutions for which Roy would have money left over. a. the total cost of the tickets plus the total cost of the meals is less than $1360. 9. 0 ≤ t ≤ 10 c. 12. MATH TIP Use a solid boundary line for inequalities that include ≥ or ≤. MATH TERMS Constraints are the conditions or inequalities that limit a situation. Unit 12. but the paper graph uses the variables t and m.indd Page 24 15/04/14 10:41 AM user-g-w-728 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . either by using paper and pencil or by using a graphing calculator. The shaded region is the intersection of the four inequalities. Graph the inequalities from Items 3 and 11 on a single grid. b. ACTIVITY 2 Continued My Notes Check Your Understanding 8. Compare and contrast the two graphs of the linear inequality: the one you made using paper and pencil and the one on your graphing calculator. 8. the total cost of the tickets plus the total cost of the meals is equal to $1360. Create Representations Students must work with the language of inequalities in such phrases as at least. b. On the remaining four days. b. m ≤ 2t 10. Describe an advantage of each graph compared to the other. Lesson 2-2 Graphing Systems of Inequalities ACTIVITY 2 continued Number of Meals Check Your Understanding . Functions t © 2015 College Board. For each pair of phrases. You can use a graph to organize all the constraints on Roy’s trip to New York City. a. 26 24 22 20 18 16 14 12 10 8 6 4 2 2 4 6 8 10 12 14 16 Number of Tickets 24 SpringBoard® Mathematics Algebra 2. at most. Sample answer: Write the related linear equation. 6 ≤ m ≤ 14 b. which is the amount of money Roy has. which is the amount of money Roy has. but he never eats more than three meals each day. a. more than. Roy eats at least one meal each day. 10. 0 ≤ t ≤ 10. not more than. Inequalities. What part of your graphs represents solutions for which Roy would have money left over? Explain. and to distinguish between at least one and more than one. a. One advantage of the paper graph is that you can add titles to the axes to show what the graph represents. but he may not be able to attend all of them. Roy realized that some other conditions or constraints apply. List the inequalities you found in Items 3 and 11. Sample answer: Both graphs show the same inequality. Roy wants the number of meals that he eats to be no more than twice the number of performances that he attends. Then choose a test point not on the boundary line and use it to determine which half-plane to shade.017-028_SB_AG2_SE_U01_A02. Unit 1 • Equations.. © 2015 College Board. and the calculator graph uses the variables x and y. Roy eats lunch and dinner the first day. m 1 • Equations. One advantage of the calculator graph is that you do not need to determine the coordinates of points on the boundary line in order to graph it. Debrief students’ answers to these items to ensure that they understand concepts related to graphing linear inequalities and interpreting solutions to situations involving linear inequalities. Ask students to distinguish between not more than three and less than three. Model with mathematics. and so on. What part of your graphs represents solutions for which Roy would have no money left over? Explain. All rights reserved. less than. Write an inequality for each constraint described below. For these solutions. Have students represent each phrase by writing an inequality. All rights reserved. Functions Answers 24 SpringBoard® Mathematics Algebra 2. 6 ≤ m ≤ 14.. For these solutions. Answers 9. 11. Explain how you would graph the inequality 2x + 3y < 12. 11 Activating Prior Knowledge. Graph the ordered pairs and draw a dashed line through them. Solutions on the boundary line represent solutions for which Roy would have no money left over. There are only 10 performances playing that Roy actually wants to see while he is in New York City. 100t + 40m ≤ 1360 b. you show that it is true or correct. he can buy either 7 or 8 tickets and still meet all of the constraints. and for points that are not solutions. identify two ordered pairs that are feasible options to all of the inequalities. © 2015 College Board. Sample answer: 14 meals and 7 tickets b. If I were Roy. some students may choose points that are not actually solutions of the system of inequalities. be sure that they use the boundary lines and shading correctly to interpret the inequalities. Roy will go to 6 performances and have 10 meals. Discussion Groups. $360 Sample answer: 1360 − (6 × 100 + 10 × 40) = 360 © 2015 College Board. Explain how you would graph this constraint on a coordinate plane: 2 ≤ x ≤ 5. so I would choose the maximum number of meals. Sample explanation: The ordered pair is in the shaded region of the graphs of all four inequalities. If Roy uses his prize money to purchase 6 tickets and eat 10 meals. 16. Then shade the region between them. I would want to eat 3 meals per day whenever possible. Confirm that these ordered pairs satisfy the inequalities listed in Item 12. For Item 13. Explain why you made the choices you did. have students write a few sentences explaining why the point is not a solution. Answers 16.. have them identify why the point is a solution or why it is not. and tell how you know that this combination of meals and tickets is feasible.8. Graph the vertical lines x = 2 and x = 5.. Have several students plot the four inequalities on the board so that any differences can be discussed and reconciled. Label the point (6. If Roy eats 14 meals. which is 14. Answers will vary based on the points that students pick. 18. ACTIVITY 2 Continued Lesson 2-2 Graphing Systems of Inequalities ACTIVITY 2 continued 13. so the number of tickets is no more than 8. and as the precision that a graphing tool allows. First ordered pair (t. Reinforce what it means to be a member of the feasible region and also the importance of verification by substituting into all four inequalities. All rights reserved. Construct viable arguments. By looking at your graph. Debriefing Have students list four other points that are solutions and four points that are not solutions. If you were Roy. 17. m): 14. In each case. 15. Interpret the meaning of this point. 18. a. My Notes ACADEMIC VOCABULARY When you confirm a statement. Activity 2 • Graphing to Find Solutions 25 .017-028_SB_AG2_SE_U01_A02. so the number of tickets is at least 6. 6. 10) on the grid in Item 6. how many tickets could Roy purchase if he buys 12 meals? Explain. Use this as an opportunity to discuss why graphically represented information is only as accurate as the care taken when graphing. a. Is this ordered pair in the solution region common to all of the inequalities? Explain. Sample explanation: Substituting 12 for m and solving m ≤ 2t for t shows that t ≥ 6. Check Your Understanding Check Your Understanding Debrief students’ answers to these items to ensure that they understand concepts related to graphing linear inequalities and interpreting solutions to situations involving linear inequalities. Tie in the graphic representation to the real-world context. I would pick 7 tickets so that I could have some money left over for souvenirs. how many meals and how many tickets would you buy during the 5-day trip? b. b. Sample answer: If I were Roy.indd Page 25 05/12/13 1:45 PM user-s068a 123 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . Substituting 12 for m and solving 100t + 40m ≤ 1360 for t shows that t ≤ 8. Activity 2 • Graphing to Find Solutions 25 17. m): b. how much money will he have left over for other expenses? Show your work. 14–15 Quickwrite. Yes. a. 7. Given the set of constraints described earlier. Second ordered pair (t. 12–13 Create Representations As students graph the various inequalities. All rights reserved. or 8 tickets. a. Functions 8 10 x . Model with mathematics. So. x ≥ 0. 12) 2. Let x represent the number of ounces of almonds in each package and y represent the number of ounces of peanuts. Graph the constraints. Answers SpringBoard Mathematics Algebra Inequalities. Reason quantitatively. x ≤ 8.20y ≤ 4.20 6 4 2 –10 –8 –6 –4 –2 –2 2 4 6 8 10 x –4 MATH TIP –6 –8 –10 20. My Notes 19. (4. Identify two ordered pairs that satisfy the constraints. 2) represents a mixture that 8 would cost $3. See the Activity Practice for additional problems for this lesson. Reinforce students’ understanding of the meaning of the solution to a system of inequalities. The table shows information about these types of nuts.40 and have 76 g of 6 protein. The company wants the nuts in each package to have at least 60 grams of protein and to cost no more than $4. Inequalities. 2 ® –6 –8 –10 26 SpringBoard® Mathematics Algebra 2. 10 LESSON 2-2 PRACTICE y 8 Nut Protein (g/oz) Cost ($/oz) Almonds 6 0. A snack company plans to package a mixture of almonds and peanuts. Sample answer: ordered pairs that satisfy the constraints: (4. (4. 12) represents both –10 –8 –6 –4 –2 2 4 6 the more expensive mixture and the –2 –4 mixture with more protein. 21. 2 20. 2) and (4. Nut Mixture y © 2015 College Board.. 1. Shade the solution region that is common to all of the inequalities. 26a. 23. All rights reserved. 16 12 8 4 4 8 12 16 20 d © 2015 College Board. 3) and (6. LESSON 2-2 PRACTICE 19. 22. ordered pairs that do not satisfy the constraints: (5. Identify two ordered pairs that satisfy the constraints in Item 19 and two ordered pairs that do not satisfy the constraints. Answers will vary depending on the answer to part a. 4) 21. 0. Watch for students who choose points that satisfy either inequality as opposed to all inequalities in a system. Write inequalities that model the constraints in this situation. will vary but should be 2. You may assign the problems here or use them as a culmination for the activity. y = − 2 x + 3 3 Sample answer: (10. ACTIVITY 2 Continued Lesson 2-2 Graphing Systems of Inequalities ACTIVITY 2 ASSESS continued Students’ answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the lesson concepts and their ability to apply their learning. b. 2).60 and have 120 g 1 2 of protein. remember that the number of ounces of each type of nut cannot be negative.30x + 0. Which ordered pair represents the more expensive mixture? Which ordered pair represents the mixture with more protein? Explain your answer. 1) and (9. and y ≤ 2 + 1 x .017-028_SB_AG2_SE_U01_A02. y = −x + 2 b.indd Page 26 05/12/13 1:45 PM user-s068a 123 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ .30 Peanuts 8 0. Sample answer: y 10 (10. Graph these inequalities on the same grid.. 6x + 8y ≥ 60. When answering Item 21. a. Unit 1 • Equations. y ≥ 0 22. Unit 1 • Equations. and shade the solution region that is common to all of the inequalities: y ≥ 2. 12) represents a mixture 4 that would cost $3. Functions 23. Ounces of Peanuts 20 Ounces of Almonds ADAPT Check students’ answers to the Lesson Practice to ensure that they understand how to graph inequalities and identify the solutions of a system of inequalities. Use this information for Items 21–23. All rights reserved. ACTIVITY PRACTICE ordered pairs in the solution region. What is the rate of change of the function. Then graph it. Show your work. h(t) 84 77 70 63 56 49 42 35 28 21 14 7 24 c(t) 19. The line with the greater positive slope is steeper. b. 26 Temperature of Bar (˚C) Cases © 2015 College Board. What is the t-intercept of the graph of the function? What does it represent in this situation? continued A college tennis coach needs to purchase tennis balls for the team. The coach has a budget of $500 to buy tennis balls. Lesson 2-1 1. So. Are all the values for t on your graph valid in this situation. 30°C. a. How many joules of heat will be required to heat the gold bar to a temperature of 260°C? Explain how you determined your answer. Is there enough money to buy the number of cases that the coach needs? Explain. and graph it. Write the equation of the line that passes through the point (−2. 49) represents in this situation. No. the ratio of the number of cases to the number of tennis balls 16. the temperature of the bar before the jeweler begins to heat it 9. Use this information for Items 12–17. Sample answer: Nine cases will cost $540. Graph the function. −3) and (−1. 648 tennis balls c. ACTIVITY 2 Continued Graphing to Find Solutions Choices ACTIVITY 2 PRACTICE Write your answers on notebook paper. 4) and has a slope of −1. No. 28°C. All rights reserved. How many cases will the coach need to purchase to have this number of tennis balls? Explain how you determined your answer. Write the equation of a function c(t) that represents the number of cases the coach will need to purchase to have a total of t tennis balls. 8. 4. 28 7. What is the actual number of tennis balls the coach will have when he buys this number of cases? c. The coach needs to purchase 600 tennis balls.indd Page 27 15/04/14 10:41 AM user-g-w-728 144 216 288 360 t Tennis Balls 14. A jeweler is heating a gold bar. 17. Write the equation of the line with y-intercept 3 and a slope of − 2 . 10. y Activity 210• Graphing to Find Solutions 8 6 4 2 –10 –8 –6 –4 –2 –2 –4 2 4 6 8 10 x 27 30 32 34 t Cases of Tennis Balls Needed 10 9 8 7 6 5 4 3 2 1 72 17. Be sure to label the axes. and 35°C. 9. Write the equation of a function h(t) that represents the amount of heat in joules required to heat the bar to a temperature of t degrees Celsius. b. so y = 2(3x − 2) represents a steeper line than y = 5 + 5(x + 4).. and the slope of y = 2(3x − 2) is 6. and each can holds 3 balls. Write the equation of the line in standard form that passes through the points (2. which is more than the $500 budgeted for the tennis balls. 27°C. What does the slope represent in this situation? 16. removing heat from the bar in order to lower its temperature 12. given that the coach can only buy complete cases of tennis balls? Explain. What is the slope of the graph? 1 A. 18 C. 3. Temperature of Bar (°C) Heat Required (joules) 26 7 27 14 28 21 29 28 30 35 35 70 12. Sample explanation: I evaluated the function for t = 600 and found that c(600) ≈ 8. 72 15. 9 cases. 1645 joules.3. This value shows that the coach needs more than 8 cases to have 600 tennis balls. 18. The initial temperature of the bar is 25°C. 6. h(t) = 7(t − 25) or equivalent 6. 29°C. the valid values of t are nonnegative multiples. Use this information for Items 4–11. © 2015 College Board. ACTIVITY PRACTICE 3. including units? ACTIVITY 2 Heat Required (joules) 017-028_SB_AG2_SE_U01_A02. 11. The only values of t that are valid are those that result in a whole number of cases. 8 D. It takes 7 joules of heat to raise the temperature of the bar 1°C. Make a table that shows how many joules of heat would be required to raise the temperature of the gold bar to 26°C. 14. Graph the function. Sample answer: The slope of y = 5 + 5(x + 4) is 5. Sample answer: I evaluated the function for t = 260. Be sure to label the axes. 72 B. 7.. Explain what a negative value of h(t) represents in this situation. 3 2. x − 3y = 11 4. A 15. 5. Without graphing the equations. All rights reserved. 18. a. 5. 7 joules/°C 8. −4). so the coach should purchase 9 cases. Explain what the ordered pair (32. c(t ) = t 72 13. explain how you can tell which one represents the steeper line: y = 5 + 5(x + 4) or y = 2(3x − 2). 10. Heating the bar to a temperature of 32°C will require 49 joules of heat. 25. A case of 24 cans costs $60. 13. 11./103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . –6 –8 –10 Activity 2 • Graphing to Find Solutions 27 . © 2015 College Board. 27. Solid line. y ≤ 4 + x c. x + y ≥ 10. a.. 21. y 12 10 (7. 28 27. b. (4. c. Did you shade above or below the boundary line? Explain your choice. 10) b. ACTIVITY 2 Continued Graphing to Find Solutions Choices ACTIVITY 2 y 20. 28. 26. x ≥ 15. 12 10 Green Fabric (yd) 24. (6. y ≥ 4. 20. (10. Unit 1 • Equations. 3 yd. Identify two ordered pairs that do not satisfy the constraints. (15. b. All rights reserved. ≥ means that points on the boundary line are included in the solution set. Graph the inequality y > 2x − 5. Functions y 28. and at least 10 yards of fabric overall. Sample answer: (15. c.indd Page 28 05/12/13 1:45 PM user-s068a 123 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . 23. 13) and (20. 8) D. 0) should be shaded. 3) MATHEMATICAL PRACTICES Reason Abstractly and Quantitatively Look back at the scenario involving the tent designer. What is the least amount of black fabric the designer can use if all of the constraints are met? Explain. 10. c. 10. Identify two ordered pairs that satisfy the constraints. 6x + 4y ≤ 60 8 6 4 2 2 4 6 8 10 12 x Black Fabric (yd) 28 SpringBoard® Mathematics Algebra 2. SpringBoard® Mathematics Algebra 2. (17. 11) 6 26 5. Which ordered pair represents Catelyn working a greater number of hours? Which ordered pair represents Catelyn earning more money? Explain your answer. Use this information for Items 26–28. a. a. x ≥ 3. The total cost of the fabric used for the tent can be no more than $60. c. and green fabric.5). Lesson 2-2 A tent designer is working on a new tent.017-028_SB_AG2_SE_U01_A02. a. All rights reserved.. Each week. 6 –6 –4 –2 2 19. y ≥ 6 − 2 x 3 b. Write inequalities that model the four constraints in this situation. Let x represent the number of hours Catelyn works at the pet store in one week and y represent the number of hours she works as a nanny in one week. What is the greatest amount of green fabric the designer can use if all of the constraints are met? Explain your answer. 10x + 8y ≥ 250 23. (20. Below the line. a. Shade the solution region that is common to all of the inequalities. What does the ordered pair you chose in Item 28 represent in the situation? b. (2. 7) C. Inequalities. continued 4 2 –2 6 4 8 x –2 –6 –8 b. For each ordered pair. 12) B. The tent will be made from black fabric. 20 15 10 5 5 10 15 20 25 30 x Pet Store Hours ADDITIONAL PRACTICE If students need more practice on the concepts in this activity. a. b. 29. Let x represent the number of yards of black fabric and y represent the number of yards of green fabric. Graph the constraints. Graph the inequality 6x − 2y ≥ 12. Use this information for Items 22–25. at least 4 yards of green fabric. Did you use a solid or dashed line for the boundary line? Explain your choice. 15) b. Unit 1 • Equations. y ≤ 11 − 8(x − 7) 22. a. so the half-plane that does not include (0. a. 25. 13) represents working 28 hours and earning $254. Sample answer: (15. she works at least 15 hours at a pet store and at least 6 hours as a nanny. B 29. which costs $4 per yard. Catelyn wants to work no more than 30 hours and earn at least $250 per week. identify the constraint or constraints that it fails to meet. a. 8) and (17. Graph the constraints. 10) represents working 30 hours and earning $280. Inequalities. Which ordered pair lies in the solution region that is common to all of the inequalities? A. Sample explanation: The points in the shaded region of the graph with the least x-value lie on the line x = 3. (0. see the Teacher Resources at SpringBoard Digital for additional practice problems. –4 21. Graph the following inequalities on the same grid and shade the solution region that is common to all of the inequalities. 10). 15) results in her working 32 hours. Functions © 2015 College Board. Catelyn’s Weekly Work Hours y 30 25 Nanny Hours Catelyn has two summer jobs. The designer will need at least 3 yards of black fabric. x + y ≤ 30. 8) results in her earning only $214. Shade the solution region that is common to all of the inequalities. Sample explanation: The point in the shaded region of the graph with the greatest y-value is (3. Sample answer: (15. She earns $10 per hour at the pet store and $8 per hour as a nanny. 0) is not a solution. 5 22 11 4 2 –2 2 4 6 8 10 12 x 24. 26. 5 8 183 . –4 25. using 4 yards of black fabric and 7 yards of green fabric for the tent b. which costs $6 per yard. Sample answer: (20. Write inequalities that model the four constraints in this situation. y ≥ 6. 2 . 22.5 yd. systems of linear equations in two variables to model • Formulate real-world situations. where Q represents millions of barrels of gasoline and P represents price per gallon in dollars. and elimination. but items that no one wants are marked down to a lower price? Suppose that during a six-month time period. people will demand 9.15. it is not a solution. Quickwrite. ThinkPair-Share. Look for a Pattern The change in an item’s price and the quantity available to buy are the basis of the concept of supply and demand in economics. SUGGESTED LEARNING STRATEGIES: Shared Reading. Find an approximation of the coordinates of the intersection of the supply and demand functions. If an ordered pair makes one equation true.30. The demand and supply functions for gasoline are graphed below. The final price that the customer sees is a result of both supply and demand.7Q + 9. ACTIVITY Systems of Linear Equations ACTIVITY 3 Monetary Systems Overload Lesson 3-1 Solving Systems of Two Equations in Two Variables Learning Targets: graphing. Activity Standards Focus My Notes Have you ever noticed that when an item is popular and many people want to buy it. They also use technology and matrices to solve systems of equations. 1–2 Shared Reading. How can you visually show all of the existing solutions for the equation? Developing Math Language Be sure students understand that a solution to a system of equations is any ordered pair that. All rights reserved. is a solution of a system of equations in two variables when the coordinates of the points make both equations true. 3 Q Bell-Ringer Activity Have students list five solutions to the equation 2x + y = 14. Close Reading. when substituted into each equation in the system.indd Page 29 05/12/13 10:31 AM user-s068a 123 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . Discussion Groups. Note Taking. Role Play. the supply and demand for gasoline has been tracked and approximated by these functions. or set of points.029-054_SB_AG2_SE_U01_A03.. and companies will be willing to supply it. substitution. All rights reserved. students write and graph systems of equations.15 million gallons of gas.2 Create equations in two or more variables to represent relationships between quantities. and elimination to solve systems of linear • Use equations in two variables. p © 2015 College Board. At a price of $3. results in a true statement for every one of the equations in the system.5Q − 10. Create Representations.4 To find the best balance between market price and quantity of gasoline supplied. Item 1 also demonstrates the limitations of graphing as a solution method. How many solutions exist for the equation? 3. A point. Activity 3 • Systems of Linear Equations 29 Common Core State Standards for Activity 3 HSA-CED. They solve the systems of equations using graphing. • Demand function: P = −0. Sample answer: (9. Demand refers to the quantity that people are willing to buy at a particular price. 6 4 2 10 Pacing: 2 class periods Chunking the Lesson #1–2 #3 Check Your Understanding #7 Example A #11 Example B Check Your Understanding Lesson Practice TEACH MATH TERMS 8 5 Guided In Activity 3. emphasize that there is more than one way to solve a system of equations and that some methods are more efficient in certain situations. Create Representations These first few items introduce solving systems of linear equations by graphing. 12 Lesson 3-1 PLAN CONNECT TO ECONOMICS The role of the desire for and availability of a good in determining price was described by Muslim scholars as early as the fourteenth century. find a solution of a system of two linear equations. graph equations on coordinate axes with labels and scales. It asks students to approximate the solution by identifying a point of intersection that is not a lattice point in the coordinate plane. Throughout this activity. Will all students have the same five solutions? 2. Gasoline (millions of barrels) 1. The phrase supply and demand was first used by eighteenth-century Scottish economists. Then pose and discuss the following questions: 1. the price goes up. substitution. Activity 3 • Systems of Linear Equations 29 . Close Reading. Make use of structure.3).7 • Supply function: P = 1. Interactive Word Wall. 10 Price (dollars) © 2015 College Board. Review with students that a lattice point is a corner or intersection of two grid lines on the Cartesian plane. 3.A. Supply refers to the quantity that the manufacturer is willing to produce at a particular price.. Explain what the point represents. but not all of the equations in the system. there are only three classifications for a solution set: (1) inconsistent.and y-intercepts. y © 2015 College Board. All rights reserved..   y = 2 x y 6 no solutions 4 2 5 –5 –2  y = 2 x + 1 c. consistent and independent b.. What problem(s) can arise when solving a system of equations by graphing? Sample answer: Graphing is not very accurate if the intersection is not on a lattice point. Lesson 3-1 Solving Systems of Two Equations in Two Variables ACTIVITY 3 continued My Notes 2. SpringBoard® Mathematics Algebra 2. Classify the systems in parts a–c as consistent and independent. Inequalities. the intersect option is found as option 5 under the 2nd CALC menu.  2 y = 2 + 4 x 4 2 5 –5 –2 d. (3) consistent and dependent. or the scaling of the graph is not accurate enough. or inconsistent. they should either write the equation in slope-intercept form or find the x. a. Inequalities. this is done under the analyze option in the Graphs&Geometry tool. Technology Tip Students can use graphing calculators to graph each system and determine its solution. (2) consistent and independent. inconsistent c. Model with mathematics. See the Teacher Resources at SpringBoard Digital for a student page for this mini-lesson. On TI calculators.  y = 5 + 2 x b. /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . y y = x +1 a. 5 –5 x –2 Developing Math Language Make sure that students understand that although there are four terms used when describing the solution set for a system of equations. Unit 1 • Equations. TECHNOLOGY TIP You can use a graphing calculator and its Calculate function to solve systems of equations in two variables. Determine the number of solutions. On a TI-Nspire. Unit 1 • Equations. • A system with exactly one solution is independent. graph each system. Functions © 2015 College Board. Functions MINI-LESSON: Solving Systems Using a Graphing Calculator If students need additional help solving systems of equations using a graphing calculator.indd Page 30 15/04/14 11:08 AM user-g-w-728 ACTIVITY 3 Continued 3 Create Representations Remind students that to graph an equation. • Systems with one or many solutions are consistent.029-054_SB_AG2_SE_U01_A03. • A system with infinite solutions is dependent. 6 infinitely many solutions Systems of linear equations are classified by the number of solutions. MATH TERMS 30 x . consistent and dependent.   y = −x + 4 6 one solution 4 2 For additional technology resources. visit SpringBoard Digital. Graphing two linear equations illustrates the relationships of the lines. consistent and dependent 30 x SpringBoard® Mathematics Algebra 2. • Systems with no solution are inconsistent. a mini-lesson is available to provide practice. All rights reserved. 3. For parts a–c. Explain your reasoning..g.8(+) Add. which means that the graphs of the equations have no points in common and the system has no solutions. The rest of the cost is usually paid in monthly installments. Write an equation that models the amount y Marlon will pay to the dealership after x months if he chooses Plan 1.indd Page 31 12/01/15 2:03 PM ehi-6 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . If the system is independent and consistent. HSN-VM. The system is inconsistent. Therefore. To reinforce Item 5. e. Write the equations as a system of equations. such as a house or car. the graph will show a pair of lines that intersect at a point. Answers 5. Classify this system. subtract.029-054_SB_AG2_SE_U01_A03.C. it will take 36 months for the accounts to be equal. 5.7(+) Multiply matrices by scalars to produce new matrices. A pair of parallel lines never intersect. as shown in the table. © 2015 College Board. 7. ACTIVITY 3 Continued Lesson 3-1 Solving Systems of Two Equations in Two Variables ACTIVITY 3 continued My Notes Check Your Understanding 4. matrix multiplication for square matrices is not a commutative operation.C.. c.6(+)) HSN-VM.6( Use matrices to represent and manipulate data. Connect the initial amounts to the y-intercept and the rates of change to the slopes when solving using analytic geometry. Describe how you can tell whether a system of two equations is independent and consistent by looking at its graph. Discussion Groups. 7 Predict and Confirm. Activity 3 • Systems of Linear Equations 31 . a. 6. 4. Describe the graph of the system and explain its meaning. The dealership offers him two payment plans.C. e. Write an equation that models the amount y Marlon will pay to the dealership after x months if he chooses Plan 2. have students make a sketch of the situation.. © 2015 College Board. Payment Plans Plan Down Payment ($) Monthly Payment ($) 1 0 300 2 3600 200 CONNECT TO PERSONAL FINANCE A down payment is an initial payment that a customer makes when buying an expensive item. and multiply matrices of appropriate dimensions. Marlon wants to answer this question: How many months will it take for him to have paid the same amount using either plan? Work with your group on parts a through f and determine the answer to Marlon’s question. Make sense of problems. All rights reserved. as when all of the payoffs in a game are doubled.. The graph of the system is a single line. A system of two linear equations is dependent and consistent. DISCUSSION GROUP TIP As you work with your group. HSN-VM. review the problem scenario carefully and explore together the information provided and how to use it to create a potential solution. unlike multiplication of numbers.600 apart and that the gap will narrow by $100 each month. Discuss your understanding of the problem and ask peers or your teacher to clarify any areas that are not clear. y = 300x b. but still satisfies the associative and distributive properties. focus student attention on the starting amounts for both plans as well as the rate of change for both accounts. 6. x {yy == 300 3600 + 200 x Activity 3 • Systems of Linear Equations 31 Common Core State Standards for Activity 3 (continued) HSN-VM. Students may note that they begin $3. All rights reserved.g.C. The graph of a system of two equations is a pair of parallel lines.C.9(+) Understand that. Look for a Pattern Prior to using analytic geometry to solve this item. HSN-VM. there are an infinite number of solutions. y = 3600 + 200x Check Your Understanding Debrief students’ answers to these items to ensure that they understand concepts related to classifying a system of equations by the number of its solutions. to represent payoffs or incidence relationships in a network. Marlon is buying a used car. 10. In 36 months. Graph the system of equations on the coordinate grid.800). the total amount paid for Plan 1 is less than the total amount paid for Plan 2. Creating and populating the tables of values often helps students who struggle with algebraic modeling to write equations correctly. Then write a second equation in terms of the two variables that models another part of the situation. Total Amount Paid ($) Differentiating Instruction continued . write the two equations as a system.10(+)) Understand that the zero and identity matrices play a role in matrix addition and HSN-VM. In how many months will the total costs of the two plans be equal? 36 months Check Your Understanding 8. Answers Used Car Payment Plans y MATH TIP 12600 10800 9000 Plan 2 7200 Plan 1 5400 3600 1800 8. To reinforce Item 9. ACTIVITY 3 Continued Lesson 3-1 Solving Systems of Two Equations in Two Variables ACTIVITY 3 Debrief students’ answers to these items to ensure that they understand concepts related to solving a system of equations by graphing. f.C. the total cost of both plans will be $10.C. How could you check that you solved the system of equations in Item 7 correctly? 9. 6 12 18 24 30 Time (months) 36 42 x e... Functions Common Core State Standards for Activity 3 (continued) HSN-VM. The graph shows that when the time is less than 36 months (or 3 years). Unit 1 • Equations. 10. Inequalities. Assign variables to these quantities. Reason quantitatively. 10. If Marlon plans to keep the used car less than 3 years. which of the payment plans should he choose? Justify your answer. HSA-REI. All rights reserved. it is a good practice to label each line with what it represents. Construct viable arguments. In this case. 32 HSA-REI. you can label the lines Plan 1 and Plan 2. 9. Write an equation in terms of the two variables that models part of the situation. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse. Finally. Plan 1. have students share their answers with a partner and discuss why they chose their answers. Inequalities. Functions © 2015 College Board.800) satisfies both of the equations in the system.9(+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater). 32 SpringBoard® Mathematics Algebra 2. 14400 When graphing a system of linear equations that represents a real-world situation.800. Explain how to write a system of two equations that models a real-world situation. Tables of values can be used to answer Item 7. SpringBoard® Mathematics Algebra 2. All rights reserved. d. What is the solution of the system of equations? What does the solution represent in this situation? (36.C.indd Page 32 05/12/13 10:31 AM user-s068a 123 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ .C. Check Your Understanding My Notes © 2015 College Board. 10. Sample answer: Identify the two quantities in the situation that can vary. Unit 1 • Equations.029-054_SB_AG2_SE_U01_A03.10( multiplication similar to the role of 0 and 1 in the real numbers. Sample answer: Check that the ordered pair (36.8(+) Represent a system of linear equations as a single matrix equation in a vector variable. 029-054_SB_AG2_SE_U01_A03. 0.5yx=+130 10. ACTIVITY 3 Continued Lesson 3-1 Solving Systems of Two Equations in Two Variables ACTIVITY 3 continued Investors try to control the level of risk in their portfolios by diversifying their investments. x + y = 5000 The money invested is $5000.50 per share. 50 shares at $10. and the other cost $10. Solve each system of equations.50.  3x + y = 16 (3. Some students may find it easier to work with whole numbers. © 2015 College Board. Then substitute that expression into the other equation to form a new equation with only one variable. 13)  x + 2 y = 14 b.  x = 25 − 3 y a. Show your work. Solve that equation.05(5000 − x) = 205 Substitute for y in the second equation. © 2015 College Board. One stock cost $18. 0. Write one equation to represent the amount of money invested. Eli invested a total of $2000 in two stocks. How much money did she invest in each fund? Step 1: Let x = money in the first fund and y = money in the second fund.40 per share. One algebraic method for solving a system of linear equations is called substitution.02x + 250 − 0. y = 3500 Solution: Sara invested $1500 in the first fund and $3500 in the second fund. 50). This guess-and-check process will ensure that they understand the problem and will also motivate students to learn a more efficient way to find the solution..02x + 0. Have students solve the problem first through guess-and-check. All rights reserved.05y = 205 The interest earned is $205.05y = 205 Write your answers on notebook paper. {18x +. using substitution.05x = 205 Solve for x. Students may struggle with this example because they fail to understand the problem.02x + 0. 80 shares at $18. 0.03x = −45 x = 1500 Substitute the value of x into one of the original equations to find y.500 for x. 7) d. Substitute the solution into one of the two original equations to find the value of the other variable. Write another equation to represent the interest earned.. All rights reserved. one earning 2 percent and another earning 5 percent annual interest. x + y = 5000 Solve the first equation for y. 3500) into the second original equation. Eli bought a total of 130 shares.02x + 0. you solve one equation for one variable in terms of another.40 Activity 3 • Systems of Linear Equations 33 Activity 3 • Systems of Linear Equations 33 . You can solve some investment problems by writing and solving systems of equations. x + y = 5000 1500 + y = 5000 Substitute 1. Try These A MATH TIP Check your answer by substituting the solution (1500. Model with mathematics.4 y = 2000 (80. Check students’ work. −0. Step 3: Example A Note Taking Walk students through the example. y = 5000 − x 0. 1)  y − x = 4 c. Have them multiply the second equation by 100 to rewrite it as 2x + 5y = 20. Sara invested $5000 into two separate funds.indd Page 33 13/01/15 12:01 PM ehi-6 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ .500. My Notes TEACHER to TEACHER Example A During one year. The interest Sara earned was $205.   4 x + 5 y = 9 (−14. Write and solve a system of equations to find how many shares of each stock Eli bought.  2 y = x − 10 (12. MATH TERMS In the substitution method. Step 2: Use substitution to solve this system. 4)] 2. Then use substitution with that value of the variable to find the value of the other variable. The total value is 400 cents. and then. ACTIVITY 3 Continued TEACHER to TEACHER Students may question why they have to learn more than one way to solve a system of equations. Multiply each equation by a number so that the terms for one variable combine to 0 when the equations are added. Example B Note Taking Work through the example with students. Inequalities. The elimination method is also called the addition-elimination method or the linear combination method for solving a system of linear equations. 7x + 5y = 9 4x − 3y = 11 [(2. 25 + 375 = 400 400 = 400 Solution: There are 5 nickels and 15 quarters in the stack of coins. 20q = 300 Solve for q. 3)] My Notes 11.. first eliminate the n variable.029-054_SB_AG2_SE_U01_A03. Look for a Pattern Have volunteers share their answers to this item. Refer to the Math Terms box for a summary of how to use the elimination method. n + q = 20 5n + 25q = 400 Step 2: To solve this system of equations. if there are no such variables. Add the two equations to eliminate n. Functions SpringBoard® Mathematics Algebra 2. Example B MATH TERMS In the elimination method. Another algebraic method for solving systems of linear equations is the elimination method. n=5 Check your answers by substituting into the original second equation.. Allow students to compare and contrast the methods by having them solve one or more of the following systems using each method. All rights reserved. 5n + 25q = 400 5(5) + 25(15) = 400 Substitute 5 for n and 15 for q. Inequalities. 7x + 5y = −1 4x − y = −16 [(−3. 11 Think-Pair-Share. −1)] Step 4: The number of coins is 20. −5(n + q) = −5(20) 5n + 25q = 400 −5n − 5q = −100 5n + 25q = 400 Step 3: 3.indd Page 34 13/01/15 12:09 PM ehi-6 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . Unit 1 • Equations. Write one equation to represent the number of coins in the stack. Unit 1 • Equations. Point out the importance of multiplying both sides of one equation by a number that will allow one variable term to be eliminated when the equations are added. you eliminate one variable. how do you decide which variable to isolate and which equation to solve? Explain. Lesson 3-1 Solving Systems of Two Equations in Two Variables ACTIVITY 3 continued © 2015 College Board. A stack of 20 coins contains only nickels and quarters and has a total value of $4. Sample answer: Choose a variable that is easy to isolate by finding the equation with a variable that has a coefficient of 1 or −1. When using substitution. 34 34 SpringBoard® Mathematics Algebra 2. Multiply the first equation by −5. q = 15 Find the value of the eliminated variable n by using the original first equation. All rights reserved. Functions © 2015 College Board. Focus a discussion on why it is helpful to look for a variable with a coefficient of 1 first. Write another equation to represent the total value. The ordered pair is the solution of the system. How many of each coin are in the stack? Step 1: Let n = number of nickels and q = number of quarters. to look for a variable with a coefficient of −1 next. 1. . 3x − 2y = −21 2x + 5y = 5 [(−5. n + q = 20 n + 15 = 20 Substitute 15 for q. ACTIVITY 3 Continued Lesson 3-1 Solving Systems of Two Equations in Two Variables ACTIVITY 3 continued My Notes Try These B Solve each system of equations using elimination. 4) d. Show your work. Ty only needs to add 2y to both sides. 000 y = 472. Make sense of problems and persevere in solving them. Write and solve a system of equations to find the cost of a single group lesson and a single private lesson. 14. solution: (6. subtract 6y from both sides. substitution. 19. Sample answer: The first equation. a level I engineer receives a salary of $56. assigning variable names and writing equations based on what you know. 25). Solve the system by graphing. and let the other variable represent the number of engineers who will be promoted to level II. In the substitution method. 6 x + 4 y = 18 Then add the equations to eliminate the variable y and get 8x = 33. ASSESS Students’ answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the lesson concepts and their ability to apply their learning. TEACHER to TEACHER In Item 18. but to solve the second equation for x.000. (−3. (−2.  −x − 5 y = −17 12. −2 x − 3 y = 5 a. Ty would need to do two steps: first.029-054_SB_AG2_SE_U01_A03. Activity 3 • Systems of Linear Equations 35 . Next year.000 for their salaries. (3. 13. −5) 5x + 6 y = −14 b. If students are having difficulty writing equations that model a situation. you start by solving for the value of one of the variables and then use that value to solve for the value of the other variable. Which method did you use to solve the system of equations in Item 18? Explain why you chose this method.  . 2 x + 9 = y  y = −4 x − 3  16. 2 x − 4 y = 15 system:  . Write and solve a system of equations to find how many of the engineers the company can afford to promote to level II. It also offers a package of 10 group lessons and 3 private lessons for $125. At one company. 19. let one variable represent the number of engineers who will stay at level I. {1210gg ++ 23pp == 110 125 . You may assign the problems here or use them as a culmination for the activity.  4 y + 19 = x 3 y − x = −13  17. (−5. x − 2 y = 8 using 13.  −5x + 3 y = −40 (5. A karate school offers a package of 12 group lessons and 2 private lessons for $110. Compare and contrast solving systems of equations by using substitution and by using elimination. review the steps of identifying what you know and what you want to know. 3x + 2 y = 9 © 2015 College Board. He will start by solving one of the equations for x. All rights reserved. 3x + 2 y = 17 4 x − 2 y = 4  See the Activity Practice for additional problems for this lesson. −6) 17. 5) 16. LESSON 3-1 PRACTICE 15.. (5. 000 where x is the number of engineers who will stay at level I and y is the number of engineers who will be promoted to level II. 4)  x + y = 8 18. Ty is solving the system   4 x + 6 y = 10 substitution. $25 for a private lesson Check Your Understanding 12. Solve the system using substitution. All rights reserved. Sample answer: In both methods. In the elimination method. and elimination. Activity 3 • Systems of Linear Equations LESSON 3-1 PRACTICE 15. 000 x + 68. $5 for a group lesson. and a level II engineer receives a salary of $68. it can afford to pay $472. 2). and then divide both sides by 4. Solve the system using elimination.000. to solve the first equation for x. Which equation should he choose? Explain your reasoning. −4) Answers −3x + 3 y = 21 c. you add equations to get rid of one of the variables. Reason abstractly. Check Your Understanding Debrief students’ answers to these items to ensure that they understand concepts related to solving a system of equations by substitution and by elimination.   x − 2 y = 10 (2. Check students’ work. The company can afford to promote 2 engineers to level II. 56. 18. you use substitution to get rid of one of the variables.. 14. 35 ADAPT Check students’ answers to the Lesson Practice to ensure that they understand basic concepts related to writing systems of equations and solving systems of equations by graphing. Sample answer: Multiply the second 2 x − 4 y = 15 equation by 2 to get  .indd Page 35 05/12/13 10:31 AM user-s068a 123 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . Explain how you would eliminate one of the variables in this © 2015 College Board. Answers will vary. The company has 8 level I engineers. What is the monetary value of each souvenir nugget? Discuss students’ answers and have them share how they solved the problem. Ordered triples are graphed in three-dimensional coordinate space. find the ordered pair (x. Just as the ordered pair (x. Bell-Ringer Activity Write the following on the board: If z = 15. Summarizing. and rooms below your classroom can be visualized to represent the negative z-axis. Unit 1 • Equations. Arizona. z © 2015 College Board. Vocabulary Organizer Solving this contextual problem is not part of this activity. Adjoining rooms on the same floor can be visualized to extend the xy-plane. –2. The system can be represented with these equations. y) is a solution of a system in two variables. Help students understand the three-dimensional coordinate system by using the physical classroom as a model. The floor of the classroom can be thought of as Quadrant I of the xy-plane and one corner of the room can be thought of as the positive z-axis. All rights reserved. Functions y © 2015 College Board. silver. Inequalities. 5 silver. Paraphrasing.. 14 g + 20s + 24b = 20 20 g + 15s + 19b = 20   30 g + 5s + 13b = 20 Introduction Close Reading.indd Page 36 05/12/13 10:31 AM user-s068a 123 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . All rights reserved. For these more complex problems. 15 silver. Although it is possible to solve systems of equations in three variables by graphing. Vocabulary Organizer. The point (3. you can buy any of these mixtures of nuggets: 14 gold. z) is a solution of a system in three variables. Think Aloud. 36 Learning Targets: . systems of three linear equations in three variables to model a • Formulate real-world situation. Note Taking..029-054_SB_AG2_SE_U01_A03. an old mining town. −2. The problem above represents a system of linear equations in three variables. Many students have trouble visualizing a three-dimensional coordinate system when it is represented with a twodimensional drawing. 20 gold. Developing Math Language (3. Group Presentation. it can be difficult. 20 silver. 4) 4 units up O 2 units left x 36 3 units forward SpringBoard® Mathematics Algebra 2. For $20. The problem is used simply to illustrate one type of situation that can be represented by three variables in a system of linear equations. or 30 gold. y) that satisfies the system of equations: x + y − z = −1 2x − 2y + 3z = 8 In Bisbee. y. Inequalities. Graphic Organizer. and bronze. and 24 bronze. 4) is graphed below. Functions SpringBoard® Mathematics Algebra 2. Identify a Subtask TEACH Sometimes a situation has more than two pieces of information. and 13 bronze. Unit 1 • Equations. the ordered triple (x. you may need to solve equations that contain three variables. and 19 bronze. SUGGESTED LEARNING STRATEGIES: Close Reading. ACTIVITY 3 Continued Lesson 3-2 Lesson 3-2 Solving Systems of Three Equations in Three Variables ACTIVITY 3 continued PLAN My Notes Pacing: 1 class period Chunking the Lesson Example A Example B #1 #2–5 #6–7 Check Your Understanding Lesson Practice systems of three linear equations in three variables using • Solve substitution and Gaussian elimination. you can buy souvenir nuggets of gold. . −3).indd Page 37 13/01/15 12:12 PM ehi-6 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . © 2015 College Board. −2 x + 3 y + z = −13  6 x + 3 y + z = −45 Step 1: Step 2: Differentiating Instruction Solve the first equation for z.. 4x − 4y − 53 = −45 4x − 4y = 8 4x − 4(−x − 10) = 8 Substitute −x − 10 for y. after the x-terms are eliminated from the three original equations. All rights reserved. Activity 3 • Systems of Linear Equations 37 Activity 3 • Systems of Linear Equations 37 . Solve the last equation from Step 1 for z. In this case. −6.50. 6x + 3y + z = −45 6x + 3y + (−2x − 7y − 53) = −45 Substitute −2x − 7y −53 for z. Use substitution to solve the third equation for x.35. students will be solving a system in two variables—y and z. All rights reserved. assign the problem of finding the monetary value of the souvenir nuggets on the previous page. As a final step. The correct solution is g = $0. 8x = −32 x = −4 © 2015 College Board. −4x − 4y − 53 = −13 Solve for y. −4y = 4x + 40 y = −x − 10 Step 3: Example A Note Taking Point out that solving systems in three variables is similar to solving systems in two variables once a variable term has been eliminated. −2x + 3y + z = −13 −2x + 3y + (−2x − 7y − 53) = −13 Substitute −2x − 7y − 53 for z. z = 8 + 42 − 53 z = −3 Solution: The solution of the system is (−4.029-054_SB_AG2_SE_U01_A03. For students who need a challenge beyond Example 3. y = −x − 10 y = −(−4) − 10 y = −6 Step 5: MATH TIP Substitute −4 for x.25. b = $0. s = $0. ACTIVITY 3 Continued Lesson 3-2 Solving Systems of Three Equations in Three Variables ACTIVITY 3 continued You can use the substitution method to solve systems of equations in three variables. Then solve for y. z = −2x − 7y − 53 z = −2(−4) − 7(−6) − 53 Substitute −4 for x and −6 for y. check your ordered triple solution in one of the original equations to be sure that your solution is correct. 2x + 7y + z = −53 z = −2x − 7y − 53 Substitute the expression for z into the second equation. Step 4: Solve the last equation from Step 2 for y. My Notes Example A 2 x + 7 y + z = −53  Solve this system using substitution. 4x + 4x + 40 = 8 Solve for x. x + y + z = 10. All rights reserved. Step 2: Use the first equation to eliminate x from the third equation.. use the values of y and z to solve the first equation for x. 0. −2 x + y + 2z = 6   x + 2 y + z = 11 Example B Step 1: Use the first equation to eliminate x from the second equation. SpringBoard® Mathematics Algebra 2. ACTIVITY 3 Continued Example B Work Backward Students often have trouble locating the source of the error when they do not arrive at the correct solution when solving a system of equations in three variables.029-054_SB_AG2_SE_U01_A03. 5)] Try These A MATH TERMS When using Gaussian elimination to solve a system of three equations in the variables x. x + y − z = 4. Then use the value of z to solve the second equation for y. 2x + y − z = 4 −2(x + 2y + z) = −2(11) © 2015 College Board. Lesson 3-2 Solving Systems of Three Equations in Three Variables ACTIVITY 3 continued My Notes Solve each system of equations using substitution. consider beginning with the following two systems. Inequalities. Show your work. Finally. 3. The first part involves eliminating variables from the equations in the system. Unit 1 • Equations. y = 6 [(4. x − y = −1. Then eliminate y from the third equation. z. 2) Universal Access For students who are struggling with solving systems in three variables in which all three variables are present in all three equations. The second part involves solving for the variables one at a time. Functions © 2015 College Board. 2 x + y − z = 4  2 y + z = 10    −3 y − 3z = −18 Replace the third equation in the system with −3y − 3z = −18. −1. 6)] 2. 2 x + y − z = 4  Solve this system using Gaussian elimination. All rights reserved. Unit 1 • Equations. 2 x + y + z = 11    4 x + y + 2z = 23 2 x − y + z = 2 (5. 2x + y − z = 4 −2x + y + 2z = 6 2y + z = 10  2 x + y − z = 4  2 y + z = 10    x + 2 y + z = 11 Add the first and second equations. 2x + y −z = 4 −2x −4y −2z = −22 −3y −3z = −18 Add the equations to eliminate x. y. The third equation now has a single variable. x + y = 5 [(2.  x + 2 y − 3z = 15 a. (3. 38 38 Multiply the third equation by −2. −4) Another method of solving a system of three equations in three variables is called Gaussian elimination.indd Page 38 23/01/15 5:21 PM ehi-6 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . . 1. you start by eliminating x from the second and third equations. solve the third equation for z. Functions SpringBoard® Mathematics Algebra 2. y + z = 12. Encourage them to write notes about what they are doing and label equations so that they can more easily retrace their steps when checking their work.. This method has two main parts. Replace the second equation in the system with 2y + z = 10. and z. Inequalities.  x + 4 y + z = 3 3x + y + z = 5   b. 6. . 4. Solve this system of equations using Gaussian elimination. 2x + 2 = 4 2x = 2 x=1 Solution: The solution of the system is (1. Point out that there are an infinite number of systems with this solution. 3x − 2y − 2z = 7. Students can critique their partner’s flowcharts based on their effectiveness for solving the given system. However. It is important to model to students that understanding others’ mathematical thinking is a good practice. Try These B © 2015 College Board. © 2015 College Board. All rights reserved. It shows how to solve a system of linear equations involving the volume of grain yielded from sheaves of rice. Show your work. Replace the third equation in the system with −3z = −6. which was written more than 2000 years ago. Make a flowchart on notebook paper that summarizes the steps for solving a system of three equations in three variables by using either substitution or Gaussian elimination. Activity 3 • Systems of Linear Equations 39 Activity 3 • Systems of Linear Equations 39 .indd Page 39 15/04/14 11:21 AM user-g-w-728 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . Have students verbally walk you through what they have done so far rather than beginning the solution process from scratch with them. −1. Differentiating Instruction a. x − 3y − 2z = 10. Challenge students to write a system of three equations in three variables that has the solution (3. If necessary. Step 6: 2y + z = 10 2y + 2 = 10 Substitute 2 for z. My Notes 3(2y + z) = 3(10) Multiply the second equation by 2(−3y − 3z) = 2(−18) 3 and the third equation by 2. The method of Gaussian elimination is named for the German mathematician Carl Friedrich Gauss (1777–1855). Check students’ flowcharts.. provide the hint that students need to work backward. TEACHER to TEACHER No two students seem to follow the exact same steps when solving a system of equations in three variables. Step 4: Solve the third equation for z. which can make it difficult when students ask you for help. CONNECT TO MATH HISTORY Step 5: −3z = −6 z=2 Solve the second equation for y. the first known use of Gaussian elimination was a version used in a Chinese work called Nine Chapters of the Mathematical Art. who used a version of it in his calculations. 1 Think-Pair-Share Have students trade flowcharts with a partner and then follow that flowchart to solve the system 2x − y + z = 10.029-054_SB_AG2_SE_U01_A03. 5). 2x + y − z = 4 2x + 4 − 2 = 4 Substitute 4 for y and 2 for z. All rights reserved. 2y = 8 y=4 Solve the first equation for x. 2). ACTIVITY 3 Continued Lesson 3-2 Solving Systems of Three Equations in Three Variables Step 3: ACTIVITY 3 continued Use the second equation to eliminate y from the third equation. 6y + 3z = 30 −6y − 6z = −36 −3z = −6 2 x + y − z = 4  2 y + z = 10   −3 z = −6  Add the equations to eliminate y. 3)  −2 x + y + 2z = 15 1.  2 x + y − z = −2   x + 2 y + z = 11 (−2. 5. Discussion Groups Ask students to consider and discuss the following questions: 4. and wheat on his farm. Write an equation in terms of c. and w that models this information. inconsistent. have them make guesses for the values of the three variables. Differentiating Instruction continued . Explain the meaning of the numbers in the ordered triple. Have students draw diagrams showing the possible ways to position three planes in three-dimensional space and then label the diagrams with the terms consistent. soybeans. 3. Write an equation in terms of c. Functions SpringBoard® Mathematics Algebra 2. This will meet his budget and the requirement to grow twice as many acres of wheat as corn. The farmer plans to grow twice as many acres of wheat as acres of corn. Unit 1 • Equations. 50. Functions © 2015 College Board. s. dependent. Let c represent the number of acres planted with corn. MATH TIP 40 A farmer plans to grow corn. soybeans. Write your equations from Items 3–5 as a system of equations. ACTIVITY 3 Continued Lesson 3-2 Solving Systems of Three Equations in Three Variables ACTIVITY 3 Tell students that an equation in three variables represents a plane in three-dimensional space. The farmer has 500 acres to plant with corn. s. Students should discuss their strategies for making educated rather than random guesses. The farmer should grow 150 acres of corn. and independent. 300). 390c + 190s + 170w = 119. and an acre of wheat costs $170. All rights reserved. Solve the system of equations. 1. 50. Would the solution to the problem still be (150. 5. an acre of soybeans costs $190. The farmer has a budget of $119. and w represent the number of acres planted with wheat.. Inequalities. SpringBoard® Mathematics Algebra 2. 000     2c = w Determine the reasonableness of your solution. Sharing and Responding After students have written their equations. w). 300) if the constraint from Item 4 were not included as part of the problem? 2c = w  c + s + w = 500   390c + 190s + 170w = 119. and wheat. c + s + w = 500 2–5 Create Representations. s represent the number of acres planted with soybeans. All rights reserved. My Notes 2. © 2015 College Board. Inequalities.indd Page 40 05/12/13 10:31 AM user-s068a 123 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . Guess and Check. Unit 1 • Equations.000 to spend on growing the crops. Write an equation in terms of c and w that models this information. Does your answer make sense in the context of the problem? 40 6..029-054_SB_AG2_SE_U01_A03. (150. and w that models this information. Growing an acre of corn costs $390. Why are three equations necessary for a word problem that contains three variables? 2. 50 acres of soybeans. Make sense of problems. and 300 acres of wheat. s. Write the solution as an ordered triple of the form (c.000 6–7 Critique Reasoning. Solve the system using substitution. Activity 3 • Systems of Linear Equations 41 . (3. Check Your Understanding 8. A small cup costs $3. The farmer should plant 150 acres of corn. Activity 3 • Systems of Linear Equations 41 10. © 2015 College Board. 2.029-054_SB_AG2_SE_U01_A03. medium. 9. a medium cup costs $4.20 Check Your Understanding Debrief students’ answers to these items to ensure that they understand concepts related to solving a system of equations with three variables. 4) 6s + 10m + 8l = 97. start by multiplying the first equation by −2. 4. and large cups of frozen yogurt. (−3. 4. Sample answer: I used Gaussian elimination because if I had used substitution instead.20 13.80 3:00–4:00 10 12 4 99.80). Solve your equation and explain what the solution means in the context of the situation. Solve the system using Gaussian elimination. Sample answer: For consistent and independent systems. See the Activity Practice for additional problems for this lesson. Explain what the solution you found in Item 6 represents in the real-world situation.00. m. and l.80  10s + 12m + 4l = 99. Compare and contrast systems of two linear equations in two variables with systems of three linear equations in three variables. Sample answer: To eliminate x from the second equation. review strategies for reordering the equations and deciding which equation to use to eliminate variables from the other equations.60 2:00–3:00 9 12 5 100. ADAPT Check students’ answers to the Lesson Practice to ensure that they understand basic concepts related to writing systems of equations and solving systems of equations using Gaussian elimination and substitution. To eliminate x from the third equation. 14. Both types of systems can be solved by using substitution or a type of elimination. Then add the resulting equation to the third equation. Sales ($) LESSON 3-2 PRACTICE 12. Which method did you use to solve the system? Explain why you used this method.20. ACTIVITY 3 Continued Lesson 3-2 Solving Systems of Three Equations in Three Variables ACTIVITY 3 continued My Notes 7.indd Page 41 15/04/14 11:24 AM user-g-w-728 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ .80. the solution of a system of two linear equations in two variables is an ordered pair. I would have ended up with equations that had fractional coefficients. Write a system of equations that can be used to determine s. the cost in dollars of small. 4. 14. Use the table for Items 12–14. and 300 acres of wheat. Answers 8. add the first and second equations in the system.. 9s + 12m + 5l = 100. −3x + 2 y + 2z = 6   x − y + 2z = 8 10.20. 0) 11. If students are having difficulty using Gaussian elimination.00. All rights reserved. (2. © 2015 College Board. 9.. ASSESS Students’ answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the lesson concepts and their ability to apply their learning. Explain how you could use the first equation in this system to eliminate x from the second and third equations in the system:  x + 2 y − z = 5 −x − y + 2z = −13. You may assign the problems here or use them as a culmination for the activity.60  12. Use appropriate tools strategically. 13. and a large cup costs $4. and the solution of a system of three linear equations in three variables is an ordered triple. All rights reserved. 50 acres of soybeans.    2 x + y − 2z = 14 LESSON 3-2 PRACTICE  x − 3 y + z = −15  2 x + y − z = −2    x + y + 2z = 1 3x + y − z = 4  11. Frozen Yogurt Sales Time Period Small Cups Sold Medium Cups Sold Large Cups Sold 1:00–2:00 6 10 8 97. .indd Page 42 05/12/13 10:32 AM user-s068a 123 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . ACTIVITY 3 Continued Lesson 3-3 Matrix Operations ACTIVITY 3 Lesson 3-3 continued PLAN WRITING MATH You can name a matrix by using a capital letter.. Inequalities. b. The address of an entry gives its location in the matrix. Paraphrasing. #1–3 #4 Example A Check Your Understanding Example B Check Your Understanding Lesson Practice Learning Targets: © 2015 College Board. Functions © 2015 College Board. b12 −6 a. Interactive Word Wall. What is the address of the entry 8 in matrix B? Explain. The numbers in a matrix are called entries.49 1–3 Close Reading. matrix C: 2 × 2 2. as is a column in a matrix. The entry 8 is in the second row and the first column. Graphic Organizer Students often confuse the terms column and row and reverse the order of the numbers in both the dimensions and the addresses of entries of a matrix. Point out that an architectural column is vertical. subtract. 3. The entries of a matrix are the numbers in the matrix. such as matrix A below.  2 4 5  A= −3 8 −2  READING MATH Bell-Ringer Activity Provide students with the tables below and have them determine the gross sales for the day. They should be careful to distinguish the row and column labels from the entries in the matrix. What are the dimensions of each matrix? matrix B: 3 × 2. The dimensions of a matrix give its number of rows and number of columns. b31 −4 d. write the lowercase letter used to name the matrix. and then write the row number and column number of the entry as subscripts. [$533. Unit 1 • Equations. To say the dimensions of matrix A. My Notes Pacing: 1-2 class periods Chunking the Lesson . Have students determine the dimensions of the matrix. Functions SpringBoard® Mathematics Algebra 2. is a rectangular arrangement of numbers written inside brackets. read 2 × 3 as “2 by 3.49 Cost for Large $4. c22 30 c. All rights reserved. Work Backward A matrix. Note Taking. and multiply matrices. All rights reserved. Make use of structure. TEACH • Add. To write the address of an entry. SUGGESTED LEARNING STRATEGIES: Interactive Word Wall. Inequalities. Matrix A has 2 rows and 3 columns. where m is the number of rows and n is the number of columns.  3 −6    B =  8 10    0  −4 14 40   C=  26 30  1. b21. Discussion Groups. Use these matrices to answer Items 1–3.49 Cost for Medium $3. 42 42 SpringBoard® Mathematics Algebra 2. The address a12 indicates the entry in matrix A in the first row and second column.” MATH TERMS A matrix (plural: matrices) is a rectangular array of numbers arranged in rows and columns inside brackets. so its dimensions are 2 × 3. Close Reading. so a12 is 4. indicated by m × n. Write the entry indicated by each address. The dimensions of a matrix are the number of rows and the number of columns. c21 26 Developing Math Language Have students practice by writing the table for Items 12–14 in Lesson 3-2 in matrix form. Unit 1 • Equations. you will learn how to use matrices to solve systems of equations. Summarizing. In the next lesson. • Use a graphing calculator to perform operations on matrices.11] Smalls Sold Mediums Sold Larges Sold 23 45 71 Cost for Small $2.029-054_SB_AG2_SE_U01_A03. A matrix with m rows and n columns has dimensions m × n. Find C + D. Note Taking Begin a graphic organizer of matrix operations with students.   C +D =  2+5 −3 + 8   10 + (−2)  =  7 −5 + (−4)  5 8+0 1+ 7 8 8  8  −9  Example A Graphic Organizer. Step 2: Use the right arrow key to select the Edit submenu. you can add or subtract them by adding or subtracting their corresponding entries. first input both matrices.. Press ENTER to show the sum. and then press the key with QUIT printed above it.indd Page 43 05/12/13 10:32 AM user-s068a 123 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . Then select [D] from the Names submenu of the Matrix menu. To do this. Technology Tip To add the matrices on a graphing calculator. All rights reserved. Step 5: Enter the entries of the matrix. Step 3: Move the cursor next to the name of one of the matrices and press ENTER to select it. a. not the subtraction key. Press + . E − F  7 −7    −1 −3  c. If two matrices have the same dimensions. For students who may also not understand the word corresponding. Include rules for determining if the operation is defined. B =  0 −4   6 Find each matrix sum or difference. a description of how to carry out the operation.  6 −2   E= 4   9 a. provide a visual cue to help them understand the phrase corresponding entries.. A =  4   5 The word matrix has many definitions that are unrelated to mathematics. and then press the key with MATRIX printed above it.029-054_SB_AG2_SE_U01_A03. My Notes TECHNOLOGY TIP When entering a negative number as an entry in a matrix. Step 4: Enter the correct dimensions for the matrix.  6 −1  a. visit SpringBoard Digital. ELL Support  3 −2 5  b. F − E −7 7     1 3  Activity 3 • Systems of Linear Equations 43 Activity 3 • Systems of Linear Equations 43 . ACTIVITY 3 Continued Lesson 3-3 Matrix Operations ACTIVITY 3 continued You can input a matrix into a graphing calculator using the steps below.  2 − 5 8 − 0 10 − (−2)  −3 8 12  =  C−D =  −3 − 8 1 − 7 −5 − (−4) −11 −6 −1 Try These A Find each matrix sum or difference. b. and a method for determining the dimensions of the answer matrix. 4. © 2015 College Board. For additional technology resources. Your screen should now show [C]+[D]. Example A  2 8 10   C= −3 1 −5 TECHNOLOGY TIP  5 0 −2   D=  8 7 −4  © 2015 College Board. To save the matrix. Find C − D. be sure to use the negative key to enter the negative sign. press 2nd . Some students may be able to make a better language connection with the word array. Check students’ work. press 2nd . All rights reserved. Then select [C] from the Names submenu of the Matrix menu. E + F  5 3   19 11  −1 5   F=  10 7  b. Step 1: Go to the Matrix menu. Input each matrix into a graphing calculator. All rights reserved. Debrief students’ answers to these items to ensure that they understand concepts related to matrices and matrix operations. Functions © 2015 College Board. take a look at the next example. All rights reserved. Express regularity in repeated reasoning.indd Page 44 25/02/15 7:44 AM s-059 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . Sample explanation: I  0 −4  wrote the opposite of each entry in the original matrix. Two matrices are additive inverses if each entry in their sum is 0. n×m 44 44 7. Functions SpringBoard® Mathematics Algebra 2. The matrix product AB is defined because A has 2 columns and B has 2 rows. Matrix addition is commutative. ask them to devise a way to change a matrix subtraction problem to a matrix addition problem by “adding the opposite. 2  −7 8. the product AB is a 3 × 1 matrix. 6. Sample answer: A table is a way of arranging information in rows and columns.  7 −2    4   0 You can also find the product of two matrices A and B if the number of columns in A is equal to the number of rows in B. Lesson 3-3 Matrix Operations ACTIVITY 3 continued © 2015 College Board..  −1 6   A+B = B+A =  −3 7  7. The matrices do not have the same dimensions. Unit 1 • Equations. To see what this means. m×p SpringBoard® Mathematics Algebra 2. Because A above is a 3 × 2 matrix and B is a 2 × 1 matrix. Inequalities. the dimensions of matrix A below are 3 × 2. Then provide an example that supports your conjecture. Unit 1 • Equations. Explain why you cannot subtract these two matrices.   . and the dimensions of matrix B are 2 × 1. For example. find the sum of the products of consecutive entries in row i of matrix A and column j of matrix B. Inequalities.  2 8   A =  −7 5    1 3   MATH TIP The inner dimensions of two matrices indicate whether their product is defined. 6. ACTIVITY 3 Continued My Notes Check Your Understanding Answers 5. 5. n×m m×p The outer dimensions indicate the dimensions of the matrix product.  2 3 4  A= −1 −6 −8  Differentiating Instruction If students need a challenge.029-054_SB_AG2_SE_U01_A03.  4 −2    B =  5 −6     6 −10  8.  4 B=  −2  The product of an n × m matrix and an m × p matrix is an n × p matrix.  −2 4   and Sample example: A =   0 3  1 2  B= −3 4  . and a matrix is a way of arranging numbers in rows and columns. Check Your Understanding . Make a conjecture about whether matrix addition is commutative.” Ask students to share their methods with the class. What is the additive inverse of the matrix shown below? Explain how you determined your answer.. To find the entry in row i and column j of the product AB. How is a matrix similar to a table? MATH TIP Recall that the Commutative Property of Addition states that a + b = b + a for any real numbers a and b. column 1 of AB. Use row 2 of A and column 2 of B. column 1 of AB. Then select [A] from the Names submenu of the Matrix menu. so AB is defined. column 2 of AB. © 2015 College Board.. Your screen should now show [A]*[B]. Find the entry in row 2. Find the entry in row 1.. Then select [B] from the Names submenu of the Matrix menu. TECHNOLOGY TIP To multiply the matrices on a graphing calculator. and the third entries. Multiply the first entries. Press ENTER to show the product. so AB is a 2 × 2 matrix. Use row 2 of A and column 1 of B. write the problem and B =  3 −4 in stacked form. 3(4) + (−2)(−1) + 2(−2) = 10 2 4  1  5 −4   17 7    3 −1 =   AB =  2    3 −2   0 10   0 −2  17 7   Solution: AB =   0 10  Activity 3 • Systems of Linear Equations 45 Activity 3 • Systems of Linear Equations 45 . 1(2) + 5(3) + (−4)(0) = 17 2 4  1  5 −4   17 _    3 −1 =   AB =  2    3 −2   _ _   0 −2  Step 3: Find the entry in row 1. Use this visual technique to help students struggling with matrix multiplication. and B is a 3 × 2 matrix. Each entry in the answer matrix is an intersection of a row and a column in the factor matrices: 1 -5 -4 3 3 -2 0 1 4 -3 P31 P22 © 2015 College Board. the second entries. Technology Tip 3(2) + (−2)(3) + 2(0) = 0 2 4  1  5 −4   17 7    3 −1 =   AB =  2    3 −2   0 _   0 −2  Step 5: For additional technology resources. column 2 of AB. All rights reserved. Use row 1 of A and column 2 of B.indd Page 45 23/01/15 5:21 PM ehi-6 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . 2 4  1  5 −4   17 7    3 −1 =   AB =  2    3 −2   _ _   0 −2  Find the entry in row 2. Use row 1 of A and column 1 of B. When finding the  3 1   product AB. where A = −2 4    0 −3  1 −5   . All rights reserved. Then add the products. 1 5 −4   A= 2   3 −2 2 4   B =  3 −1    0 −2  TEACHER to TEACHER Step 1: Determine whether AB is defined.029-054_SB_AG2_SE_U01_A03. Identifying the address of each entry in the product matrix helps students to focus on the correct row and column to use in the factor matrices. ACTIVITY 3 Continued Lesson 3-3 Matrix Operations ACTIVITY 3 continued My Notes Example B Find the matrix product AB. A has 2 rows and B has 2 columns. Press x . first input both matrices. Step 2: A is a 2 × 3 matrix. visit SpringBoard Digital. 1(4) + 5(−1) + (−4)(−2) = 7 Step 4: Example B Close Reading Emphasize the importance of determining the dimensions of the product matrix prior to performing the operation. ED 21 −37   −11 −47  not defined Check Your Understanding 9. −7 14. What are the dimensions of A? 13. Find AB if it is defined. B + C =  −4 −1 −1 17   15. What is the entry with the address b12? LESSON 3-3 PRACTICE 12. ACTIVITY 3 Continued My Notes Find each matrix product if it is defined. Look for and make use of structure. BC =  −14 −36  Try These B SpringBoard® Mathematics Algebra 2. not defined  27 71  17. You may assign the problems here or use them as a culmination for the activity. The number of rows of S must equal the number of columns of R. 15. 11. Functions © 2015 College Board. so S has 4 columns. All rights reserved.. 4  3  1 4  C= −7 6  12   12   −8  −80 c. Find B + C.  2 −14   KL =  −4 −14   9 3  14. Answers Lesson 3-3 Matrix Operations ACTIVITY 3 Check Your Understanding . what are the dimensions of S? Explain your answer. 10.  3 −6 1  A= 2 0  −8  5 −7   B= 2  −3  4 10   C= −1 −3 12. multiply the first row times the first column. The matrix product RS is a 3 × 4 matrix.029-054_SB_AG2_SE_U01_A03. Unit 1 • Equations. Find C − B. AB =  and B =  −3 4   −9 continued © 2015 College Board. Sample example: A =   0  1 2 −14  . CD −2 10   and BA =   6 0  10.. C − B =   2 −5 16. Debrief students’ answers to these items to ensure that they understand concepts related to matrix products. 16. 2 × 3 13.  −2 9. Functions Check students’ answers to the Lesson Practice to ensure that they understand basic concepts related to Matrix operations. 46  8 5 −1  E= −4 4 −9  b. LESSON 3-3 PRACTICE Use these matrices to answer Items 12–17. If R is a 3 × 2 matrix.” while you circle the first row of the first factor and the first column of the second factor. Find BC if it is defined. have them recite with you. Instead. Inequalities. 17. 11. Rebekah made an error when finding the matrix product KL. What mistake did Rebekah make? What is the correct matrix product?  2 1 8 5   K= L= −4 −2   0 −3  2(1) 8(5)  2 40  =  KL =  −4(0) −2(−3)  0 6  Students’ answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the lesson concepts and their ability to apply their learning. she should have found each entry klij in KL by finding the sum of the products of consecutive entries in row i of matrix K and row j of matrix L. Her work is shown below. Inequalities. The number of columns of S must equal the number of columns of RS. CE −16 −12    −24 −86  ASSESS ADAPT  0 8  D= −4 −5 a. All rights reserved. See the Activity Practice for additional problems for this lesson. 46 SpringBoard® Mathematics Algebra 2. so S has 2 rows. If students are having difficulty multiplying matrices. Rebekah incorrectly found the entries of KL by multiplying the corresponding entries of K and L. 2 × 4. “To find the entry in the first row and first column.indd Page 46 05/12/13 10:32 AM user-s068a 123 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . Unit 1 • Equations. Is matrix multiplication commutative? Provide an example that supports your answer. No. Critique the reasoning of others. Answers 2. 1 0 0   I = 0 1 0    0 0 1  The product of a square matrix and its multiplicative inverse matrix is an identity matrix I. Yes. A multiplicative identity matrix is often called an identity matrix and is usually named I. the matrix product AB is the  1 0 identity matrix  . Identify AB and BA as multiplicative identity matrices. 2 4  1. or inverse matrix. and the products AB and BA as square matrices. and all the rest of the entries are 0. Yes. systems of three linear equations in three variables by using • Solve graphing calculators with matrices. So. A multiplicative identity matrix is a square matrix in which all entries along the main diagonal are 1 and all other entries are 0. Is B the inverse of A? Explain. 5 −2    A= B= 1  1 3  −0.indd Page 47 23/01/15 5:21 PM ehi-6 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ .5 Debrief students’ answers to these items to ensure that they understand concepts related to identity and inverse matrices. by definition. In an identity matrix.2 −0. all of which are 1. The matrix product AI 2. Bell-Ringer Activity  0. all entries on the main diagonal are 0. 2 0  2     0. The entries in blue are on the main diagonal. 2 4  . © 2015 College Board. B. Marking the Text. MATH TERMS A square matrix is a matrix in which the number of rows equals the number of columns. Identify A.3 0   0 1 Developing Math Language Use the Bell-Ringer Activity to emphasize the terms square matrix.  0 1 Activity 3 • Systems of Linear Equations 47 . Debriefing. Construct viable arguments. multiplicative identity matrix. Identify a Subtask. Create Representations.  0 1 4. A−1 is the inverse of A if A • A−1 = I. [I3×3] An identity matrix is the product of a square matrix and its multiplicative inverse matrix. Multiply A by a 2 × 2 identity matrix. all entries on the main diagonal are 1.. In this matrix.. A 3 × 3 identity matrix is shown below. or identity matrix. The multiplicative inverse of matrix A is often called the inverse of A and may be named as A−1. The main diagonal of a square matrix is the diagonal from the upper left to the lower right. 2     1  0. B =  − 0. 2 3 0 0. Write matrices A and B (shown below) on the board and have students find AB and BA. Identify B as the multiplicative inverse matrix of A and B as the multiplicative inverse matrix of A.  1 0  Not all square matrices have inverses. Check Your Understanding 0 1 1. But if a square matrix has an inverse. AI =   1 3 is the same as A. it must be a square matrix. and all the rest of the entries are 1. You can use matrices and inverse matrices to solve systems of linear equations.029-054_SB_AG2_SE_U01_A03. Check Your Understanding Use these matrices and a graphing calculator to answer Items 2–4. My Notes Pacing: 1-2 class periods Chunking the Lesson Check Your Understanding #5–7 Example A #8 #9–10 #11–12 #13–14 #15 Check Your Understanding Lesson Practice SUGGESTED LEARNING STRATEGIES: Vocabulary Organizer. All rights reserved. Discussion Groups. Explain why   is not an identity matrix. ACTIVITY 3 Continued Lesson 3-4 Solving Matrix Equations ACTIVITY 3 continued Lesson 3-4 PLAN Learning Targets: systems of two linear equations in two variables by using graphing • Solve calculators with matrices. TEACH A multiplicative identity matrix. 3 1  A =  2 0 −2  . 3. All rights reserved. MATH TIP © 2015 College Board. and multiplicative inverse matrix. the matrix product BA is the  1 0 identity matrix  . Activity 3 • Systems of Linear Equations 47 1. 3. 4. is a square matrix in which all entries are 0 except the entries along the main diagonal. Is A the inverse of B? Explain. Note Taking. Describe the relationship between the matrix product AI and A. Look for a Pattern A square matrix is a matrix with the same number of rows and columns. indd Page 48 15/04/14 11:31 AM user-g-w-728 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . Create Representations The notation A−1 that is used to denote the multiplicative inverse of a matrix A can seem strange to students. • X is the variable matrix. the matrix formed by the coefficients of the system of equations. In Items 5–7.. What is A−1? 4 3  11 A−1 =  11 1 −2  11 11 6. Then select [A] from the Names submenu of the Matrix menu.029-054_SB_AG2_SE_U01_A03. Press ENTER to show the inverse matrix. Input matrix A.. The first step in solving a system of linear equations by using matrices is to write the system as a matrix equation. Thus. use the matrix equation   1 −4   y  −2  5. Your screen should now show [A]−1. Unit 1 • Equations. Then press x−1 . variable matrix. To solve for X.  44 33   11 22 −11 11  77 = 11 =   A− B= = 11 A B −22 11  11 − − 22  − 11 11 11 11 48 48 SpringBoard® Mathematics Algebra 2. Be certain that students can properly identify the coefficients. you first need to find A−1. constant matrix B variable matrix X To solve a matrix equation AX = B for X. that A−1 does not mean that students should take the multiplicative inverse of each entry in the matrix. the coefficient matrix. Find the matrix product A−1B. ACTIVITY 3 Continued 5–7 Activating Prior Knowledge. a column matrix that represents all the variables of the system of equations. • A is the coefficient matrix. and constant in the equation 2x − 4y = 12 prior to having them learn the terms coefficient matrix. 8 Point out. can be written as 1x + (−4)y = −2. Lesson 3-4 Solving Matrix Equations ACTIVITY 3 continued © 2015 College Board. To solve ax = b. All rights reserved. try converting them to fractions by pressing MATH and selecting 1: Frac. visit SpringBoard Digital. system   x − 4 y = −2 A •X= B 2 3  x   7     =    1 −4   y  −2  coefficient matrix A MATH TERMS A matrix equation is an equation of the form AX = B. which more clearly shows that the coefficient of x is 1 and the coefficient of y is −4. Functions SpringBoard® Mathematics Algebra 2. variables. which is equal to 8−1. however. TECHNOLOGY TIP If the entries in the inverse matrix are decimals. into a graphing calculator. Likewise. and constant matrix. to solve the matrix equation AX = B. the solution of AX = B is X = A−1B. x − 4y = −2. you can multiply both sides of the equation by the multiplicative inverse matrix of A. All rights reserved. you use a process similar to what you would use when solving the regular equation ax = b for x. The diagram shows how to write the 2 x + 3 y = 7 as a matrix equation. You can use a graphing calculator to help you find A−1. My Notes MATH TIP The second equation in the system. Remind them that the multiplicative inverse of the number 8 is 1 . Functions © 2015 College Board. Unit 1 • Equations. Developing Math Language . 2 3  x   7    =  . Inequalities. you could multiply both sides of the equation by the multiplicative inverse of a. Technology Tip For additional technology resources. Inequalities. a column matrix representing all the constants of the systems of equations. • B is the constant matrix.    =  . MATRIX[B] 2 × 1 [82 ] [160 ] Use the calculator to find A−1B. All rights reserved. Step 1: Use the system to write a matrix equation. Solution: The matrix product A−1B is equal to the variable matrix X. ACTIVITY 3 Continued Lesson 3-4 Solving Matrix Equations ACTIVITY 3 continued 7.   5 6   y  13 Example A The hourly cost to a police department of using a canine team depends on the hourly cost x in dollars of using a dog and the hourly salary y of a handler. Use a matrix equation to  solve the system.. Direct these students to research the Augmented Matrix Method and the Adjoint Method. Ask students to predict how this change might affect the variable matrix found in Step 3. TEACHER to TEACHER Some students may question whether it is possible to find a multiplicative inverse matrix without using a calculator. x must equal 2 and y must equal 1. © 2015 College Board. and C are real numbers. and explain what the solution means. (−3. (5. so x = 2 and y = 38. (1. and the hourly cost for a team of four dogs and four handlers is $160.  5x + 6 y = 13 6 x − 3 y = −18 b. 8) 4   y   34   2  1 −2   x  −23   =  . [A]-1 * [B] [ [2] [38] ] Step 4: Identify and interpret the solution of the system. All rights reserved. The solution of the system is (2. Then use the matrix equation to solve the system. What are the values of x and y in the variable matrix X? How do you know? x My Notes x = 2. 10) c.029-054_SB_AG2_SE_U01_A03. Try These A Write a matrix equation to model each system. MATRIX[A] 2 × 2 [3 2 ] [4 4 ] © 2015 College Board.  1 Example A Note Taking Walk students through the example. The solution shows that the hourly cost of using a dog is $2 and the hourly salary of a handler is $38. Try These A Answers  2 1  x   8     =   .. Because X =   and  y  2 A−1B =   . Research on these methods may certainly make students appreciate their graphing calculators or other technology. y = 1. B. each equation in the system must be written in standard form Ax + By = C.  3x + 3 y = 21 Activity 3 • Systems of Linear Equations 49 Activity 3 • Systems of Linear Equations 49 . The hourly cost for a team of three dogs and two handlers is $82. The 3x + 2 y = 82 system  4 x + 4 y = 160 models this situation. 38). X is equal to A−1B. Step 3:  6 −3  x  −18  b. 2 x + y = 8 a.  2 x + 4 y = 34  x − 2 y = −23 c.  3  y   21  3 MATH TIP Before you can write a matrix equation to model a system of two linear equations in two variables.indd Page 49 05/12/13 10:32 AM user-s068a 123 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . where A. Have half of the class use the matrix equation as written and the other half reverse the order of the system of equations in the matrix equation.  3 2   x   82    =   4 4   y  160  Step 2: Enter the coefficient matrix A and the constant matrix B into a graphing calculator. Make sense of problems. −2) a. Functions © 2015 College Board. Model with mathematics. The matrix will have 3 rows because the system has 3 equations. and z is the number of large packs. It will have 3 columns because each equation has 3 coefficients. Unit 1 • Equations. Karen makes handmade greeting cards and sells them at a local store. A solution you obtain from a graphing calculator (either from graphing the system or from solving a matrix equation) is likely to be more accurate. What is an advantage of using a graphing calculator to solve a system of two linear equations in two variables as opposed to solving the system by making a hand-drawn graph? Sample answer: A solution you obtain from a hand-drawn graph is likely to be an estimate and may not be very accurate. Lesson 3-4 Solving Matrix Equations ACTIVITY 3 continued My Notes 8. Inequalities. The cards come in packs of 4 for $11. You can also use a matrix equation to solve a system of three linear equations in three variables.029-054_SB_AG2_SE_U01_A03. 9–10 Debriefing. although the x. the store sold 16 packs containing 92 of Karen’s cards for a total of $223.indd Page 50 05/12/13 10:32 AM user-s068a 123 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . ask students how they would go about finding the exact answers to a system of equations if they found that the calculator was providing them with rounded answers. Write a matrix equation to model the system. © 2015 College Board. Functions SpringBoard® Mathematics Algebra 2. or 10 for $20. 3 × 3.  x + y + z = 16  4 x + 6 y + 10z = 92   11x + 15 y + 20z = 223 9.. what would be the dimensions of the coefficient matrix? How do you know?  1 1 1  x   16        4 6 10   y  =  92  11 15 20   z   223        50 50 SpringBoard® Mathematics Algebra 2. 6 for $15. Last month. Create Representations Reinforce the structure of the matrix equation by having students manually multiply the coefficient matrix by the variable matrix. The following system models this situation where x is the number of small packs. Inequalities. All rights reserved. All rights reserved. Unit 1 • Equations.and y-coordinates may be rounded. . 10.. If you were to model the system with a matrix equation. Use this information to complete Items 9–15. y is the number of medium packs. ACTIVITY 3 Continued 8 Debriefing As a follow-up to this item. My Notes 11–12 Create Representations. 14. © 2015 College Board. Ask questions to clarify ideas and to gain further understanding of key concepts. All rights reserved. Activity 3 • Systems of Linear Equations 51 Activity 3 • Systems of Linear Equations 51 .029-054_SB_AG2_SE_U01_A03. Compare and contrast using a matrix equation to solve a system of two linear equations in two variables with using a matrix equation to solve a system of three linear equations in three variables. the variable matrix is 3 × 1. 13–14 Create a Plan. and the constant matrix is 2 × 1. The store sold 8 small packs of Karen’s cards. and 3 large packs. 3). Use a graphing calculator to find A−1. the variable matrix is 2 × 1. except for the dimensions of the matrices. 5 medium packs. the coefficient matrix is 2 × 2. Use a graphing calculator to find A−1B. Debriefing Be certain that students know how to toggle between fraction and decimal representation on their calculators. Create Representations. How can you check that you found the solution of the system correctly? Sample answer: I could substitute the solution into one of the original equations of the system to check that it satisfies the equation. Identify a Subtask These items continue to walk students through the steps for solving the word problem presented prior to Item 9. Work with your group.indd Page 51 25/02/15 7:46 AM s-059 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . Find the solution of the system of equations and explain the meaning of the solution. ACTIVITY 3 Continued Lesson 3-4 Solving Matrix Equations ACTIVITY 3 continued 11. All rights reserved. Sample answer: Both processes are the same. take notes to aid comprehension and to help you describe your own ideas to others in your group. (8. 13. DISCUSSION GROUP TIP As you listen to the group discussion.  5 −2  5 6 3   −1  3 1 A = −5 − 2   2 − 1  1  3 3  12. Point out that students will want to follow these same steps to solve subsequent word problems even though those items no longer reference these steps. the inverse of the coefficient matrix.. Each group should prepare a final answer to the question and present it to the class. the coefficient matrix is 3 × 3. 15. Use appropriate tools strategically. For a system of two equations. and the constant matrix is 3 × 1. 5. © 2015 College Board. For a system of three equations. 8 A−1B =  5  3   15 Discussion Groups Have students share their answers within small groups.. 18.029-054_SB_AG2_SE_U01_A03.indd Page 52 05/12/13 10:32 AM user-s068a 123 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ .62). CONNECT TO ECONOMICS The euro is the unit of currency used in many of the nations in the European Union. Doug incorrectly solved the  3 2  18  matrix equation in Item 17 by finding the matrix product    . 6 © 2015 College Board. Functions LESSON 3-4 PRACTICE − 1 1  8 4   19. Use a graphing calculator to find the inverse of the matrix  . The British pound is the unit of currency used in the United Kingdom. Emily has 3 euros and 1 British pound worth a total of $5. What happens when you try to solve the system   4 x + 2 y = 16 by writing and solving a matrix equation? What do you think this result indicates about the system? Confirm your answer by graphing the system and using the graph to classify the system.  2 8 3  y  =  80  . and a British pound is worth $1. LESSON 3-4 PRACTICE y  2 8  19. 1. If students are having difficulty setting up the matrix equation. a. You may assign the problems here or use them as a culmination for the activity. which confirms that the system is inconsistent and has no solutions. What is the relationship between AX and the system of equations? Answers  3x + 2 y   . Unit 1 • Equations. All rights reserved. 4x + 2y = 16 2 x + 4 y = 22 20. Answers will vary. have them check their equation by manually finding the matrix product AX = B. ACTIVITY 3 Continued Lesson 3-4 Solving Matrix Equations ACTIVITY 3 Check Your Understanding Debrief students’ answers to these items to ensure that they understand concepts related to solving systems of equations with matrices. 23.  2 4   20  What mistake did Doug make? What should he have done instead? 2 x + y = 6 18. I think this probably indicates that the system has no solutions. Write the system of equations as a matrix equation. b. Model with mathematics. 17. Inequalities. 2 x + 8 y + 3z = 80   4 x − 6 y + 7 z = −62 22. where x represents the value of 1 euro in dollars and y represents the value of 1 British pound in dollars. A euro is worth $1..31.55 c. −2)       4 −6 7   z  −62  2 x + 4 y = 9. 10. Doug tried to solve for X by finding the matrix product AB. SpringBoard® Mathematics Algebra 2. 23.31. A system of equations and the matrix equation that models it are shown below.55  2 4   x   9. c. write a matrix equation to model each system. Then interpret the solution.10     = b. (1. Which solution method for solving systems of equations do you find easiest to use? Which method do you find most difficult to use? Explain why.62. All rights reserved. See the Activity Practice for additional problems for this lesson.. AX =   2 x + 4 y  are the same as the variable expressions in the system of equations.  x + y + z = 11  21. A graph of the system shows two parallel lines.  5 4  10 8 For Items 20–21. (1. The entries in AX 16. 5) 20.10. the product of the coefficient matrix and the variable matrix of the matrix equation.  −3 2   y   7  1 1 1  x   11      21. The result should be the original system of equations. Instead. Write a system of equations to model this situation.   3 1  y   5.  5 − 1  32 16  2 4   x   22     =   . ADAPT Check students’ answers to the Lesson Practice to ensure that they understand basic concepts related to writing and solving matrix equations. 3x + 2 y = 18 2 x + 4 y = 20  17. 2x + y = 6  3 2   x  18     =    2 4   y   20  . Then use the matrix equation to solve the system. 52 52 SpringBoard® Mathematics Algebra 2. Find AX. Functions © 2015 College Board. continued My Notes Check Your Understanding 16.55 a. he should have found A−1B. Sample answer: I get an error message when I try to find A−1B on a graphing calculator. (3. Unit 1 • Equations. −3x + 2 y = 7  4 2 –6 –4 2 –2 4 6 x –2 ASSESS Students’ answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the lesson concepts and their ability to apply their learning. Use the matrix equation to solve the system. Steve has 2 euros and 4 British pounds worth a total of $9. Critique the reasoning of others.  3x + y = 5. Ensure students are using appropriate and precise mathematical terminology. Inequalities.10 22. The chemist should mix 375 milliliters of the 2% solution and 225 milliliters of the 10% solution. which cost $1 each. (3. where s is the  number of songs and m is the number of music videos. (15. Solve the system using substitution. 5. A small furniture factory makes three types of tables: coffee tables. and granola in each jar. Write and solve a system of equations to find the number of tables of each type the factory should make each day. dining tables. The number of dining tables made per day should equal the number of coffee tables and end tables combined. and tell how many solutions it has.. 4 –6 Calories per Ounce –6 53 –4 2 –2 4 6 x –2 –4 –6 6. It has infinitely many solutions. a. y is the number of dining tables. grape jelly. Activity 3 • Systems of Linear Equations 53 . 10) D a. The factory needs to make 54 tables each day. (3. The system is consistent and dependent. 3. and z is the number of end tables made each day. 11. All rights reserved. (9. Each jar of the sandwich spread will contain 9 ounces of peanut butter.029-054_SB_AG2_SE_U01_A03. and 12 end tables each day. The number of coffee tables made each day should be three more than the number of end tables. 6 4 9. Show your work.  x + y = 600 0. Graph the system  3 y = −3 − 6 x Granola b.02 x + 0. ACTIVITY PRACTICE 1. (−40. where x is the number   x = z + 3 of coffee tables. All rights reserved. 5 y − x = −5 7 y − x = −23 y 8 8. y is the number of ounces of grape jelly. To solve a system involving 3 variables. 15  y = −2 x − 1 . Solve the system using substitution. −3) 8.  x + 2 y + 3z = 8 3x + 4 y + 5z = 10  12. 27. The system of equations  has 10 x + cy = 3 solutions for all values of c except: Peanut butter Grape jelly A. grape jelly. Write and solve a system of equations to find the number of ounces of peanut butter. 7. (375. 2). 225). 9. Solve the system using elimination. 10 B.10 y = 0. −15 C.indd Page 53 05/12/13 10:32 AM user-s068a 123 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . The factory should make 15 coffee tables. Can you solve this system? Explain. −9) (29. Sample explanation: The system includes 3 variables. 10 Lesson 3-2 Write your answers on notebook paper. 6. but there are only 2 equations. Solve the system by graphing. 7. Grams of Fat per Ounce 168 14 71 0 4 132 12 2 y  6 An 18-ounce jar of the sandwich spread will have a total of 2273 calories and 150 grams of fat. Classify the system. Solve the system using Gaussian elimination. Mariana bought 10 songs and 5 music videos. 7 ounces of grape jelly. (10. A chemist needs to mix a 2% acid solution and a 10% acid solution to make 600 milliliters of a 5% acid solution. you need at least 3 equations. 27 dining tables. and granola as a sandwich spread. 12). 10. 5. −2. Activity 3 • Systems of Linear Equations 2 –4 10. Mariana had a $20 gift card to an online music store. and end tables.05(600) .  s + 2m = 20  s − 5 = m . Mariana bought five more songs than music videos. and 2 ounces of granola. and music videos. 6 8 10 x –8 –10 2. A snack company plans to sell a mixture of peanut butter. −3 D. She spent the entire amount on songs. No..  y = −3x + 6  y = −1 x −2  3  2 –10 –8 –6 –4 –2 –2 Ingredient 2 x + 3 y = 7 4. 2 x − 5 y = 8  x − 3 y = −1  © 2015 College Board. The table gives information about each ingredient. 7. Write and solve a system of equations to find the number of songs and the number of music videos Mariana bought.  y = x + z .  x + y + z = 6 2 x + y + 2z = 14   3x + 3 y + z = 8 Lesson 3-1 © 2015 College Board. 12. b. where  x is the number of milliliters of the 2% solution and y is the number of milliliters of the 10% solution. 3. Write and solve a system of equations to find the volume of the 2% solution and the volume of the 10% solution that the chemist will need. 5) (7. and z is the number of ounces of granola. 5). 1)  x + y + z = 18 168 x + 71 y + 132z = 2273. where   14 x + 12z = 150 x is the number of ounces of peanut butter. ACTIVITY 3 Continued Systems of Linear Equations Monetary Systems Overload ACTIVITY 3 continued ACTIVITY 3 PRACTICE 1. 4.  x + y + z = 54 11. 4. which cost $2 each. 2 x + 4 y + z = 31 −2 x + 2 y − 3z = −9    x + 3 y + 2z = 21 2. 48 x + 2. What are the dimensions of matrix A? 14.62  (4. Write the system of equations represented by the matrix equation below. 19.  3  2 0   2 23. Then solve the matrix equation. 18. Systems of Linear Equations Monetary Systems Overload ACTIVITY 3 continued Lesson 3-3 Use the given matrices for Items 13–20. Then solve the equation to find the prices. (−2. see the Teacher Resources at SpringBoard Digital for additional practice problems. 5)      1 −4   z  −30   8  x + y = 6 26. Which expression gives the value of P12? A. Inequalities.62 the number of pounds of beef and y is the number of pounds of pork. 1(3) + 5(1) C.50 Andrew 2 0 1 $17. not defined 3  0 −3    2  −1  1 10 12    −7 26 17  20. which means  0 1 that B is the inverse of A.  3 4 5  k   87. Compare and contrast solving an equation of the form ax = b for x with solving a matrix equation of the form AX = B for X.029-054_SB_AG2_SE_U01_A03. mugs are $5. 3x + 2 y − 7 z = −29  4 x − 6 y + 5z = −19   8 x + y − 4 z = −30 13. −2(3) + 5(−1) B. Functions ADDITIONAL PRACTICE If students need more practice on the concepts in this activity.48/lb and the pork costs $2.  0 1 which means that A is the inverse of B. Find ED if it is defined.  7 11 12 . 15. 17. The product of 2 matrices is defined if the number of columns in the first matrix is equal to the number of rows in the second matrix.00 MATHEMATICAL PRACTICES Look For and Make Use of Structure 28.50 John 1 1 2 $31.48 2.  b. 54 Key Chains Lesson 3-4 SpringBoard® Mathematics Algebra 2. −3 −2   x   1    =   4   y  −3  5 25.  1  3  − 1 0 − 2  4 22.  3 1 −2 −1   a.5. To solve ax = b. To solve AX = B.  m =  5  . 4(0) + 1(2) 21.5 .           2 0 1  g   17   g  12  Key chains are $2. Inequalities. and Andrew sold key chains.  b. 26.5).indd Page 54 05/12/13 10:32 AM user-s068a 123 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . Then use a matrix equation to solve the system. The matrix product AB is the  1 0 identity matrix   . Yes. Find A + D. Explain how to determine whether the product of two matrices is defined and how to determine the dimensions of a product matrix. 22. 4(0) + (−2)(2) D. 6..5 pounds of beef and 1. 13.64   y  19. a.  . and gift wrap is $12.5  k   2.  .  4 −6 5  y  = −19  . The product matrix has the same number of rows as the first matrix and the same number of columns as the second matrix. where x is 3.  5 7   1 14 −1 17   18. ACTIVITY 3 Continued 21. 2 × 3 14. The  1 0 matrix product BA is also  . which is the reciprocal of a. Write a matrix equation to model the system. which you can determine by using a graphing calculator.  3 0 1  A= −1 2 6   3 −1 4   D= 3 1  2  1 2 3  4 −2     C =  3 2 1 B= 5  1  −2 4 3     4 1  E=  2 3 24. Write a system of equations that can be used to determine how many pounds of each type of meat Guillermo bought.5 pounds of pork. The solution of ax = b is x = b .  1 1 2   m =  31.  6 −1 5   16. Guillermo bought ground beef and ground pork for a party. a you multiply both sides by the multiplicative inverse matrix of A. 1. 20. All rights reserved. 1 1   x  6  . Find AB if it is defined. you multiply both sides by the multiplicative inverse of a. Sample answer: To solve both types of equations. Are these matrices inverses of each other? Explain.64/lb.  4 1 − 3 1  1   2 2  2  23. All rights reserved. John. Find B − E. =  5x + 4 y = −3  y  −2  3 2 −7   x  −29       25. C © 2015 College Board. Guillermo bought 4. Unit 1 • Equations.    =  3. The beef costs $3.. 16. mugs. 0 4  −2 1   A= B= 1 8   0. 17. Guillermo bought 6 pounds of meat for a total of $19. The table below shows the number of items that each person sold and the amount of money collected from the sales.64 y = 19. 19. Then use the matrix equation to solve the system. 28. 54 SpringBoard® Mathematics Algebra 2.25 0  Mugs Gift Wrap Amount of Sales Dean 3 4 5 $87. Dean. and gift wrap for a school fundraiser.50. Functions © 2015 College Board. 27. Unit 1 • Equations. Find the inverse of each matrix. Let P equal the matrix product BA. The solution of AX = B is X = A−1B. What is the entry with the address c13? 15. Find AC if it is defined. you multiply both sides by a multiplicative inverse.5          27. −3x − 2 y = 1  x   1 24. Write a matrix equation that can be used to find the price for each item in the table.62. 1400 1200 1000 800 600 400 200 200 400 600 800 1000 1200 1400 x Gaming Systems from Plant 1 Unit 1 • Equations. x ≥ 0. Let x represent the number of gaming systems that will be made in Plant 1. For both plants combined.. The company will lose $50 for each console it sells and earn a profit of $15 for each game sold for the system. a.000. y Number of Gaming Systems per Week 1600 Gaming Systems from Plant 2 © 2015 College Board. 900). and Systems Embedded Assessment 1 Use after Activity 3 GAMING SYSTEMS A gaming manufacturing company is developing a new gaming system. To produce the new system. Inequalities.055-056_SB_AG2_SE_U01_EA1. Solve your equation. Inequalities. the company has allocated the following resources on a weekly basis: no more than 8500 hours of motherboard production. Include equations arising from linear functions. In addition to a game console. 4x + 8y ≤ 12.. Sample answer: (600. b. Embedded Assessment 1 Assessment Focus • Systems of equations • Systems of inequalities • Absolute value equations Answer Key 1. a. Functions 55 . and interpret solutions as viable or nonviable options in a modeling context. y ≥ 0 Plant 1 Plant 2 (hours per system) (hours per system) 9 1 9 3 4 8 b. d = 1. 9x + y ≤ 8500. Graph the constraints. The greatest distance a player can stand from the Jesture and be recognized is 3. 3. a.1 Create equations and inequalities in one variable and use them to solve problems. Identify an ordered pair that satisfies the constraints. HSA-CED. Shade the solution region that is common to all of the inequalities. It will make up for the loss from profits made from the sales of games for the system.2. b.A. c. t = 15g − 50 b. HSA-CED.A. 40 20 0 1 2 3 4 5 6 g 7 –40 –60 Games Purchased c. c.A. Solve the following problems about the gaming system. Functions 55 Common Core State Standards for Embedded Assessment 1 HSA-CED.3 Represent constraints by equations or inequalities. no more than 9000 hours of technical labor. $55 2. the company plans on using resources in two manufacturing plants.000 hours of general manufacturing.4 meters. c. Unit 1 • Equations. The table gives the hours needed for three tasks. where d represents the extreme distances in meters a player can stand from the Jesture and still be recognized b. 3. What is the total amount the company will earn from the average customer who buys a game console and seven games? 100 Total Earnings ($) 80 © 2015 College Board. All rights reserved. Explain what the ordered pair represents in the context of the situation. The company plans to sell the video game console at a loss in order to increase its sales. |d − 2.6 meters. and interpret the solutions. a. Motherboard production Technical labor General manufacturing 60 –20 2. and let y represent the number of gaming systems that will be made in Plant 2. graph equations on coordinate axes with labels and scales. All rights reserved. Graph the equation on a coordinate grid.2 Create equations in two or more variables to represent relationships between quantities. b. Equations.6.2 or equivalent. Show your work. the company will also produce an optional accessory called a Jesture that allows users to communicate with the game console by using gestures and voice commands. a. and no more than 12. and by systems of equations and/or inequalities. 1.4| = 1. Write inequalities that model the constraints in this situation. the total amount the company will earn from a customer who buys a console and g games.indd Page 55 05/12/13 9:58 AM user-s068a 123 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . The player’s distance from the Jesture can vary up to 1. 9x + 3y ≤ 9000. Inequalities. The company could meet the constraints by making 600 gaming systems per week at Plant 1 and 900 gaming systems per week at Plant 2. The Jesture accessory can recognize players when they are within a certain range. Resources Total Earnings per Customer t a.2 meters from the target distance of 2. The company predicts that the average customer will buy seven games for the video game console. and the least distance the player can stand from the Jesture and be recognized is 1. Write an absolute value equation that can be used to find the extreme distances that a player can stand from the Jesture and still be recognized. Write an equation that can be used to determine t. d = 3.2 meters. Check that students understand the terms used. Discuss multiple representations. 3b. 4b) Reasoning and Communication Proficient The solution demonstrates these characteristics: • Clear and accurate (Items 2b. turn to Embedded Assessment 2 and unpack it with them. and Systems Embedded Assessment 1 unnecessary steps but results in a correct answer real-world scenarios using linear equations. y is the number of fitness points earned per minute of aerobics. The Jesture will come with a fitness program. The program allows players to earn fitness points depending on the number of minutes they spend on each activity. 2. All rights reserved. Embedded Assessment 1 . and absolute value equations Partially accurate creation of graphs of equations and inequalities • Little difficulty in explaining • Partially correct interpreting solutions in the context of a real-world scenario explanations and interpretations of solutions in the context of a real-world scenario solving problems • of how to represent real-world scenarios using linear equations. 3a. and absolute value equations Clear and accurate understanding of creating graphs of equations and inequalities • Ease and accuracy in explaining interpreting solutions in the context of a real-world scenario SpringBoard® Mathematics Algebra 2 SpringBoard® Mathematics Algebra 2 • A functional understanding • Partial understanding of strategy that results in a correct answer (Items 1a.. 2a. 4. systems of equations and inequalities. 4b) Unpacking Embedded Assessment 2 56 Yoga (minutes) 30 15 10 Play Tester TEACHER to TEACHER Once students have completed this Embedded Assessment. and z is the number of fitness points earned per minute of jogging. and absolute value equations Mostly accurate creation of graphs of equations and inequalities some incorrect answers • how to represent real-world scenarios using linear equations. 1b. 4b) Emerging • An appropriate and efficient • A strategy that may include • A strategy that results in (Items 1c.055-056_SB_AG2_SE_U01_EA1. Cassie Clint Kian Aerobics (minutes) 10 20 25 Jogging (minutes) 20 10 20 Fitness Points 130 95 140 a. All rights reserved. Inequalities.. Use a graphic organizer to help students understand the concepts they will need to know to be successful on Embedded Assessment 2. Write a system of three equations that can be used to determine the number of points a player gets for 1 minute of each activity. and absolute value equations Inaccurate or incomplete creation of graphs of equations and inequalities • Incomplete or inaccurate explanations and interpretations of solutions in the context of a real-world scenario © 2015 College Board. Use this first Embedded Assessment as an opportunity to focus on your expectations for what student work should look like as the course progresses. b. systems of equations and inequalities. b. The table shows how many minutes three players spent on each activity and the total number of fitness points they earned. emphasizing that students should show their work on each part of a question when asked.indd Page 56 05/12/13 9:58 AM user-s068a 123 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . Solve your system. Use after Activity 3 GAMING SYSTEMS 4. systems of equations and inequalities. 3b. Mathematical Modeling / Representations Scoring Guide Exemplary understanding of solving systems of equations and inequalities. 2c. A player earns 1 point per minute of yoga. and interpret the solution. 2 points per minute of aerobics and 4 points per minute of jogging. where x is the number of fitness points earned per minute of yoga. a. and absolute value equations 56 Incomplete of solving systems of equations and inequalities and absolute value equations solving systems of equations and inequalities and absolute value equations • Little or no understanding of solving systems of equations and inequalities and absolute value equations • No clear strategy when • Fluency in representing • Little difficulty representing • Partial understanding of • Little or no understanding • • real-world scenarios using linear equations. 4a) (Items 2c. systems of equations and inequalities. 4). 2b. Mathematics Knowledge and Thinking TEACHER to TEACHER Problem Solving You may wish to read through the scoring guide with students and discuss the differences in the expectations at each level. or equivalent. 3b. Equations. (1. 30 x + 10 y + 20z = 130 15x + 20 y + 10z = 95   10 x + 25 y + 20z = 140 © 2015 College Board. • Write inequalities. there is only one y-value. My Notes Learning Targets: 4 Graph functions expressed symbolically and show key features of the graph. their descriptions will provide formative information about their understanding. Ask for clarification of their meaning. Work with your group on this item and on Items 2–4. Sample answer: It is a function because for each x-value. © 2015 College Board. y  6 4 2 5 –5 x –2 –4 –6 Activity 4 • Piecewise-Defined Functions 57 Common Core State Standards for Activity 4 HSF-BF. they transform various parent piecewise functions. Students may describe the graphs as “pieces” that are connected or “pieces” that are not connected. Quickwrite If students have previous experience with piecewise-defined functions. note nothing is below the x-axis] Have students discuss the methods they used to graph these functions. including step functions and absolute value functions. by hand in simple cases and using technology for more complicated cases. and with vertex at (0. and f(x + k) for specific values of k (both positive and negative). and make notes to help you remember and use those words in your own communications. Finally.3 Identify the effect on the graph of replacing f(x) by f(x) + k. opening upward. −6)] 3. Quickwrite. All rights reserved. it passes the vertical line test. the domain and range of functions using interval notation. Why is the graph a function? As you listen to your group’s discussions as you work through Items 1–4.057-072_SB_AG2_SE_U01_A04. and set notation. you may hear math terms or other words that you do not know. Marking the Text. Create Representations. f(kx). it then shows a decreasing linear function with a slope of −2 for x-values from 3 to infinity. Use your math notebook to record words that are frequently used. Think-Pair-Share. Sample answer: The graph shows an increasing linear function with a slope of 1 for x-values from negative infinity to 3. Make use of structure.B. y = x2 − 6 [graph of parabola. Describe the graph as completely as possible. They look at the absolute value and step functions.7 Guided In Activity 4. and graphically. slope of −3. Activity Standards Focus SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge. HSF-IF. Create Representations. students identify and graph various piecewise-defined functions. y-intercept of 5] 2. All rights reserved. Activity 4 • Piecewise-Defined Functions 57 . 1–2 Activating Prior Knowledge. 1). emphasize the use of technology to graph piecewise-defined functions as well as how changes in coefficients and constants affect the graphs of functions. © 2015 College Board. 3. y = −3x + 5 [graph of line. with y-axis as line of symmetry. Graph y = x2 − 3 for x ≤ 0 and y = 1 x + 1 for x > 0 on the same 4 coordinate grid. It is imperative that students recognize the graphs on each grid as functions. Interactive Word Wall.7b Graph piecewise-defined functions. TEACH Bell-Ringer Activity Ask students to graph the following and be ready to share a description of the features of each graph.. They explore functions made up of parts of linear functions. Students are asked to graph two equations on the same grid and recognize the combined graphs as a function. as this provides the access to defining piecewise-defined functions. y = x+ 1 [V-shaped with the point of the “V” at (0.C. y  Lesson 4-1 6 PLAN 4 Pacing: 1 class period Chunking the Lesson 2 5 –5 x #1–2 #3–4 #5 Check Your Understanding #9 #10 Check Your Understanding Lesson Practice –2 –4 –6 1. find the value of k given the graphs. Throughout this activity. HSF-IF.indd Page 57 25/02/15 7:50 AM s-059 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . DISCUSSION GROUP TIP 2.. kf(x). Discussion Groups The graphs of both y = x − 2 for x < 3 and y = −2x + 7 for x ≥ 3 are shown on the same coordinate grid below. ACTIVITY Piecewise-Defined Functions ACTIVITY 4 Absolutely Piece-ful Lesson 4-1 Introduction to Piecewise-Defined Functions • Graph piecewise-defined functions.C. 1. make sure that they are using the correct rule for the given domain values. Functions 5 –5 –2 x © 2015 College Board. take special care when discussing the values of the functions at x = 0. Press the y = key. Describe the graph in Item 3 as completely as possible. 3–4 Create Representations The graph shown in Item 3 is discontinuous at x = 0. followed by the MATH key to access the “Test. 2. type in (2x + 1)(−3 ≤ x) (x < 5). press the 2nd key. Functions MINI-LESSON: Recognizing and Evaluating Functions If students need additional help identifying functions. The graph is a function because for 4 each x-value. or evaluating functions. MATH TERMS A piecewise-defined function is a function that is defined using different rules for the different nonoverlapping intervals of its domain. when graphing. 58 SpringBoard® Mathematics Algebra 2. Note: In order to access the < symbol. ACTIVITY 4 Continued Lesson 4-1 Introduction to Piecewise-Defined Functions ACTIVITY 4 continued My Notes Sample answer: The graph has a break at x = 0. it is an increasing linear function with a slope of 1 . For Y1.057-072_SB_AG2_SE_U01_A04. if − 3 ≤ x < 5  if x ≥ 5 8. Complete the table of values. Inequalities. −2 if x < −1 h( x ) =  x 2 if − 1 ≤ x < 1  if x ≥ 1 1 58 if x < −1 if x ≥ −1 y  x g(x) −4 6 −3 4 −2 2 −1 2 0 3 –4 1 4 –6 2 5 6 4 2 SpringBoard® Mathematics Algebra 2. type in (8)(x ≥ 5). 5 Create Representations As students complete the table.” 3. equations. The pieces can be straight or curved. Unit 1 • Equations. To the left of x = 0.. Note: You key in the domain of Y2 separately. The rules may change as the domain changes. Additionally. Technology Tip 1. but not including x = 0.. If this occurs. Inequalities. type in (x2 − 2)(x < −3). All rights reserved. which is included. Developing Math Language Single Brace A piecewise-defined function may have more than two rules. Model with mathematics. can be graphed on a graphing calculator by following the steps below: 4. The functions in Items 1 and 3 are piecewise-defined functions . Each set of rules provides a different “piece” of the function. 4. It is important for students to note the endpoints of the intervals of the piecewise definitions. Press the GRAPH key. . −2 x − 2 g (x ) =   x + 3 For additional technology resources. Then graph the function. consider the function below. See the Teacher Resources at SpringBoard Digital for a student page for this mini-lesson. there is only one y-value.indd Page 58 05/12/13 1:48 PM user-s068a 123 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . Why is the graph a function? © 2015 College Board. In Item 4. For example. visit SpringBoard Digital. Piecewisedefined functions are written as follows (using the function from Item 3 as an example): Name of function Function Rules 5. Piecewise-defined functions exist because in the real world not every situation can be modeled using only a single function. students may stop the graph for the left-hand piece before x = −1 and not continue to show x = −1 with an open circle. it is a decreasing quadratic function with a minimum at y = −3. For Y2. ask them to consider the domain of the left part of the graph and then ask them which rule applies to points between −2 and −1. For Y3. To the right of x = 0. Domain Restriction  x 2 − 3 if x ≤ 0 f (x ) =  1  x + 1 if x > 0  4 5. and tables. A piecewise-defined function such as  x 2 − 2. a mini-lesson is available to provide practice. Unit 1 • Equations. MATH TIP A piecewise-defined function is so called because it is a function that follows a different set of rules as the domain changes. if x < −3  f (x ) = 2 x + 1. All rights reserved. recognizing functions from graphs. or elements. 9 Activating Prior Knowledge Defining the domain and range for the function will require students to look not only at the function rules and their restricted domains. 8. My Notes y  ACTIVITY 4 Continued x Set notation is a way of describing the numbers that are members. { x | x ∈ R }. Critique the reasoning of others. Interval notation is a way of writing an interval as a pair of numbers. range: y > −1. ∞. The set notation also uses the symbols ∈ (element) and  (set of all real numbers). Graph the first function rule for the values of x given after the first “if ” statement. Domain: −∞ < x < ∞. and the range is all real numbers greater than zero. Domain: −∞ < x < ∞. 10 Activating Prior Knowledge Be sure that students recognize that the function is not defined for x = 1 in Item 10b. Write the domain and range of g(x) in Item 5 by using: a. In Item 9. range: −∞ < y < ∞. (−∞. All rights reserved. Formative information can be gathered as a result of this exercise. (−1. 1) and (1. 2 < x ≤ 6}. and write its domain and range using inequalities. You can represent the domain and range of a function by using inequalities. y > −1} Activity 4 • Piecewise-Defined Functions 59 6. ∞). b. { y | y ∈ R . 8. each takes on a different look in expressing them. Activity 4 • Piecewise-Defined Functions 59 . Use these items to assess understanding and to start a class discussion of domain and range. range: y > 0 b. ∞). the variables x and y are used to represent the domain and range. { y | y ∈ R } b. Domain: (−∞. interval notation MATH TIP c. use a closed circle for the endpoint. are not included as endpoints. inequalities MATH TERMS The domain of a function is the set of input values for which the function is defined. Explain how to graph a piecewise-defined function. { x | x ∈ R . and negative infinity. (−∞. While all three can be written to represent the same domain and range. In interval notation. which represent the endpoints. −2 x + 2 if x < 1  x + 2 if x < 0 g (x ) =  f (x ) =  if x > 1  x − 2 2 x − 1 if x ≥ 0 y  6 6 4 4 2 2 5 –5 x 5 –5 –2 –2 –4 –4 –6 –6 Check Your Understanding Debrief students’ answers to these items to ensure that they understand concepts related to graphing a piecewise-defined function. 2 < x ≤ 6 is written in set notation as {x | x ∈ R . TEACHER to TEACHER Note that there are three different ways to express the domain and range of a function: (1) interval notation. Set notation is the only format that uses braces { } and the vertical bar  to represent the words such that. Use additional practice as needed. ∞) © 2015 College Board. Look back at Item 5.Lesson 4-1 Introduction to Piecewise-Defined Functions ACTIVITY 4 continued Check Your Understanding Answers 6. Yes. Substituting −1 for x in the expression x + 3 gives −1 + 3 = 2. infinity. and (3) set notation. Domain:{ x | x ∈ R }. whereas in interval notation. use a bracket if an endpoint is included. Domain: x ≠ 1. 2 < x ≤ 6 is written in interval notation as (2. −∞. and set notation.” 10. (2) inequalities. note the domain is all real numbers. Use a parenthesis if an endpoint is not included. (−∞. ∞). The function rule that applies when x = −1 is x + 3. g(−1) = 2. Repeat this process if there are more than 2 function rules. x ≠ 1}. 6]. range: (0. ∞). set notation. Likewise. If the restriction on x for a function rule includes ≤ or ≥. how do you know whether to use an open circle or a closed circle for the endpoints of the function’s graph? The domain of a piecewise-defined function consists of the union of all the domains of the individual “pieces” of the function. the range of a piecewise-defined function consists of the union of all the ranges of the individual “pieces” of the function. Esteban says that g(−1) = 2. 7. For example. All rights reserved. but also at the graph for the range values. use an open circle for the endpoint. Graph each function. y > 0} 10. range: { y | y ∈ R. interval notation. The range of a function is the set of all possible output values for the function. Is Esteban correct? Explain. of a set. a. Show your work. 9. which is read “the set of all numbers x such that x is an element of the real numbers and 2 < x ≤ 6. © 2015 College Board. If the restriction on x for a function rule includes < or >. ∞). the ±∞ (infinity symbol) and parentheses are used. Then graph the second function rule for the values of x given after the second “if ” statement. If a piecewise-defined function has a break. You can also use interval notation and set notation to represent the domain and range. For example. 7. a. In the inequalities format and the set notation format. g(−2) = −4. –4 Lesson 4-1 Introduction to Piecewise-Defined Functions ACTIVITY 4 Check Your Understanding . (−∞. Unit 1 • Equations. How could you represent this domain using set notation? 12. ∞).04 x 17. x > 0} 12. 16. a. so x = 613. (−∞. b. f (x ) =  40 26 + 0. Debrief students’ answers to these items to ensure that they understand different notations for the domain and range of piecewise functions. range: y ≥ 0. LESSON 4-1 PRACTICE 14. which is not included. Explain how to use interval and set notation to represent the range y ≥ 3. g(0) = 2. {x | x ∈ }. $35. 60 ® 5]. −4 x if x < −2  g (x ) = 3x + 2 if − 2 ≤ x < 4   x + 4 if x ≥ 4 17. All rights reserved. [−5. This value of x satisfies the domain restriction of the second rule of the piecewise function. domain: −∞ < x < ∞. ASSESS Students’ answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the lesson concepts and their ability to apply their learning. An electric utility charges residential customers a $6 monthly fee plus $0. ≤ y ≤ 5. to ∞.  x 2 if x ≤ 0 a. The interval notation for the range is [3. Functions y {y | −5 ≤ y ≤ 5} 17. SpringBoard® Mathematics Algebra 2.indd Page 60 05/12/13 1:48 PM user-s068a 123 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ .04.. What can you conclude about the graph of a piecewise-defined function whose domain is {x | x ∈ R. 3]. so I evaluated the second rule for x = 613. Then write its domain and range using inequalities. g(4) = 8 48 if 0 ≤ x ≤ 500 6 + 0. range: y ≤ 3. Sample explanation: The customer uses 613 kWh. {y | y ≥ 0} y  4 2 2 –2 x 4 continued My Notes Check Your Understanding WRITING MATH You can use these symbols when writing a domain or range in set notation. Write a piecewise function f(x) that can be used to determine a customer’s monthly bill for using x kWh of electricity. 13. [0. b. See the Activity Practice for additional problems for this lesson. Inequalities. a. f (x ) =  1  x if x > 0  2 if x < −1 3x b. Unit 1 • Equations. The domain of a function is all positive integers. The range includes the interval from 3.08/kWh for usage over 500 kWh. c. ∞).08(x − 500) if x > 500 Monthly Bill ($) –4 32 24 16 8 100 300 500 700 Electricity Used (kWh) x c. Sample answer: The graph has a break or hole at x = 2. which is included. and x = 4. Write the range of the function using an inequality.04 per kilowatt hour (kWh) for the first 500 kWh and $0. Inequalities. domain: −∞ < x < ∞. Model with mathematics. In set notation. 60−5SpringBoard Mathematics Algebra 2.. x ≠ 2}? LESSON 4-1 PRACTICE 14. (−∞. Functions © 2015 College Board. Graph each piecewise-defined function. y ≥ 3} 13. ACTIVITY 4 Continued Answers 11. interval notation. Monthly Electric Bill 15. encourage them to evaluate the function at x = −1 to determine which endpoint is closed and which is open. You may assign the problems here or use them as a culmination for the activity. ∞). Evaluate the piecewise function for x = −2.057-072_SB_AG2_SE_U01_A04. interval notation. {x | x ∈ }. © 2015 College Board. and set notation. the range is {y | y ∈ . How much should the utility charge the customer? Explain how you determined your answer. The range of a function is all real numbers greater than or equal to −5 and less than or equal to 5. A customer uses 613 kWh of electricity in one month. All rights reserved. If students have difficulty with the endpoints in Item 14b. 56 16. Graph the piecewise function. f (x ) =  −x + 2 if x ≥ −1 15. b. a. and set notation. x = 0. {x | x ∈ . {y | y ≤ 3} y  4 2 –4 2 –2 4 x –2 –6 ADAPT Check students’ answers to the Lesson Practice to ensure that they understand how to graph and evaluate piecewisedefined functions and how to identify the domain and range. | such that ∈ is an element of R the real numbers Z the integers N the natural numbers 11. ∞). MATH TERMS TEACH A piecewise-defined function with a constant value throughout each interval of its domain is called a step function. a step function is easy to recognize because it looks like a series of horizontal steps. Because the pieces of a step function remain constant. it would not be a function. 1. This is not the case with a step function because if there were any vertical pieces. has a constant y-value of 1. The interval of the graph to the right of and including x = 2 has a constant value of 3. −2 if x < −3  if − 3 ≤ x < 2.indd Page 61 27/01/14 8:23 PM ehi-6 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . for each value. Why do you think the type of function graphed in Item 1 is called a step function? While it is a specific type of piecewisedefined function. The interval of the graph to the right of x = −3 and including x = −3. Describe the graph in Item 1 as completely as possible. Each break in the graph is a “jump” up or down to the next step. x = −3 [f(x) = −5] 3. Pacing: 1 class period Chunking the Lesson #1–3 #4–5 #6–7 #8–9 Check Your Understanding Lesson Practice SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge. 2. x = −4 [f(x) = 4] 2. Create Representations. Look for a Pattern. • Describe the attributes of these functions. Think-PairShare Ask students to reflect back on the word constant from earlier algebra courses and define in their own words what this means and how it applies to the step function definition. The interval of the graph to the left of (but not including) © 2015 College Board. Think-Pair-Share A step function is a piecewise-defined function whose value remains constant throughout each interval of its domain. ACTIVITY 4 Continued Lesson 4-2 Step Functions and Absolute Value Functions ACTIVITY 4 continued PLAN My Notes Learning Targets: • Graph step functions and absolute value functions. Sample answer: The graph looks like a series of steps.057-072_SB_AG2_SE_U01_A04. each “step” is what type of line segment? When students are finished discussing. Interactive Word Wall. Activity 4 • Piecewise-Defined Functions 61 Activity 4 • Piecewise-Defined Functions 61 .. Discussion Groups. Reason abstractly. if − 3 ≤ x < 2  if x ≥ 2  x + 3. Point out that a real staircase also has vertical pieces connecting the horizontal landings. Developing Math Language 3. All rights reserved. © 2015 College Board. ask a representative from one group or pair to share their responses with the class and ask other students for feedback. x = 2 [f(x) = 5] 5. Sample answer: The domain of the graph is the set of all real numbers. x = 0 [f(x) = 1] 4. 1. if x < −3  f (x ) = 2 x + 1. Quickwrite. { x | x ∈ R }. and to the left of (but not including) x = 2. Graph the step function f (x ) = 1  if x ≥ 2 3 y  6 4 –4 2 –2 4 6 x –2 –4 –6 1–3 Activating Prior Knowledge.. x = 5 [f(x) = 8] 2 –6 Lesson 4-2 x = −3 has a constant y-value of −2. All rights reserved. Bell-Ringer Activity Have students evaluate the piecewise function −x . . f(2. Responses will vary. highlight Dot. 4. Then have the students follow the steps given to them to graph this function. visit SpringBoard Digital.1_ = −4 because the greatest integer less than or equal to −3. Finally. Complete the table and graph the piecewise-defined function.. Ask them to think about the places.” in order to keep the calculator from connecting the breaks between the steps and making it appear that this is not really a function at all. Now take a look at a different type of piecewise-defined function.indd Page 62 05/12/13 1:48 PM user-s068a 123 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . . that will be included (closed circle plot) and not included (an open circle plot). Be sure to point out the Technology Tip in the margin for changing mode from “Connected” to “Dot. Graphing a step function in dot mode will prevent the calculator from connecting breaks in the graph with line segments. Unit 1 • Equations. Then. included but the right endpoint not included. Inequalities. 2. Graph the greatest integer function on a graphing calculator. Unit 1 • Equations. ask for some of their examples. For example. −x if x < 0 f (x ) =  if x ≥ 0  x x f(x) −3 3 −2 2 −1 1 0 0 1 1 2 2 3 3 SpringBoard® Mathematics Algebra 2.7) = _2.7_ = 2 because the greatest integer less than or equal to 2. Check students’ work. Below are two examples: 1. So. go to the Mode window. but not exceeding. Functions SpringBoard® Mathematics Algebra 2. All rights reserved. The step from x = 0 (included) to x = 1 (not included) has a y-value of 0. which yields a value f(x) that is the greatest integer less than or equal to the value of x. The cost to ship a parcel has a flat rate of $5. the cost per ounce increases to $0.1 is −4. Differentiating Instruction Extend students’ learning by asking students to journal or write some situation in the real world that could represent a step function. and so on. To do so.1) = _−3. Before they use their calculators.7 is 2. select 5: int(. After giving them some time to think and write. To locate int on the calculator. Inequalities. or parts of the steps. press MATH to reach the Math menu.20 per ounce up to. 62 62 Lesson 4-2 Step Functions and Absolute Value Functions ACTIVITY 4 continued © 2015 College Board. and press ENTER . Describe the graph of the greatest integer function as completely as possible. Predict and Confirm. All rights reserved. the first 12 ounces. The electrician charges a fee of $75 per hour (or any fractional part of one hour) for labor. Sample answer: Each step is a horizontal segment with the left endpoint For additional technology resources. when the weight exceeds 12 ounces.30 per ounce. the fee for 0 ≤ h ≤ 1 = $75. 6. 4–5 Chunking the Activity. the fee for 1 < h ≤ 2 = $150. and f(−3. Have them compare their predictions to the results on the calculator. Make sense of problems. ACTIVITY 4 Continued My Notes One step function is the greatest integer function. Each step is 1 unit long and 1 unit above the step to its left. TECHNOLOGY TIP Before graphing a step function on a graphing calculator. written f(x) = _x_. Then use the right arrow key to access the Number submenu. Discussion Groups Have students work with a partner. plus an additional cost of $0. you will need to enter the function as y = int(x).057-072_SB_AG2_SE_U01_A04. ask students to predict what the function of y = int(x) will look like. Functions y  4 2 5 –5 –2 –4 x © 2015 College Board. 5. The labor cost of hiring an electrician to come to one’s home to do some work. Activity 4 • Piecewise-Defined Functions 8–9 Marking the Text. if x < 0 of f (x ) =  . range: { y | y ∈ R. Could you have determined the values of the function in Item 8 another way? Explain. Quickwrite.057-072_SB_AG2_SE_U01_A04. Then use the definition of the absolute value function to make the necessary transformation: x = −x = −1 for x < 0 and x x x = x = 1 for x > 0.indd Page 63 05/12/13 1:48 PM user-s068a 123 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . 8. All rights reserved. © 2015 College Board. The notation for the function is f(x) = |x|. The sharp change in the graph at x = 0 is the vertex. f(−14) = 14 b. ACTIVITY 4 Continued Lesson 4-2 Step Functions and Absolute Value Functions ACTIVITY 4 continued 7. MATH TIP The absolute value function f(x) = |x| is defined by f (x) = {−xx if x < 0 if x ≥ 0 ) d. What are the x-intercept(s) and y-intercept of the function? x-intercept: 0. y-intercept: 0 d. 63 Activity 4 • Piecewise-Defined Functions 63 . Differentiating Instruction Students may encounter the function x f (x ) = in this or future math x classes. if x > 0  f (x ) = 1 c. what is it? Yes. Think-Pair-Share In graphing this piecewise-defined function. determine the distance from zero. Look back at the graph of f(x) shown in Item 6. © 2015 College Board. Reason quantitatively. f 2 − 5 = −2 + 5 CONNECT TO AP 9. Note that the x x function is undefined at x = 0. a. f(8) = 8 c. The vertex of an absolute value function is an example of a cusp in a graph. f(0) = 0 ( 6–7 Create Representations. but understanding this concept in terms of piecewise-defined functions is likely to be new. The function f(x) in Item 6 is known as the absolute value function. students are introduced to the absolute value function. The graph is symmetrical about the y-axis. a. Describe the symmetry of the graph of the function. but it is imperative that they also understand absolute value as a piecewise-defined function for future studies in mathematics. Use this opportunity to create the piecewise representation  f (x ) = −1. the function has a minimum of 0. Item 9 addresses that issue. Use the piecewise definition of the absolute value function to evaluate each expression. A graph has a cusp at a point where there is an abrupt change in direction. Interactive Word Wall Evaluating absolute values should not be a new experience for students. Does the function have a minimum or maximum value? If so. y ≥ 0} b. Sample answer: Yes.. All rights reserved. What are the domain and range of the function? My Notes domain: { x | x ∈ R }. with an overall “V” shape.. It is likely that students will have an informal understanding of absolute value. A step function known as the ceiling function. Unit 1 • Equations. The rules that define a step function are all constant functions. h(x ) =  if x ≥ −2  x + 2 SpringBoard® Mathematics Algebra 2. Sample answer: Two equations must be solved. watch for students who use closed circles for all of the endpoints.indd Page 64 05/12/13 1:48 PM user-s068a 123 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ .7). Explain why the absolute value function f (x) = |x| is a piecewise-defined function. 17. f (x ) =  45  60 if 6 ≤ x < 13 if 13 ≤ x < 19 if x ≥ 19 y  6 4 y 15. yields the value g(x) that is the least integer greater than or equal to x. 18. When graphing step functions.057-072_SB_AG2_SE_U01_A04. ASSESS Make sense of problems and persevere in solving them. f(x) = x. A day ticket for students at least 13 years old and less than 19 years old costs $45. they must pass the vertical line test – no vertical line can intersect the graph in more than one point. the graph will not pass this test. 12. Inequalities. Cost of Day Ticket ($) Check students’ answers to the Lesson Practice to ensure that they understand concepts related to step functions and absolute-value functions. LESSON 4-2 PRACTICE 16. Write the equation of a step function f(x) that can be used to determine the cost in dollars of a day ticket for the ski lift for a person who is x years old. What are the domain and range of the function? y  © 2015 College Board. For the interval x < 0. You may assign the problems here or use them as a culmination for the activity. The absolute value function can be defined using different rules for 2 nonoverlapping intervals of its domain. 14. Write the equation for the function using piecewise notation. How is a step function different from other types of piecewise-defined functions? 12. 15. a. (−2. A day ticket for adults at least 19 years old costs $60..4) = 3.13). Construct viable arguments. 0) if x < −2 −x − 2 19. g(−8.13) = 1. Remind these students that in order for these graphs to represent functions. Graph this step function. 2 60 50 –6 40 30 20 10 4 8 12 16 20 24 Age x –4 –2 2 17. Use this information for Items 14 and 15. g(0. Demonstrate that if all of the endpoints are closed. g(2.. Find g(2. range: {y | y ∈ . g(0. Functions 16. continued My Notes Check Your Understanding Answers 10. Use the absolute value function h(x) = |x + 2| for Items 16−19. 3 2 1 –4 –3 –2 1 –1 2 3 4 x © 2015 College Board. Inequalities. Unit 1 • Equations. b. Functions 4 6 x . A day ticket for a ski lift costs $25 for children at least 6 years old and less than 13 years old.7) = −8 ADAPT 64 64 SpringBoard® Mathematics Algebra 2. See the Activity Practice for additional problems for this lesson. a. 10. All rights reserved. y ≥ 0} 18. All rights reserved. –1 –2 –3 –4 b. 13. representing the two intervals of the domain. For the interval x ≥ 0. Students’ answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the lesson concepts and their ability to apply their learning. 11. f(x) = −x. and g(−8. Graph the absolute value function.4). domain: {x | x ∈  }. 11. What are the coordinates of the vertex of the function’s graph? 4 19. 25  14. How does the definition of absolute value as a piecewise-defined function relate to the method of solving absolute value equations? LESSON 4-2 PRACTICE 13. Graph the step function you wrote in Item 14. ACTIVITY 4 Continued Lesson 4-2 Step Functions and Absolute Value Functions ACTIVITY 4 Check Your Understanding Debrief students’ answers to these items to ensure that they understand concepts related to step functions and absolute value functions. written g(x) = x . This lesson refers to the parent absolute value function as f(x) = |x|. Think-Pair-Share. Transformations may be performed on a parent function to produce a new function. k ⋅ f(x). a mini-lesson is available to provide practice. x vertical translation down 2 units c. f(x) = |x + 1| for x = 9 [f(x) = 10] 4. h(x) = |x| – 2 d. or y = x Quadratic: f(x) = x2. Model with mathematics. which stretch a graph away from the x-axis or shrink a graph toward the x-axis • horizontal stretches or horizontal shrinks.indd Page 65 15/04/14 11:37 AM user-g-w-728 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . or y = x Each of these parent functions can be altered to change its basic position. or y = x2 Cubic: f(x) = x3. Debriefing. 65 A parent function is the most basic function of a particular type. Transformations that students are expected to recognize from earlier math courses include the following: vertical translations. size. f(x) = |x| for x = 9 [f(x) = 9] 2. All rights reserved. and shape. g(x) = |x| + 1 b. k(x) = 3|x| f(x) = |x| MATH TIP –5 4 4 2 2 5 –5 x –5 –2 –2 –4 –4 vertical stretch by a factor of 3 f(x) = |x| 5 q(x) = –|x| Lesson 4-3 x Transformations include: • vertical translations. • Find the value of k. and reflections over the x-axis. which shift a graph left or right • reflections. 1 Activating Prior Knowledge. students are asked to identify transformations of the parent absolute value function. known as a transformation. graph the function and identify the transformation of f(x) = |x|. or y = 1 x Radical: f (x )= x . h(x) = |x| − 2 y  f(x) = |x| g(x) = |x| + 1 y  f(x) = |x| 4 4 2 2 5 –5 x 5 –2 –2 –4 –4 vertical translation up 1 unit © 2015 College Board. ACTIVITY 4 Continued Lesson 4-3 Transforming the Absolute Value Parent Function ACTIVITY 4 continued PLAN My Notes Learning Targets: the effect on the graph of replacing f(x) by f(x) + k. which shift a graph up or down • horizontal translations. f(x) = |x| for x = −9 [f(x) = 9] 3. a new function. a.057-072_SB_AG2_SE_U01_A04. which produce a mirror image of a graph over a line • vertical stretches or vertical shrinks. f(x) = |x + 1| for x = −9 [f(x) = 8] 1. All rights reserved.. Recall that a parent function is the most basic function of a particular type. Create Representation. Pacing: 1 class period Chunking the Lesson #1 #2–3 #4–5 #6 #7–8 #9–10 #11 Example A Check Your Understanding Lesson Practice SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge. which stretch a graph away from the y-axis or shrink a graph toward the y-axis Students should discuss the impact of the “+ 1” in Items 3 and 4. The cause of a transformation involves some type of arithmetic operation(s) to the parent function. Here are a few more examples of common parent functions: Linear: f(x) = x. Activity 4 • Piecewise-Defined Functions 65 . given these graphs. and f(x + k). See the Teacher Resources at SpringBoard Digital for a student page for this mini-lesson. q(x) = −| x | y  y k(x) = 3|x| © 2015 College Board. is produced. Look for a Pattern. When this occurs. For each function below. Developing Math Language reflection over the x-axis Activity 4 • Piecewise-Defined Functions MINI-LESSON: Vertical Translations and Vertical Stretch/Shrink If students need additional help graphing vertical translations of parent functions or vertical stretches or shrinks of a parent function. Have students evaluate each function for the given value: 1. or y = x3 Inverse: f (x )= 1x .. TEACH Bell-Ringer Activity Students should be familiar with how to evaluate absolute value functions. Identify a Subtask The absolute value function f(x) = |x| is the parent absolute value function. • Identify f(kx). vertical stretches/shrinks. Create Representations With text-based reminders about parent functions and transformations. then y = f(x) = f(0). Unit 1 • Equations. In the absolute value function f(x) = |x − k| with k > 0. Functions MINI-LESSON: Reflections over the x-axis If students need additional help graphing reflections of parent functions over the x-axis. When x = 0. the y-value for the graph of y = f(x − a) at b + a is the same y-value that y = f(x) has at x = b. When x = a. All rights reserved. See the Teacher Resources at SpringBoard Digital for a student page for this mini-lesson. point out that the shift is a units right if a > 0 and left if a < 0. SpringBoard® Mathematics Algebra 2. In the absolute value function f(x) = |x + k| with k > 0. Point out that those functions show a horizontal shift in the x-values from 0 to a. Again. then y = f(x − a) = f(a − a) = f(0). a mini-lesson is available to provide practice. Graph the parent function f(x) = |x|. Sample prediction: horizontal translation 2 units to the left y  4 TEACHER to TEACHER 2 h(x) = |x + 2| Here is one way of explaining horizontal translations given by x − a: Show students that the y-value for the graph of y = f(x − a) at a is the same y-value that y = f(x) has at 0. x − a moving to the right and x + a moving to the left. All rights reserved. vertical stretch of f(x) by a factor of 2. 3. The y-values are equal. Reason abstractly and quantitatively.. Functions © 2015 College Board. Unit 1 • Equations. . Differentiating Instruction /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . h(x) is a 4–5 Look for a Pattern The counterintuitive nature of this translation. x –2 –4 d. A horizontal translation occurs when the independent variable. What transformations do your graphs show? g(x) is a horizontal translation of f(x) 3 units to the right. 66 2 f(x) = |x| Predictions may vary. describe how the graph of the function changes. The graph moves k units to the right. then giving them additional opportunities to graph transformations may be appropriate. is replaced with x + k or with x − k. Items 4 and 5 provide the opportunity to reason abstractly to generalize the effects of x ± a. compared to the parent function. 4. –5 f(x) = |x| 5 x –2 –4 The functions in Items 2 and 3 are examples of horizontal translations. The function moves k units to the left. Predict and Confirm Horizontal translations are introduced in Item 2 and then expanded in Items 3–5. a. Use the results from Item 2 to predict the transformation of h(x) = |x + 2|. compared to the parent function. describe how the graph of the function changes. x. If a is positive. The practice exercises on this page and the next may be helpful in reinforcing understanding. Predict the transformation for g(x) = |x − 3| and h(x) = |−2x|. 5.057-072_SB_AG2_SE_U01_A04. this shift is to the left a units. In general. Use the coordinate grid at the right. this shift is to the right a units. 66 SpringBoard® Mathematics Algebra 2.. Then graph the function to confirm or revise your prediction. and if a is negative.indd Page 66 16/01/15 6:02 PM ehi-6 ACTIVITY 4 Continued Lesson 4-3 Transforming the Absolute Value Parent Function ACTIVITY 4 My Notes TECHNOLOGY TIP Many graphing calculators use a function called “abs” to represent absolute value. 4 g(x) = |x – 3| 5 –5 c. 2. continued © 2015 College Board. or if they have limited experience with transforming parent functions. y  h(x) = |–2x| b. 2–3 Create Representations. may be a surprise to students. Inequalities. If students experience difficulties in describing transformations. Inequalities. Graph the function g(x) = |x − 3| and h(x) = |−2x|. Sample prediction: a horizontal stretch by a factor of 2 y  6 f(x) = |x| 4 h(x) = 2 5 –5 Technology Tip 1 x 2 Note: Use this basic process with the transformations of the parent absolute value function. f(x) = |x − 4| 2 x 5 f(x) = |x – 4| x –5 f(x) = |x + 5| –2 –4 –4 y  7. x –2 Activity 4 • Piecewise-Defined Functions 67 Activity 4 • Piecewise-Defined Functions 67 . Critique Reasoning. y) on the graph of the transformed function. Similarly. have students work in small groups or with a partner to try to predict how the value of k in f (x ) = kx affects the graph of f(x) = |x|. and ask students to write the equations of the functions you graphed. what if g (x ) = 3x and h(x ) = 1 x ? Have students collaborate 3 and share their findings. Describe the graph of g(x) as a horizontal stretch or horizontal shrink of the graph of the parent function. For additional technology resources. 2. Item 7 to predict how the graph of h(x) = 1 x is transformed from the © 2015 College Board. a. 4. follow these basic steps: My Notes 6. Graph the parent function f(x) = |x| and the function g(x) = |2x|.n key.. visit SpringBoard Digital. “abs(” for absolute value. by pressing ENTER . © 2015 College Board. Then graph h(x) to confirm or revise your prediction. Press the CLEAR key to clear any previous functions. Press the MATH key. ky) on the graph of the transformed function. a. y) on the graph of the original function to the point (x. Think-Pair-Share After presenting Items 7 and 8. In other words. Create Representations. y) on the graph of the original function to the point (kx. g(x) = |2x| 4 2 8.. Use the results from A horizontal stretch or shrink by a factor of k maps a point (x. Use the coordinate grid at the right.T.057-072_SB_AG2_SE_U01_A04. Chunking the Activity Before students graph the functions given in Item 6.indd Page 67 27/01/14 8:31 PM ehi-6 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . 7–8 Discussion Groups. 6 Predict and Confirm. Press the y = key in the upper left corner. Press the right arrow one time to highlight the NUM command. 5. Press the GRAPH key. b. f(x) = |x + 5| y  4 2 –2 1. 3. Predict and Confirm. Have them confirm their predictions by graphing. ACTIVITY 4 Continued Lesson 4-3 Transforming the Absolute Value Parent Function ACTIVITY 4 continued b. Press the ) key to close parentheses. Press the X. 6. have them write down their predictions for the transformation that will take place with the parent absolute value graph. All rights reserved. Select option 1. y  4 –5 To graph the parent absolute value function on a TI graphing calculator. 8. a vertical stretch or shrink by a factor of k maps a point (x. Before moving on to the next items. 6 MATH TIP f(x) = |x| 5 –5 x –2 a horizontal shrink by a 1 factor of 2 7. Graph each function. 2 graph of the parent function. sketch a couple of absolute value horizontal transformations. All rights reserved.Θ. Express regularity in repeated reasoning. Item 10 is written in such a way that you may want to emphasize some points. based upon the previous items.. Lesson 4-3 Transforming the Absolute Value Parent Function ACTIVITY 4 continued My Notes 9. describe how the graph of the function changes compared to the graph of the parent function. 11. y  a. f(x) = |x| 6 4 2 5 –5 x –2 f(x) = |x| 4 g(x) 2 5 –5 x –2 –4 vertical translation down 3 units. g(x) = 3|x| or g(x) = |3x| . g(x) 11 Summarizing Point out that the phrases “vertical stretch” and “horizontal shrink” can both be used to describe what is taking place in Item 11a. Marking the Text Students probably will not have too much difficulty expressing regularity in repeated reasoning and coming up with a generalization for Item 9. Each graph shows a transformation g(x) of the parent function f(x) = |x|. However. ACTIVITY 4 Continued 9–10 Close Reading. The functions in Items 9 and 10 are written the same. k k when k is a fraction. Functions SpringBoard® Mathematics Algebra 2. the difference between these descriptions is the values of their factors. y  b. In the absolute value function f(x) = |kx| with 0 < k < 1. but students should look closely at the values of k. What if −1 < k < 0? The graph of f(x) is a horizontal stretch of the graph of the parent 1 function by a factor of k when 0 < k < 1 or when −1 < k < 0. However.indd Page 68 14/01/15 10:21 AM ehi-6 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . Unit 1 • Equations.. Inequalities. k is really a fractional value. is the inverse of the fractional value.057-072_SB_AG2_SE_U01_A04. In the absolute value function f(x) = |kx| with k > 1. 1 vertical stretch by a factor of 3 or horizontal shrink by a factor of 3 . The reason why the factors are both 1 is because 1 . All rights reserved. Because of the way this is written. 10. Functions © 2015 College Board. What if k < −1? The graph of f(x) is a horizontal shrink of the graph of the parent 1 function by a factor of k when k > 1 or when k < −1. g(x) = |x| − 3 68 68 SpringBoard® Mathematics Algebra 2. Inequalities. Describe the transformation and write the equation of g(x). Unit 1 • Equations. © 2015 College Board. the only difference between the answers for Items 9 and 10 is the words shrink and stretch. All rights reserved. describe how the graph of the function changes compared to the graph of the parent function. In Item 10. x –2 Try These A For each absolute value function. Step 2: Apply the horizontal translation first. Furthermore. Its equation is h(x) = |x + 3|. Apply the horizontal translation. Now stretch each point on the graph of h(x) vertically by a factor of 2. explain that an absolute value represents a distance. reflection in the x-axis and/or vertical shrink or stretch 4. ACTIVITY 4 Continued Lesson 4-3 Transforming the Absolute Value Parent Function ACTIVITY 4 continued My Notes Example A Describe the transformations of g(x) = 2|x + 3| from the parent absolute value function and use them to graph g(x). All rights reserved. reflection in the y-axis and/or horizontal shrink or stretch 3. vertically stretch by a factor of 4. Then shift each point on the graph of f(x) by 3 units to the left. vertical translation Answers a. or measure. y  4 2 –2 6 –4 f(x) = |x| b. apply the transformations of f(x) = |x| in this order: 1. Activity 4 • Piecewise-Defined Functions 69 x . Step 3: x 5 –5 h(x) = |x + 3| 4 –8 h(x) = –|x – 1| + 2 4 2 Apply the vertical stretch. followed by a vertical stretch by a factor of 2. reflect over the x-axis. of a number from the origin of a number line.. this is why they remain above the x-axis unless written with a negative sign preceding the absolute value bars. Solution: The new function is g(x) = 2|x + 3|. a measurement cannot be a negative value. k(x) = 4|x + 1| − 3 translate 1 unit to the left. y  g(x) = 2|x + 3| TECHNOLOGY TIP You can check that you have graphed g(x) correctly by graphing it on a graphing calculator. g(x) is a horizontal translation of f(x) = |x| by 3 units to the left. Step 1: Describe the transformations. Graph f(x) = |x|. then translate 3 units down Activity 4 • Piecewise-Defined Functions 69 Keeping that in mind. horizontal translation 2. 5 –5 –2 k(x) = 4|x + 1| – 3 –4 h(x) = |x + 3| 6 4 Universal Access 2 –8 –6 –4 2 –2 4 For students having difficulty with the concept of absolute value. 2 –6 –4 2 –2 y  x 4 –2 © 2015 College Board. To do so. All rights reserved.057-072_SB_AG2_SE_U01_A04. © 2015 College Board. subtract 3 from the x-coordinates and keep the y-coordinates the same. and then apply the vertical stretch.indd Page 69 07/12/13 4:06 AM s-059 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . Name the new function h(x). and translate up 2 units b. a.. when applying this concept to absolute value functions and their graphs. To do so. keep the x-coordinates the same and multiply the y-coordinates by 2. y  Example A Activating Prior Knowledge This concept of unraveling multiple transformations to a parent absolute value function is similar to following orders of operations with arithmetic operations. describe the transformations represented in the rule and use them to graph the function. h(x) = −|x − 1| + 2 translate to the right 1 unit. MATH TIP Try These A To graph an absolute value function of the form g(x) = a|b(x − c)| + d. 057-072_SB_AG2_SE_U01_A04. The vertex of the graph of the absolute value parent function is (0. y) on the original graph to point (x. so the domain is all real numbers. Sample explanation: The graph of f(x) is a translation of the graph of the absolute value parent function by 2 units left and 5 units down. a reflection over the x-axis and a vertical stretch by a factor of 5 17. Functions . a. y) on the original graph to point (4x.. so the range is all real numbers ≥ −1. g(x) = −3|x + 2| + 4 19. f(x) = 5|x + 9| − 23 1 b.• Equations. a. The graph of f(x) opens upward. 12. Attend to precision. 18. g(x) = |x − 6| 16. stretch horizontally by a factor of 4. Functions y  4 4 4 2 2 2 5 –5 x y  b. (−2. y) on the transformed graph. f (x ) = − 4 (x + 12) c. Write the equation for each transformation of f(x) = |x| described below. y  c. g(x) and f(x) are the same function. a. y ≥ −1}.indd Page 70 05/12/13 1:49 PM user-s068a 123 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . −5). 14. 4 2 2 4 6 x LESSON 4-3 PRACTICE 13. You may assign the problems here or use them as a culmination for the activity. stretch vertically by a factor of 5. Sample explanation: The function is defined for all real values of x. Sample answer: Both transformations stretch points on the original graph away from an axis. 2 ASSESS 5 © 2015 College Board. All rights reserved. ACTIVITY 4 Continued Check Your Understanding Debrief students’ answers to these items to ensure that they understand concepts related to transformations of functions. Lesson 4-3 Transforming the Absolute Value Parent Function ACTIVITY 4 continued My Notes Check Your Understanding Answers 12. −5). Domain: {x | x ∈  }. All rights reserved. 4 2 5 –5 –2 x x –5 x –5 –2 –2 –4 –4 19. Unit 1 • Equations. 16. Explain how you determined your answer. Inequalities. range: {y | y ∈ . Without graphing the function. –6 –4 –2 x –2 Students’ answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the lesson concepts and their ability to apply their learning. −1 is the minimum value of f(x). f(x) = 3|x − 2| − 4 or f(x) = |3(x − 2)| − 4 SpringBoard® Mathematics Algebra 2. –4 © 2015 College Board. 17. The graph of g(x) is the graph of f(x) = |x| translated 6 units to the right. allow them to graph by generating ordered pairs. SpringBoard® Mathematics Algebra 2. y  18. so the vertex of the graph of f(x) must be (−2. Write the equation of g(x). Describe the graph of h(x) = −5|x| as one or more transformations of the graph of f(x) = |x|. y  70a. What are the domain and range of f(x) = |x + 4| − 1? Explain. and reflect over the x-axis. ADAPT Check students’ answers to the Lesson Practice to ensure that they understand transformations of the absolute value parent function. 15. g(x) = |2x| − 3 c. 0). g(x) = −|x + 4| + 3 d. Graph each transformation of f(x) = |x|. Translate left 12 units. b. If students have trouble graphing a function by using the values in the equation. Compare and contrast a vertical stretch by a factor of 4 with a horizontal stretch by a factor of 4. y  6 14. Inequalities. 70 d. determine the coordinates of the vertex of f(x) = |x + 2| − 5. What is the relationship between g(x) and f(x) = |x|? Why does this relationship make sense? 13. Unit 1c. 4y) on the transformed graph. Graph the function g(x) = |−x|. A vertical stretch maps a point (x. −1). and translate down 23 units. and its vertex is at (−4. Translate left 9 units. g(x) = |x − 4| + 2 b. See the Activity Practice for additional problems for this lesson. LESSON 4-3 PRACTICE 15. A horizontal stretch maps a point (x.. This relationship makes sense because |x| = |−x|. and set notation. Write your answers on notebook paper. {y | y ≥ −1} 6 4 2 9. f(−4) = 20. 2 80 Cost ($) © 2015 College Board. 7. {x | x ∈ }. domain: {x | x ∈ }. b. range: −∞ < y < ∞. 2  4 if x < −2  function f (x ) = 1 if − 2 ≤ x < 3 . Explain why the graph shown below does not represent a function. 8]. ∞). range: y ≥ −1. f(1) = 1. a. $1440 2 –2 x 4 –2 –4 b. {y | y ∈ . B 6. Write a piecewise function f(x) that can be used to determine the welder’s earnings when she works x hours in a week. or y = −3} 1 2 3 4 5 6 7 x Days Rented Activity 4 • Piecewise-Defined Functions 71 . and x = 4. Graph the step 2 4 –4 –4 3. 3. the relationship is not a function. (−∞. 2. –6 y  4  x 2 if x ≤ 1 f (x ) =  −2 x + 3 if x > 1 2 –2 x 4 –2 2. [−1. How much does the welder earn when she works 48 hours in a week? A. b. Evaluate f(x) for x = −4. a. {x | x ∈ . f (x ) = 60  if 2 < x ≤ 7 90 y  4 –4 y 2 –2 4 –4 5.. Write the domain using an inequality. (−∞. (−∞. 30 if 0 < x ≤ 1 Functions Activity  4 • Piecewise-Defined if 1 < x ≤ 2 11. Graph the step function. y if 0 ≤ x ≤ 40 if x > 40 Weekly Earnings 800 Earnings ($) 6. f (x ) = 20 x {800 + 30( x − 40) 4.indd Page 71 15/04/14 11:39 AM user-g-w-728 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . and y = −2. Show your work. 5. Customers are given the weekly rate if it is cheaper than using the daily rate. because the input −2 has 2 outputs. Graph the piecewise function. y = 1. f (x ) =   x − 2 if x < 1 if x ≥ 1 100 x –2 71 Cost of Renting Wallpaper Steamer b. When x = −2. The domain of a function is all real numbers greater than −2 and less than or equal to 8. $1200 D. it is rounded up to the nextgreatest day. interval notation. Use a domain of 0 < x ≤ 7.  −3 if x ≥ 3 x –2 © 2015 College Board. range: {y | y = 4. a. All rights reserved. All rights reserved. y = 2. 11. Graph each of the following piecewise-defined functions. interval notation. y ≥ 4. a. {y | y ∈  } 2 –4 –4 60 40 20 b. −2 < x ≤ 8. Then write its domain and range using inequalities. What are the domain and range of the step function? A welder earns $20 per hour for the first 40 hours she works in a week and $30 per hour for each hour over 40 hours. domain: −∞ < x < ∞. y ≥ 4} 8. Write the equation of the piecewise function f(x) shown below. ∞). domain: −∞ < x < ∞. Use this information for Items 3–5. x 6 4 4 –2 4 y  –4 y  –4 2 –2 b. −2 < x ≤ 8} 7.057-072_SB_AG2_SE_U01_A04. ∞). a. It costs $30 per day or $90 per week to rent a wallpaper steamer. ACTIVITY 4 Continued Piecewise-Defined Functions Absolutely Piece-ful ACTIVITY 4 continued ACTIVITY 4 PRACTICE 8. The range of a function is [4. 4. $1040 C. If the time in days is not a whole number. y  if x < −3 −5x  f (x ) =  x 2 if − 3 ≤ x < 4  2 x + 4 if x ≥ 4 Lesson 4-1 1. {x | x ∈  }. a. x = 1. −3x − 4 if x < −1 f ( x ) =   x if x ≥ −1 ACTIVITY PRACTICE 1. Write the range using an inequality and set notation. and set notation.. Write the equation of a step function f(x) that can be used to determine the cost in dollars of renting a wallpaper steamer for x days. Lesson 4-2 2 10. f(4) = 12 −2 x 9. ∞). ∞). 640 480 320 160 8 16 24 32 40 x Hours Worked 10. $990 B. (−2. When k < 0. and then a translation 3 units up. followed by a horizontal shrink by a factor of 1 . Lesson 4-3 2 1 18. and f(3. domain: {x | x ∈ }. and describe the graph as a transformation of the graph of f(x) = |x|. f(x) = |x − 2| + 1 B. a translation 1 unit right. and then a translation 1 unit down d. Find f(−2. –4 b. A step function called the integer part function gives the value f(x) that is the integer part of x. 17. ACTIVITY 4 Continued 12. g(x) = 1 |x| + 2 3 c.1). A step function called the nearest integer function gives the value g(x) that is the integer closest to x. the 2 a graph of g(x) is the graph of f(x) translated x k units down. followed by a reflection across the x-axis and a vertical stretch by a factor of 2. Inequalities.057-072_SB_AG2_SE_U01_A04. followed by a vertical stretch by a factor of 2. a. Consider the absolute value function f(x) = |x + 2| − 1. b. g(x) = |x + 3| − 1 b. 0. −2. 17. a horizontal translation 3 units left and a vertical translation 1 unit down b. Use the definition of f(x) = |x| to rewrite |x| f (x ) = as a piecewise-defined function. such as 1. g(0. the graph of h(x) is the graph of –2 f(x) vertically stretched by a factor of k.5. translated right 7 units. b.5. g(x) = −2|x − 1| − 1 –4 2 –2 4 x –2 A. and translated up 5 units b. f(x) = |x + 1| + 2 MATHEMATICAL PRACTICES Reason Abstractly and Quantitatively 20. a. Describe how the graph of j(x) = |kx| changes compared to the graph of f(x) = |x| when k < 0. review them carefully to ensure you understand all the terminology and what is being asked. d. range: {y | y ∈  . Write the equation of the function g(x) shown in the graph. f(x) = |x − 1| + 2 C. The graph of g(x) is a reflection of the graph of f(x) across the x-axis. stretched horizontally by a factor of 5. Functions © 2015 College Board. 2. Piecewise-Defined Functions Absolutely Piece-ful ACTIVITY 4 y continued 4 4 –2 13. c. the nearest integer function gives the value of g(x) that is the even integer closest to x. a. Unit 1 • Equations.5. Then identify the transformations. and translated down 10 units c. x Then graph the function. a. f (x ) =  if x > 0 1 y 1 2 x 16. Before answering each part. What are the domain and range of the function? c. 4 −1 if x < 0 14.6). the graph of h(x) is the graph of f(x) –4 vertically shrunk by a factor of k.5). y ≥ −1} c. When k > 0. Describe the symmetry of the graph. −2. All rights reserved. and 3. a. Graph the integer part function. y-intercept: 1 d. When 0 < k < 1. Check students’ graphs. 0. see the Teacher Resources at SpringBoard Digital for additional practice problems. and then a 5 translation 4 units down ADDITIONAL PRACTICE If students need more practice on the concepts in this activity. 2 . b. y  –2 15. the graph of j(x) is f(x) horizontally –6 c stretched or shrunk by a factor of 1 . the graph of g(x) is the graph of f(x) translated k units up. f(x) = |x + 2| + 1 D.5|x − 7| + 5 1 x − 10 f ( x ) = − b. a. 3 13. SpringBoard® Mathematics Functions 18. f(0. Graph the function.1). Unit 1 • Equations. Graph the nearest integer function. f(x) = 0. For half integers.5. a translation 1 unit right. The graph is symmetric about the vertical line x = −2. g(x) = 5|x − 1| − 4 y  Algebra 2. Find g(−2. shrunk vertically by a factor of 0. What are the x-intercept(s) and y-intercept of the function? d. A 4 20. c. 16.. a. a. When k > 1. 2 –4 x © 2015 College Board. g(x) = −2|x| + 3 or g(x) = −|2x| + 3. Which function is shown in the graph? y  6 14.. a.Inequalities. and g(3. Describe how the graph of g(x) = |x| + k changes compared to the graph of f(x) = |x| when k > 0 and when k < 0. a. k SpringBoard® Mathematics Algebra 2. a. 2 4 6 –6 –4 –2 b. f(x) = |x − 9| − 6 b 19.indd Page 72 25/02/15 7:51 AM s-059 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . a. Graph the following transformations of f(x) = |x|. b. b. All rights reserved. x-intercepts: −3 and −1. 72 19. reflected over the x-axis.5). 72a–d. a vertical shrink by a factor of 1 followed by a translation 3 2 units up c. –4 –2 –1 –1 12.6). 4 2 –4 2 –2 4 x –2 –4 17. translated right 9 units and translated down 6 units 4 2 15. Describe how the graph of h(x) = k|x| changes compared to the graph of f(x) = |x| when k > 1 and when 0 < k < 1. Write the equation for each transformation of f(x) = |x| described below. y  4 2 –4 2 –2 x 4 –2 b. d 6 5 c. When k < 0. 1c is introduced in this activity but is also addressed in higher level mathematics courses. Evaluate (m + 2)2 for m = −5. and division are operations on real numbers. subtraction. Add the functions t (h) and s (h) to find (t + s)(h). 1. Model with mathematics. emphasize that when evaluating functions combined with operations. Combining functions ahead of time is efficient when evaluating many input values. with a y-intercept at the origin and a slope of 10 . 3. All rights reserved. (t + s)(h) = 10h + 8h = 18h b. Close Reading.A. Paraphrasing. HSF-BF. 5. Find t(4) + s(4) and tell what it represents in this situation. and Stephan’s earnings for the 4-hour job are $32.1c(+) Compose functions. t(4) + s(4) = 40 + 32 = 72. Simplify 5a2 + 2a − 4a − 6a2. How does the answer compare to t(4) + s(4)? (t + s)(4) = 18(4) = 72. and answer any questions they may have prior to moving forward with the lesson. Lesson 5-1 PLAN t(h) = 10h. In other words. Group Presentation Ensure students understand this application by placing them in small groups and having each group create a scenario with adding two real-world functions. Find t(4) and s(4) and tell what these values represent in this situation. • Build functions that model real-world scenarios. s(h) = 8h Pacing: 1 class period Chunking the Lesson 2.1b Combine standard function types using arithmetic operations. Find (t + s)(4). and Stephan earns $8 per hour.A. Write a function t(h) to represent Tori’s earnings in dollars for working h hours and a function s(h) to represent Stephan’s earnings in dollars for working h hours. Tori earns $10 per hour. HSF-BF. SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge.. My Notes Learning Targets: • Combine functions using arithmetic operations. ACTIVITY Function Composition and Operations ACTIVITY 5 New from Old Lesson 5-1 Operations with Functions 5 Investigative Activity Standards Focus In Activity 5. multiplication. (f + g)(x) = f (x) + g(x). [9] Ask students to share their responses. Encourage them to use the hourly earnings functions as a template but also to feel free to use a variable other than hours. t(4) = 40. What does the function (t + s)(h) represent in this situation? WRITING MATH The notation (f + g)(x) represents the sum of the functions f (x) and g(x). MATH TIP Addition. Paraphrasing Ask students questions like: What would the graph of these functions look like? [t(h) would be linear. Tori’s earnings for the 4-hour job are $40. because Tori and Stephan will not receive pay for negative hours] 4–7 Discussion Groups. #1–3 #4–7 #8–11 #12 #13–14 #15 #16–17 #18–20 #21 #22 Check Your Understanding Lesson Practice 4. and s(h) 1 would be linear. You can add two functions by adding their function rules. You can also perform these operations with functions. Then simplify the function rule. Jim sends Tori and Stephan on a job that takes them 4 hours. Debriefing.indd Page 73 05/12/13 4:07 AM s-059 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . a. How much will Jim spend on Tori and Stephan’s earnings for the 4-hour job? $72 Activity 5 • Function Composition and Operations 73 Common Core State Standards for Activity 5 HSF-BF. s(4) = 32.1 Write a function that describes a relationship between two quantities. Throughout this activity. © 2015 College Board. The sum of Tori and Stephan’s earnings for the 4-hour job is $72. Think-Pair-Share. the amount in dollars Jim must spend on Tori and Stephan’s earnings for a job that takes h hours Have students review some of the concepts they will need to apply in this lesson.073-088_SB_AG2_SE_U01_A05.] 1–3 Activating Prior Knowledge. Quadrant I only. students perform operations on functions. Students then write composite functions.A. Summarizing.] What does the 1 y-intercept represent? [zero pay for zero hours worked] What does the slope represent? [the rate of pay per hour] Would you be interested in looking at the entire coordinate plane? [No. Discussion Groups. All rights reserved. Quickwrite Jim Green has a lawn service called Green’s Grass Guaranteed. [−a2 − 2a] 2. Chunking the Activity. 6. [Note: HSF-BF. TEACH Bell-Ringer Activity © 2015 College Board. 1. (t + s)(4) has the same value as t(4) + s(4). the value of an input evaluated first in the separate functions and then operated is equal to the value of the combined function with that input. with a y-intercept at the origin and a slope of 8. Tori and Stephan are two of his employees. Activity 5 • Function Composition and Operations 73 .. Ask students to complete the following exercises.A. Inequalities. charges customers a fixed fee of $75 plus $175 per tree for the same service. ( f − g)(x) = x + 3 e. Press the y = key. How much would Jim spend on Tori and Stephan’s earnings for a job that takes 6 hours? Explain how you determined your answer.] 7. (g + h)(x) = x2 + 7 c. ( f + g)(x) = 5x + 1 b. Look for and make use of structure. in other words. key in 4 ENTER . For example. 5. j(t) = 25 + 150t. One of Jim’s competitors. I evaluated (v − j )(t) for t = 8: (v − j )(8) = 50 + 25(8) = 250. For a basic tree-trimming job. All rights reserved. 10x − (6x + 2) = 10x − 6x − 2 = 4x − 2 WRITING MATH The notation (f − g)(x) represents the difference of the functions f(x) and g(x). Then simplify the function rule. not just the first term. Vista Lawn & Garden. 6. (h + f )(x) = x + x + 10 d. key in 6 ENTER . b. Unit 1 • Equations. My Notes © 2015 College Board. Set TblStart to =0. 4. Functions © 2015 College Board. and h(x) = x2 − 2x + 8. (h − g)(x) = x2 − 4x + 9 2 74 SpringBoard® Mathematics Algebra 2. Access the table by pressing 2nd GRAPH . SpringBoard® Mathematics Algebra 2.indd Page 74 05/12/13 4:07 AM s-059 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . The calculator is waiting for you to enter the x-value (in this case. Beside the function. Set the change in the table. Jim charges customers a fixed $25 fee plus $150 per tree. Chunking the Activity. or cost. How much will a customer save by choosing Jim’s company to trim 8 trees rather than choosing Vista? Explain how you determined your answer. Find (v − j )(5). and students will use the same structure to combine like terms. You could do this for the function addition represented in the previous items by following these steps: For additional technology resources. (g − f )(x) = −x − 3 f. 1. Predict and Confirm Lead a discussion about these items by asking the following: • How do these functions differ from those presented in Items 1−7? [These functions have a y-intercept other than (0. Notice the table is blank. ACTIVITY 5 Continued Lesson 5-1 Operations with Functions ACTIVITY 5 You can use the table feature of a graphing calculator to find specific function values. 8. they apply this knowledge to adding and/or subtracting a linear function to a quadratic function. I evaluated (t + s)(h) for h = 6: (t + s)(6) = 18(6) = 108. 12. 8. Unit 1 • Equations. remember to subtract each term of the expression. This should give the corresponding y-value of $72. $108. subtract 6x − 2 from 10x as follows. Debriefing Once students know how to add and subtract linear functions. Technology Tip continued . In other words. the number of hours) for which you would like to know the corresponding y-value. Write a function j(t) to represent the total charge in dollars for trimming t trees by Jim’s company and a function v(t) to represent the total charge in dollars for trimming t trees by Vista. Functions 12 Activating Prior Knowledge. At x=. 8–11 Activating Prior Knowledge. type in 10x + 8x. $250. to 1. find each function and simplify the function rule.] • What is one thing you have to be cautious about when subtracting expressions? [Subtract each term of the expression. Subtract j(t) from v(t) to find (v − j )(t). Now you can continue this by trying other numbers of hours that were not already in the examples. What does the function (v − j)(t) represent in this situation? the amount in dollars a customer will save by choosing Jim’s company to trim t trees rather than Vista 10. At x=. or ∆Tbl.] • What makes these functions have these y-intercepts? [the fixed fees charged by each company] • Why is subtraction being used rather than addition? [It is basically an example of comparison shopping. Inequalities. g (x) = 2x − 1. v(t) = 75 + 175t MATH TIP 74 9. 3. All rights reserved. 0). 7.. (v − j)(t) = 75 + 175t − (25 + 150t) = 75 + 175t − 25 − 150t = 50 + 25t When subtracting an algebraic expression.073-088_SB_AG2_SE_U01_A05. 11. the subtraction sign is distributed throughout the subtrahend expression. The same function rules apply.. What does this value represent in this situation? (v − j )(5) = 50 + 25(5) = 175. Press 2nd WINDOW to look at the table setup. where one wants to know how much will be saved by using one company instead of the other. Set the Indpnt: to Ask. a. visit SpringBoard Digital. (f − g)(x) = f(x) − g(x). This should give the corresponding y-value of $108. 2. A customer will save $175 by choosing Jim’s company rather than Vista to trim 5 trees. a. Given f (x) = 3x + 2. in terms of a given number of trees. have them construct a table of values for h and n(h). because h represents a number of hours. Then simplify the function rule. its value cannot be negative. 14. he needs to include the number of shrubs he can install in an 8-hour day. In Item 15b. c(h) = 16 + 65h 15. b. Jim will charge $16 for each shrub. which they learned in Algebra 1. Write a function n(h) to represent the number of shrubs Jim can install in an 8-hour day when it takes him h hours to install one shrub. explain the following: My Notes MATH TIP When considering restrictions on the domain of a real-world function. All rights reserved. engage them by asking them why the h-value must be positive. What are the restrictions on the domain of (n c)(h)? A linear function is an algebraic equation in which the greatest degree of a variable term is 1. consider both values of the domain for which the function would be undefined and values of the domain that would not make sense in the situation. The notation (f g)(x) represents the product of the functions f (x) and g(x). (f g)(x) = f (x) g(x). n( h) = 8 h b. In the bid. All rights reserved. For Item 14. © 2015 College Board.indd Page 75 05/12/13 4:07 AM s-059 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . • The standard form of a linear function is Ax + By = C. and the total cost of his services for an 8-hour day. ⋅ ⋅8 ( n c )( h) = (16 + 65 h) = 128 + 520 h h WRITING MATH ⋅ ⋅ ⋅ ⋅ The value of h must be positive. a monomial is being multiplied by a binomial. the total number of hours in the workday. 13.073-088_SB_AG2_SE_U01_A05. and c are constants. where A. Ask: If it takes Jim one hour to install one shrub. Attend to precision. • The y-intercept form of a linear equation is y = mx + b.. b. Also. What are the restrictions on the domain of n(h)? Explain. In Item 13b. The total cost of Jim’s services for an 8-hour day is equal to the number of shrubs he can install times the charge for each shrub. and C are constants. A quadratic function is an algebraic equation in which one or more of the variable terms is squared. Find the total cost of Jim’s services using the functions n(h) and c(h) to find (n c)(h). where m is the slope and b is the y-intercept. Write a function c(h) to represent the amount Jim will charge for a shrub that takes h hours to install. a. Marking the Text. ACTIVITY 5 Continued Lesson 5-1 Operations with Functions ACTIVITY 5 continued Jim has been asked to make a bid for installing the shrubs around a new office building. To ensure that students understand the correct response to 15b. © 2015 College Board. giving the function a degree of 2. Differentiating Instruction. • The point-slope form of a linear equation is y − y1 = m(x − x1). or the function would be undefined. where m is the slope and (x1. Activity 5 • Function Composition and Operations 75 . • The general form is ax2 + bx + c = 0. Differentiating Instruction Activity 5 • Function Composition and Operations 75 13–14 Close Reading. the greatest exponent of a variable term is 1. 15 Activating Prior Knowledge Explain to the students that multiplying functions will require them to multiply polynomials. For those students who need additional explanation of the functions used in Item 12. Simplify the Problem To help students understand the function in Item 13. y1) are the coordinates of a point through which the line passes. The types of polynomials being multiplied will obviously vary with the functions. Be sure to use the Distributive Property. the number of hours it takes Jim to install one shrub. a squared power is the greatest degree a quadratic function can have. tell students who are struggling to refer back to either function from Item 8 because they are the same type. In other words. students again encounter the topic of domain restrictions. The value of h cannot be 0. In Item 15a. how many shrubs can Jim install in an 8-hour day? What if it takes Jim 2 hours to install one shrub? 3 hours? 4 hours? Elicit from students the operation of division between 8. Support students whose first language is not English by further explaining the word restriction. highlight restrictions and domain. and h. a. In other words.. where a. the cost per shrub including installation. He will also charge $65 per hour for installation services. However. B. and (n c)(h) to determine the following values for Jim’s bid.5) = 16 + 65(0. where h is the number of hours the job takes. Use the functions n(h).] Lesson 5-1 Operations with Functions ACTIVITY 5 continued © 2015 College Board.073-088_SB_AG2_SE_U01_A05. 0 . n(h). however. The answer of 16 in Item 16b represents that predetermined cost per shrub.5: (n⋅c)(0.5) = 128 0 . ⋅ c)(h) for h = 0. a.5) = 48. including installation $48. Write a function c(h) to represent the total charge for applying compost to a lawn. Then have students make a conjecture as to whether they think it will cost more or less if Jim estimates that it will take him 40 minutes to install each shrub. the number of shrubs Jim can install in an 8-hour day ⋅ 16 shrubs. Reason quantitatively. Explain how you could check your answer to Item 16c.5 hour to install each shrub. ensure students are not confused by the solution to 16a being 16 shrubs and the predetermined cost per shrub (listed in Item 14) being the same. After discussing. ACTIVITY 5 Continued ⋅ My Notes CONNECT TO BUSINESS When a company makes a bid on a job. Why is the total cost of Jim’s services for an 8-hour day less? [because he is getting less work done per hour] Jim offers two lawn improvement services. Present a table with four column titles—h. The fact that the number of shrubs and the cost per shrub are the same is a mere coincidence. I evaluated n(h) for h = 0.5 to h = 2 (a longer amount of time per 3 shrub). and explain how you determined your answers. the cost per shrub. Lawn Improvement Services Service Hourly Charge ($) Material Cost for Average Yard ($) Compost 40 140 Fertilizer 30 30 18. Differentiating Instruction Ask students to discuss whether their conjectures were correct or incorrect when they altered the value of h in Items 16 and 17 from h = 0. and (n c)(h)—and place values for h = 0. the total cost of Jim’s services for an 8-hour day $776. c(h). I evaluated c(h) for h = 0. Jim estimates that it will take 0. Functions © 2015 College Board. Unit 1 • Equations. 16–17 Create Representations. 3 [Answer should be approximately $712.5 17.5: n(0. including installation: 16($48. Inequalities. Inequalities. .50. a. All rights reserved.5) = 8 = 16. c. 16.. If it bids too high.5: c(0. the company states the price at which it is willing to do the job.50. c(h) = 40h + 140 b. Functions SpringBoard® Mathematics Algebra 2.5 b. All rights reserved.50) = $776.5 in each column. the job may be offered to one of its competitors. The company must make its bid high enough to cover all of its expenses. Fill in values 3 for h = 2 in a new row of the table.. Write a function f (h) to represent the total charge for applying fertilizer to a lawn. as described in the table. I evaluated (n + 520 = 776. Sample answer: Multiply the number of shrubs Jim can install in an 8-hour day by the cost per shrub. c(h). Unit 1 • Equations.indd Page 76 05/12/13 4:07 AM s-059 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . f (h) = 30h + 30 76 76 SpringBoard® Mathematics Algebra 2. where h is the number of hours the job takes. Debriefing In Item 16. have students try h = 2 in the functions. x ≠ −3 18–20 Predict and Confirm. and h(x) = 2x + 6. (h ÷ g)(x). For a job that takes 4 hours. The coefficient 2 of 2x in the numerator cancels with the common factor of 2 in the denominator. d. a. the (x + 3)’s in the numerator and denominator can be entirely canceled. ( g ÷ f )(x) (g ÷ f )( x ) = x + 3 . g(x) = x + 3. 21 Activating Prior Knowledge.indd Page 77 05/12/13 4:07 AM s-059 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . g(x) ≠ 0. Furthermore. find each function and simplify the function rule. 10y + 15 = 5(2y + 3) 1 3( p + 4 ) 1 3 p + 12 = = . and the only way to cancel would be if there were a term of (x + 3) in the numerator. Note any values that must be excluded from the domain. What does this value represent in this situation? 40(4 ) + 140 = 2. ( h ÷ g )( x ) = 2 x + 6 = 2. 6 p + 24 2 2 6( p + 4 ) p ≠ −4 Activity 5 • Function Composition and Operations 77 . and e by factoring the expression’s numerator and denominator and dividing out common factors. g(x) ≠ 0 represents the quotient of the functions f (x) and g(x) given that g(x) ≠ 0.. 77 For struggling students. All rights reserved. there is no common factor. ( c ÷ f )(4 ) = 21. All rights reserved. and 21e. Divide c(h) by f (h) to find (c ÷ f )(h) given that f (h) ≠ 0. x2 + 5x = x(x + 5) 2. some students are going to want to cancel out the x’s.] d. a. ACTIVITY 5 Continued Lesson 5-1 Operations with Functions ACTIVITY 5 continued 19. x ≠ −3 x+3 2 x + 6 = 2( x + 3 ) = 2 x +3 ( x + 3) In the numerator. 3. Look for and make use of structure. (f ÷ g)(x) = f (x) ÷ g(x).. See the Teacher Resources at SpringBoard Digital for a student page for this mini-lesson. Since factoring has not been covered in Algebra 2 at this point. g(x) ≠ 0 Discuss that the two terms in the denominator have a common factor of 2. Be prepared to explain that this is not possible because the x in the denominator is part of the term (x + 3). 1. ( c ÷ f )( h) = 40 h + 140 30 h + 30 My Notes WRITING MATH b. ⋅ ⋅ g)(x) = 2x(x + 3) = 2x + 6x b. h(x) ≠ 0 © 2015 College Board. In Item 21d. In Item 21c. there is a common factor of 2 that can be factored out. Find (c ÷ f )(4). a mini-lesson is available to provide practice.073-088_SB_AG2_SE_U01_A05. What does the function (c ÷ f )(h) represent in this situation? the ratio of the cost in dollars of applying compost to the cost in dollars of applying fertilizer for a job that takes h hours The notation (f ÷ g)(x). it may be helpful to take some extra time to review factoring out a common factor before moving forward. Given f (x) = 2x. the cost of 30(4 ) + 30 applying compost is 2 times the cost of applying fertilizer. Here are some suggestions of samples you might use. 21d. multiplying. or dividing functions. you may wish to review with students the following: 20. ( f g)(x) (f (g © 2015 College Board. Activating Prior Knowledge Ask students to make a conjecture as to the number of hours (if any) that it would take for the total charge of applying compost to equal the total charge of applying fertilizer. In other words. c. Debriefing Note that the Math Tip refers to factoring expressions in the numerator and denominator in Items 21c. (g h)(x) (f ÷ h)( x ) = 2x = x 2 x + 6 x + 3 . 2 x = /2 x = x 2 x + 6 /2(x + 3) x + 3 2 ⋅ ⋅ h)(x) = (x + 3)(2x + 6) = 2x + 12x + 18 MATH TIP 2 You may be able to simplify the function rules in Items 21c. Differentiating Instruction Activity 5 • Function Composition and Operations MINI-LESSON: Function Operations If students need additional help with adding. (f ÷ h)(x). e. x ≠ 0 2x In Item 21e. After doing so. [Students will hopefully realize the impossibility of this because both the hourly charge and material cost are greater for the compost service. subtracting. x ≠ 2 . All rights reserved.073-088_SB_AG2_SE_U01_A05. ( f + g)(x) 27. All rights reserved. ACTIVITY 5 Continued Lesson 5-1 Operations with Functions ACTIVITY 5 continued My Notes 22.. Functions LESSON 5-1 PRACTICE (f + g)(x) = 5x + 1 + 3x − 4 = 8x − 3 (f − g)(x) = 5x + 1 − (3x − 4) = 2x + 5 (f g)(x) = (5x + 1)(3x − 4) = 15x2 − 17x − 4 ( f ÷ g ) = 5x + 1 . the student should have written the rule for f(x) in parentheses so that both terms of the rule would be subtracted. 22 Debriefing This item guides students toward the conclusion that operations with functions follow similar processes and rules as operations with numbers. Addition of functions is commutative. LESSON 5-1 PRACTICE Answers 23. so j(x) = h(x) − (h − j)(x) = 4x + 5 − (x − 2) = 3x + 7. Lastly. the function rule that is the divisor cannot be equal to 0. See the Activity Practice for additional problems for this lesson. Rather than simple addition. 2 is excluded from the domain 3 of (f ÷ g)(x). and what is the correct answer? (g − f )(x) = 3x − 4 − 5x + 1 = −2x − 3 31. The function (f ÷ g)(x) is 3 undefined when g(x) = 0. 31. Sometimes students understand the concepts but are confused by the notation. Note any values that must be excluded from the domain. For Items 26–30. 27. Explain how you determined your answer. With both real numbers and function division. function division may require knowledge of factoring polynomials. Operations on functions involve function rules. The function f (t) = 264t gives the cost of running the ad t times on a more popular FM station. 25. How are operations on functions similar to and different from operations on real numbers? Sample answer: Operations on real numbers involve only numbers. a. Unit 1 • Equations. The main differences when performing function operations are the use of function notation and variables. ADAPT Check students’ answers to the Lesson Practice to ensure that they understand function operations. ASSESS b. Otherwise. If this happens. whereas subtraction and division do not. The function a(t) = 800 + 84t gives the cost of making the ad and running it t times on an AM station. Make a conjecture about whether addition of functions is commutative. 30. ( f − g)(x) 28. Find (a + f )(12) and tell what it represents in this situation. a. the processes of addition. Because g(x) = 3x − 2. g(x) ≠ 0 ⋅ 30. © 2015 College Board. Sample explanation: I know that h(x) − j(x) = (h − j)(x).. Inequalities. Find each function and simplify the function rule. what value(s) of x are excluded from the domain of ( f ÷ g)(x)? Explain your answer. the divisor cannot be 0. 78 78 SpringBoard® Mathematics Algebra 2. and division are essentially the same. and for division of functions. ( f ÷ g)(x). What mistake did the student make. multiplication. Inequalities. (a + f)(t) = 800 + 84t + 264t = 800 + 348t. Find (a + f )(t) and tell what it represents in this situation.indd Page 78 05/12/13 4:07 AM s-059 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . Jim plans to make a radio ad for his lawn company. subtraction. The function (a + f)(t) represents the cost of making the ad and running it on both stations t times. Give an example that supports your conjecture. (f + g)(x) = (g + f)(x) = 2x + 7. b. x ≠ 43 3x − 4 When subtracting the rule for f(x) from the rule for g(x). g(x) = 0 when x = 23 . As with numbers. It will cost $4976 to make the ad and run it on both stations 12 times. encourage them to begin by rewriting an expression such as (f + g)x as f(x) + g(x) and then substitute expressions for f(x) and g(x). 24. ⋅ SpringBoard® Mathematics Algebra 2. Sample example: Given that f(x) = 4x + 2 and g(x) = −2x + 5. find j(x). ( f g)(x) 29. 28. addition and multiplication of functions follow the commutative properties. The correct answer is (g − f)(x) = 3x − 4 − (5x + 1) = −2x − 5. You may assign the problems here or use them as a culmination for the activity. function operations involve combining like terms. j(x) = 3x + 7. For division of real numbers. 29. A student incorrectly found (g − f )(x) as follows. 24. . Unit 1 • Equations. Given that h(x) = 4x + 5 and (h − j )(x) = x − 2. Given that f (x) = 2x + 1 and g(x) = 3x − 2. Students’ answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the lesson concepts and their ability to apply their learning. (a + f)(12) = 800 + 348(12) = 4976. Check Your Understanding Debrief students’ answers to these items to ensure that they understand concepts related to function operations. 26. Make sense of problems and persevere in solving them. the divisor cannot equal zero. Check Your Understanding 23. Functions © 2015 College Board. 25. So. f(x) = 5x + 1 g(x) = 3x − 4 26. use the following functions. ACTIVITY 5 Continued Lesson 5-2 Function Composition ACTIVITY 5 continued Learning Targets: • Write functions that describe the relationship between two quantities. • Explore the composition of two functions through a real-world scenario. 3–5 Quickwrite.. All rights reserved. Graphic Organizer.. Model with mathematics. and describe the mathematical concepts your group uses to create its solutions. The term composition will be introduced later in the activity. what will he charge for a mowing job? Write your answer as a cost function where c(t) is Jim’s charge for t hours of work. Group Presentation Obtaining a cost estimate provides a numerical example for a two-step process that involves composing one function with another. All rights reserved. © 2015 College Board. PLAN My Notes Pacing: 1 class period Chunking the Lesson #1–2 #3–5 #6–8 #9 #10 #11–14 Check Your Understanding Lesson Practice SUGGESTED LEARNING STRATEGIES: Create Representations. Self Revision/Peer Revision Recall that Jim has a lawn service called Green’s Grass Guaranteed. Be sure that all students understand the numerical process before they encounter the symbolic function notation later on.073-088_SB_AG2_SE_U01_A05. reread the problem scenarios in this lesson as needed. Make notes on the information provided in the problems. Write the equation of a function in terms of a for the number of hours t it will take Jim to mow the property. Jim charges a fixed $30 fee to cover equipment and travel expenses plus a $20 per hour labor charge. Jim prepares a cost estimate for each customer based on the size (number of acres) of the property. $190. Another customer has a acres of property. t(a) = 4a 5. 3. Work with your group on Items 1–14. The APCON company is one of Jim’s customers. Lesson 5-2 8 hours Bell-Ringer Activity Write the equations y = 6x and y = 6x + 11 on the board. What was the total charge for this job? $150. numerical check for student understanding of the scenario. How much will Jim charge APCON to mow its property? Justify your answer. Use the discussion as an opportunity to review independent and dependent variables in equations. Debriefing Item 1 serves as a concrete. It takes Jim 4 hours to mow 1 acre. 4. Jim worked for 6 hours. Identify a Subtask. On a recent mowing job. Group Presentation. Activity 5 • Function Composition and Operations 79 Activity 5 • Function Composition and Operations 79 . c(t) = 30 + 20t TEACH DISCUSSION GROUP TIP With your group. Respond to questions about the meaning of key information. and how they relate to domain and range of functions. How many hours does that job take? © 2015 College Board. Use Item 2 as a quick assessment of understanding of function notation. Sample explanation: Use the number of hours (8) to substitute into the cost function: c(t) = 30 + 20t = 30 + 20(8) = 190 dollars. Item 2 asks students to write an algebraic rule relating cost and time. 1–2 Create Representations. APCON has 2 acres that need mowing. 1. Have students discuss what kind of equations these represent and their similarities and differences. Summarize the information needed to create reasonable solutions. If Jim works for t hours. On every mowing job. Identify a Subtask. 30 + (20 × 6) = 150 2.indd Page 79 25/02/15 7:53 AM s-059 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . Debriefing. 073-088_SB_AG2_SE_U01_A05.indd Page 80 05/12/13 4:07 AM s-059 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ ... ACTIVITY 5 Continued 6 Think-Pair-Share Use this item to assess students’ understanding of slope. It will also help them to focus on the quantities represented by each variable. Lesson 5-2 Function Composition ACTIVITY 5 continued My Notes TEACHER to TEACHER In the linear equation y = mx + b, m is the constant rate of change of y with respect to x. When that equation is graphed, m indicates the slope of the line. Therefore, slope is the graphical representation of rate of change. The rate of change has two representations—a numerical one with units of measure (e.g., $20/h) and a graphical one (e.g., slope of 20). 7–8 Think-Pair-Share, Graphic Organizer, Debriefing Domain and range of the functions are represented in a table and in a graphic organizer. These items set the stage for understanding how the domain and range of a composite function are related to the domain and range of the functions from which it is created. The functions in Items 2 and 4 relate three quantities that vary, based on the needs of Jim’s customers: • The size in acres a of the property • The time in hours t needed to perform the work • The cost in dollars c of doing the work. 6. Attend to precision. Complete the table below by writing the rate of change with units and finding the slope of the graph of the function. MATH TIP In a linear function f (x) = mx + b, the y-intercept is b. The variable m is the rate of change in the values of the function—the change of units of f (x) per change of unit of x. When the function is graphed, the rate of change is interpreted as the slope. So y = mx + b is called the slopeintercept form of a linear equation. Function Rate of Change (with units) Slope c(t) = 30 + 20t $20 per hour 20 t(a) = 4a 4 hours per acre 4 7. Complete the table below by naming the measurement units for the domain and range of each function. Function Notation Description of Function Domain (units) Range (units) c(t) cost for job hours dollars t(a) time to mow acres hours 8. Calculating the cost to mow a lawn is a two-step process. Complete the graphic organizer below by describing the input and output, including units, for each part of the process. Output: Time (in hours) Input: Time (in hours) Cost for Job Output: Cost (in dollars) 80 SpringBoard® Mathematics Algebra 2, Unit 1 • Equations, Inequalities, Functions MINI-LESSON: Linear Functions and Slope If students need additional help with identifying the slope and y-intercept of a linear function, as well as interpreting slope and rate of change, a mini-lesson is available to provide practice. See the Teacher Resources at SpringBoard Digital for a student page for this mini-lesson. 80 SpringBoard® Mathematics Algebra 2, Unit 1 • Equations, Inequalities, Functions © 2015 College Board. All rights reserved. Time to Mow © 2015 College Board. All rights reserved. Input: Area (in acres) 073-088_SB_AG2_SE_U01_A05.indd Page 81 05/12/13 4:07 AM s-059 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ ... ACTIVITY 5 Continued Lesson 5-2 Function Composition ACTIVITY 5 continued The graphic organizer shows an operation on two functions, called a composition. The function that results from using the output of the first function as the input for the second function is a composite function. In this context, the composite function is formed by the time-to-mow function and the cost-for-job function. Its domain is the input for the time function, and its range is the output from the cost function. 9. Make sense of problems. The cost to mow is a composite function. Describe its input and output as you did in Item 8. Input: Area (in acres) MATH TERMS A composition is an operation on two functions that forms a new function. To form the new function, the rule for the first function is used as the input for the second function. A composite function is the function that results from the composition of two functions. The range of the first function becomes the domain for the second function. Area (in acres) Input: My Notes Time to Mow Number of Hours Cost (in dollars) When a composite function is formed, the function is often named to show the functions used to create it. The cost-to-mow function, c(t(a)), is composed of the cost-for-job and the time-to-mow functions. 10 Marking the Text, Graphic Organizer This item allows for a different representation to help students identify the domain and range of the composite function. The representations in Items 9 and 10 parallel the representations in Items 7 and 8. The c(t(a)) notation implies that a was assigned a value t(a) by the timeto-mow function. Then the resulting t(a) value was assigned a value c(t(a)) by the cost-to-mow function. © 2015 College Board. All rights reserved. © 2015 College Board. All rights reserved. 10. Complete the table by writing a description for the composite function c(t(a)). Then name the measurement units of the domain and range. Function Notation Description of Function c(t(a)) cost to mow Domain (units) Range (units) acres dollars The word composition means the act of combining parts to form a whole. Pertaining to math functions, composition is the act of forming a new function by involving two or more functions in succession. It involves taking the output of the first function and using it as the input for the second function, and so on. The result of following these steps creates an entirely new function, called a composite function. It is important to note that when working with composite functions, students start from the inside and work their way outward, similar to order of operations with arithmetic. 9 Think-Pair-Share, Graphic Organizer Students will recognize that composition of the two functions creates a new function, the cost-to-mow function, and will identify the domain and range of the composite function. Cost for Job Output: Developing Math Language Activity 5 • Function Composition and Operations 81 Activity 5 • Function Composition and Operations 81 073-088_SB_AG2_SE_U01_A05.indd Page 82 05/12/13 4:07 AM s-059 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ ... ACTIVITY 5 Continued 11–14 Quickwrite, Group Presentation, Debriefing Work with students to analyze what happens in this situation. Students should see that the time function has been substituted into the cost function. The resulting composite is a function of a acres. Students will practice function composition later in this activity. Lesson 5-2 Function Composition ACTIVITY 5 continued My Notes Jim wants to write one cost function for mowing a acres of property. To write the cost c as a function of a acres of property, he substitutes t(a) into the cost function and simplifies. c(t) = c(t(a)) c(t(a)) = c(4a) Use Item 11 to determine whether or not students understand the process of composition. = 30 + 20(4a) Substitute t(a) for t in the cost function. t(a) = 4a, so write the function in terms of a. Substitute 4a for t in the original c(t) function. c(t(a)) = 30 + 80a Item 12 brings students back to the context. During debriefing, make sure that the point is made that composition is an efficient process if you need to repeatedly use the two-step process represented by composition. 11. Attend to precision. Write a sentence to explain what the expression c(t(2)) represents. Include appropriate units in your explanation. It represents the cost in dollars to mow 2 acres. Use student communication to reinforce understanding of the meaning of composition. Be sure that students see the different meanings of the value 50 in Items 13 and 14. 12. Construct viable arguments. Why might Jim want a single function to determine the cost of a job when he knows the total number of acres? Sample response: If Jim needs to determine the cost for different-sized properties, this function will be useful. 13. Explain what the expression c(t(50)) represents. Include appropriate units in your explanation. It represents the number of acres that can be mowed for $50. 82 SpringBoard® Mathematics Algebra 2, Unit 1 • Equations, Inequalities, Functions MINI-LESSON: Composition Function Notation If students need additional help with understanding the meaning of composite function notation, a mini-lesson is available to provide practice. See the Teacher Resources at SpringBoard Digital for a student page for this mini-lesson. 82 SpringBoard® Mathematics Algebra 2, Unit 1 • Equations, Inequalities, Functions © 2015 College Board. All rights reserved. 14. Explain what information the equation c(t(a)) = 50 represents. Include appropriate units in your explanation. © 2015 College Board. All rights reserved. It represents the cost in dollars to mow 50 acres. 5r) and highlight 0. What is the slope of this function? Interpret the slope as a rate of change. Be sure students understand what the domain and range of the composition are in the context of the situation. ACTIVITY 5 Continued Lesson 5-2 Function Composition ACTIVITY 5 continued My Notes Check Your Understanding 15. 21. This can help students see what should be substituted into the function (0. Activity 5 • Function Composition and Operations 83 . See the Activity Practice for additional problems for this lesson. Students’ answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the lesson concepts and their ability to apply their learning. For example. Check Your Understanding Debrief students’ answers to these items to ensure that they understand concepts related to composition of functions. ASSESS Hannah’s Housekeeping can clean one room every half hour.5r) and where it should be substituted. The slope is 12. It means the cost will increase by $12 for each additional hour of cleaning time. encourage the use of colored pencils or highlighters. a(b(−2)) = −2 16.. 24. Write a function c(h(r)) to represent the cost of cleaning r rooms. LESSON 5-2 PRACTICE Model with mathematics. 22. The domain of the second function in the composite function is the range of the first function. Write a function c(h) for the cost to clean a house for h hours. The first function used to form a composite function has a domain of all real numbers and a range of all real numbers greater than 0. Look for and make use of structure. and then write c(0. Write a function h(r) for the hours needed to clean r rooms. highlighting both h’s. What is the value and meaning of c(h(12))? 24. 18. domain: hours. All rights reserved. If students have difficulty substituting algebraic expressions into functions. Sample answer: The composite function is composed of the functions f and g.. h(r) = 0. The variable x represents the input. c(h) = 20 + 12h 19. Explain what this notation indicates about the composite function. Explain how a composition of functions forms a new function from the old (original) functions. Hannah’s Housekeeping charges a $20 flat fee plus $12 an hour to clean a house.indd Page 83 05/12/13 4:07 AM s-059 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ .5r. 17. The composite function first evaluates the function g for the value of x to give a value for g(x).5r) = 20 + 6r 23. in Item 22. 19. If students are confused by the notation. range: dollars 20. © 2015 College Board. All rights reserved. they can begin by writing c(h(r)) = c(0. 21. encourage them to write each step. LESSON 5-2 PRACTICE 18. 17. What are the units of the domain and range of this function? 20. Possible answer: The output of one function becomes the input of another function. ADAPT Activity 5 • Function Composition and Operations 83 Check students’ answers to the Lesson Practice to ensure that they understand basic concepts related to function composition. What is the domain of the second function in the composite function? Explain. write the equation for the composite function a(b(c)) and evaluate it for c = −2. c(h(r)) = 20 + 12(0.5r 22. $92 is the cost to clean 12 rooms. students can write c(h) = 20 + 12h. 15. Given the functions a(b) = b + 8 and b(c) = 5c. © 2015 College Board. a(b(c)) = 5c + 8. Answers 16.073-088_SB_AG2_SE_U01_A05. 23. Then it evaluates the function f for the value of g(x) to give a value for f(g(x)).5r). For example. The notation f ( g (x)) represents a composite function. All real numbers greater than 0. You may assign the problems here or use them as a culmination for the activity. Debriefing A composition of functions forms a new function by substituting the output of the inner function into the outer function. of the first (inner) function becomes the domain of the second (outer) function. g ( f (3)) 4 d. c(4) 110. Unit 1 • Equations. and h as functions of x. Find 3r − 4 in terms of x. [37] 2. [−4b − 48] Have students discuss their results prior to moving forward with the lesson. provides a visual way of demonstrating how the output. Reason quantitatively. Think-Pair-Share. in order to help them with the composite functions. c. Inequalities. the use of the two tables. [3x − 22] 3. SUGGESTED LEARNING STRATEGIES: Note Taking. TEACH 1. They also learn an alternate notation for composition and work with more typical representations of functions using f. Students compose functions numerically and algebraically. g(3) 2 c. It takes 16 hours to mow 4 acres. 84 d.. 84 x f(x) x g(x) 1 3 1 4 2 2 2 3 3 1 3 2 4 4 4 1 a. 1 Create Representations In Item 1. It costs $110 for 4 hours of mowing. Lesson 5-3 . All rights reserved. Functions © 2015 College Board. y = g(f (x)) and y = f (g(x)) are two different functions. It costs $110 to mow 1 acre. 1. f (3) 1 b.. f (g(3)) 2 SpringBoard® Mathematics Algebra 2. © 2015 College Board. Group Presentation. Create Representations. Inequalities. c(t(4)) 350. Unit 1 • Equations. Then tell what the expression represents. c(t(1)) 110. Functions SpringBoard® Mathematics Algebra 2. Use the tables of values below to evaluate each expression. if r = x − 6. a.indd Page 84 05/12/13 4:07 AM s-059 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . ACTIVITY 5 Continued Lesson 5-3 More Function Composition ACTIVITY 5 continued PLAN My Notes Pacing: 1 class period Chunking the Lesson #1 #2–3 Check Your Understanding #7–11 Check Your Understanding Lesson Practice Learning Targets: • Write the composition of two functions. as column 2 from the first table is mirrored as column 1 in the second table. It costs $350 to mow 4 acres. The order matters when you compose two functions. Bell-Ringer Activity Have students review (without function notation) substituting values from one expression into another and simplifying. Find −4a in terms of b. one for each function. The function y = f (g(x)) is a composition of f and g where g is the inner function and f is the outer function. The tables show information about Jim’s mowing service. Find 2x + 9. or range. MATH TIP 2. Area of Property a (acres) Time to Mow t(a) (hours) Time to Mow t (hours) Cost to Mow c(t) ($) 1 4 4 110 2 8 8 190 3 12 12 270 4 16 16 350 TEACHER to TEACHER b. t(4) 16. if a = b + 12. The main focus of the remainder of this activity is practice with composition of functions. and the second table represents the c function. Use the tables to evaluate each expression. if x = 14. g. All rights reserved. The first table represents the t function. • Evaluate the composition of two functions.073-088_SB_AG2_SE_U01_A05. students will need to recognize the inner and outer parts of the composite function in order to correctly apply the chain rule when they study calculus. 8. f (g (2)) 9 For example. f(g(x)) = (2x − 1)2 = 4x2 − 4x + 1 9. (g o h)(t) can also be written as g(h(t)). 5. Create Representations.073-088_SB_AG2_SE_U01_A05. Given that p(t) = t2 + 4 and q(t) = t + 3. ⋅ ⋅ ⋅ 7–11 Think-Pair-Share. g(f(2)) = 2(2)2 − 1 = 7 f(g(2)) = (2(2) − 1)2 = 9 Activity 5 • Function Composition and Operations 85 CONNECT TO AP In order to decompose a function. use these three functions: © 2015 College Board. Answers 4. Debriefing These items provide an opportunity for students to compose two functions algebraically. (p o q)(t) = (t + 3)2 + 4 = t2 + 6t + 13. (f g)(x) is the product of the functions f and g: (f g)(x) = f(x) g(x). Sample answer: (g o h)(t) represents a composition of the functions g and h in which h is the inner function and g is the outer function. If students need help. (f o g)(x) is a composition of the functions f and g: (f o g)(x) = f(g(x)). Students are not expected to represent either function or the composite functions algebraically. Sample explanation: To find the rule for (p o q)(t). write the equation for ( p  q)(t). I replaced t in the rule for p(t) with the rule for q(t). Explain how you determined your answer. you will identify the “inner” function and the “outer” function that form a composite function. Evaluate each expression. Write each composite function in terms of x. Check Your Understanding Check Your Understanding • • • 2–3 Create Representations. 2 1 2 3 (f  g)(x) = f (g)(x) 3 2 3 4 4 3 4 1 Read the notation as “f of g of x. By contrast.indd Page 85 15/04/14 11:41 AM user-g-w-728 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . Group Presentation. Explain how (f  g)(x) is different from (f g)(x). Predict and Confirm. a. ACTIVITY 5 Continued Lesson 5-3 More Function Composition ACTIVITY 5 continued 3. Students should also gain familiarity with the composition notation f o g and g o f. You may find it necessary to provide further practice with composing different types of functions as they are introduced later in the year. Reason abstractly. 4. All rights reserved. Using f and g from Item 2. 6. All rights reserved. y = f ( g (x)) g(f(x)) = 2x2 − 1 The composite functions that students write in Item 3 will be numeric only. Compare your answers with those from Item 7. complete each table of values to represent the composite functions ( f  g)(x) and ( g  f )(x). a. x (g  f )(x) = g(f(x)) My Notes WRITING MATH 1 4 1 2 The notation (f  g)(x) represents a composition of two functions. 6. 7.. Debriefing Give students an opportunity to work through Item 2 on their own. © 2015 College Board. a. y = g ( f (x)) b. Most of these items use only linear or simple quadratic functions. the function h(x) = f (g(x)) = (2x + 3)2 could be the composition of the inner function g(x) = 2x + 3 and the outer function f (x) = x2. Verify that you composed g and f correctly by evaluating g ( f (2)) and f ( g (2)) using the functions you wrote in Item 8. CONNECT TO AP f (x) = x2 g (x) = 2x − 1 h(x) = 4x − 3 In AP Calculus. For Items 7–11. Examples: Composite f(g(x)) = 5 x − 3 Inner Function g(x) = 5x − 3 Outer Function Composite f(g(x)) = 4x2 − 1 Inner Function g(x) = x2 Outer Function f(x) = 4x − 1 f (x ) = ( x ) Activity 5 • Function Composition and Operations 85 . g ( f (2)) 7 b. x ( f  g)(x) = f(g(x)) b.” Debrief students’ answers to these items to ensure that they understand concepts related to function compositions. What is does the notation ( g  h)(t) represent? What is another way you can write ( g  h)(t)? ⋅ 5.. within a table of values. demonstrate each step in the process of composing the functions numerically. ACTIVITY 5 Continued Check Your Understanding Debrief students’ answers to these items to ensure that they understand concepts related to evaluating the composition of functions.08p) = 0. All rights reserved.864p gives the total cost of the jeans if the discount is applied before the sales tax is added on. Then I substituted 8 for (p o q)(n) and solved for n. if necessary. Use this information for Items 16–18. Given that p(n) = 4n and q(n) = n + 2. Both composite functions have the same rule.8(1. f(g(2)) = 11. f(2) = 11. LESSON 5-3 PRACTICE LESSON 5-3 PRACTICE 14. h(g(x)) = h(2x − 1) = 4(2x − 1) − 3 = 8x − 4 − 3 = 8x − 7 MATH TIP 5. h(x) = g(f(x)) = 3(5x + 1) − 4 = 15x − 1.08p 18. and ( g  f )(2). 15. t(p) = 1. Students who make errors in evaluation may be evaluating the functions in the wrong order. All rights reserved. s(p) = 0. 17. Write the composition (h  g)(x) in terms of x. Write the composition ( g  g)(x) in terms of x. Construct viable arguments. 16. Write a function t(p) that gives the total cost including tax for a pair of jeans priced at p dollars.08(0. Unit 1 • Equations. Sample explanation: The composite function s(t(p)) = 0. No. Sample explanation: First I wrote the rule for (p o q)(n): (p o q)(n) = 4(n + 2) = 4n + 8. 18. Evaluate f (2). 2x − 1.8p 17. Inequalities. You may assign the problems here or use them as a culmination for the activity. For Items 14 and 15. and substituted it for x in the rule for g: 2(2x − 1) − 1. 12. Functions ADAPT Check students’ answers to the Lesson Practice to ensure that they understand how to write and evaluate function compositions. See the Activity Practice for additional problems for this lesson. The jeans at a store are on sale for 20% off. ASSESS b. a.indd Page 86 23/01/14 9:08 PM ehi-6 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . Sample answer: I took the rule for g. Functions © 2015 College Board.. ( f  g)(2). 86 SpringBoard® Mathematics Algebra 2. g(f(2)) = 29 15. and the sales tax rate is 8%. n = 0. for what value of n is (p  q)(n) = 8? Explain how you determined your answer. Does it matter whether the sales clerk applies the sale discount first or adds on the sales tax first to find the total cost? Use compositions of the functions s and t to support your answer.8p) = 0. Write a function s(p) that gives the sale price of a pair of jeans regularly priced at p dollars. Lesson 5-3 More Function Composition ACTIVITY 5 continued My Notes 10. Unit 1 • Equations. so both give the same total cost for the jeans: 0. (g  g)(x) = g(2x − 1) = 2(2x − 1) − 1 = 4x − 3 Check Your Understanding Students’ answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the lesson concepts and their ability to apply their learning. The composite function t(s(p)) = 1. Evaluate h( g (3)). Then I simplified the resulting expression: 2(2x − 1) −1 = 4x − 3. Write the composite functions h(x) = g ( f (x)) and k(x) = f ( g (x)).073-088_SB_AG2_SE_U01_A05. 86 SpringBoard® Mathematics Algebra 2. g(2) = 2. g(g(2)) = g(2(2) − 1) = g(3) = 2(3) − 1 = 5 As shown in Item 11. use the following functions: • • f (x) = 5x + 1 g (x) = 3x − 4 14.60. 13. and remind them to begin in the innermost parentheses and work outward. a.864($25) = $21.. © 2015 College Board. the inner and outer functions that form a composite function can be the same function. (p o q)(n) = 4n + 8 8 = 4n + 8 0 = 4n 0=n . 12. Explain how you found the rule for the composition ( g  g)(x) in Item 11b. Have them write the composition without using the notation o. Evaluate g ( g (2)).864p gives the total cost of the jeans if the sales tax is added on before the discount is applied. 13. h(g(3)) = h(2(3) − 1) = h(5) = 4(5) − 3 = 17 Answers b. A customer wants to buy a pair of jeans regularly priced at $25. g(2). Inequalities. k(x) = f(g(x)) = 5(3x − 4) + 1 = 15x − 19 16. 11. ( g ÷ h)(x ) = 3x − −3 ⋅ ⋅ 9.. C 11. g (x) = 3 − x. (f − g)(x) = 5x + 2 − (3 − x) = 6x − 1 4. The mowers consume 2. range: dollars 15. h(x) ≠ 0 50 yd continued 11. and tell what it represents in this situation. Given that p(n) = 4n2 + 4n − 6 and q(n) = n2 − 5n + 8. For $600. ( f + g)(x) 2. ( f − g )(x) ACTIVITY 5 ⋅ ⋅ 15. d.5h. (l w)(x) = (x + 60)(x + 50) = x2 + 110x + 3000 d. Make a conjecture about whether subtraction of functions is commutative. 40 D. © 2015 College Board. 1. domain: hours. (h − f )(x) 5. range: gallons 14. 12. The range is the outer function’s range. 14. where l(x) is the new length in yards and x is the increase in length in yards b. (f − g)(x) ≠ (g − f)(x). c. w(x) = x + 50. 42 Activity 5 • Function Composition and Operations 87 ACTIVITY PRACTICE 1.75(12) = 105.. (f + g)(x) = 5x + 2 + (3 − x) = 4x + 5 2.5 gallons of gasoline each hour. The domain of this new function is the inner function’s domain. ACTIVITY 5 Continued Function Composition and Operations New from Old Write your answers on notebook paper. (h + g)(x) 4. ( f g)(x) 6. the cost in dollars of renting both a car and a hotel room for d days.indd Page 87 05/12/13 4:07 AM s-059 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . A. a. What does (l w)(x) represent in this situation? Write and simplify the equation for (l w)(x). Skate Park © 2015 College Board. It will cost $105 to run the lawn mowers for 12 hours. The cost in dollars of renting a hotel room for d days is given by h(d) = 74d. c(g(h)) = 3. 38 C. Identify the units of the domain and range. 10.5g.99. Gasoline costs $3. ⋅ Jim wants to calculate the cost of running his lawn mowers. (h + g)(x) = x − 3 + 3 − x = 0 3. and h(x) = x − 3 to answer Items 1–8. All rights reserved.073-088_SB_AG2_SE_U01_A05. Use composition of functions to create a function for the cost c in dollars of gasoline to mow h hours. Subtraction of functions is not commutative. Lesson 5-1 Use f (x) = 5x + 2. ACTIVITY 5 PRACTICE 3. a. 16. What is the slope of the composite function. ( g ÷ h)(x). you can rent both a car and a hotel room for about 6 days. The cost in dollars of renting a car for d days is given by c(d ) = 22d + 25. All rights reserved. Find each function and simplify the function rule. Identify the units of the domain and range. Show your work. the new area of the skate park in square yards. x yd a. A rectangular skate park is 60 yards long and 50 yards wide. g(h) = 2. x ≠ 3 3− x x = −1. Show your work. 17. Write a function w(x) that gives the new width of the skate park in terms of x. Write a function c ( g) for the cost c in dollars for g gallons of gasoline. c(g) = 3. a. 16. b. 13. Write a function g (h) that gives the number of gallons g that the mowers will use in h hours. Plans call for increasing both the length and the width of the park by x yards. Write a function l (x) that gives the new length of the skate park in terms of x.50 per gallon. Use the composite function in Item 15 to determine the cost of gasoline to mow 12 hours. 8. l(x) = x + 60. (f − g)(x) = −2x + 2 and (g − f)(x) = 2x − 2. where w(x) is the new width in yards and x is the increase in width in yards c. g(x) ≠ 0 9. ( f ÷ g)(x). x ≠ 3 8.5(2. range: dollars.5h) = 8. What does (c + h)(d ) represent in this situation? Write and simplify the equation for (c + h)(d). domain: hours. The new area of the skate park will be 3575 square yards if its length and width are each increased by 5 yards. Then explain how the domain and range of the composite function are related to the domain and range of g (h) and c (g). 17. c(g(12)) = 8. ( g h)(x) ⋅ ⋅ 7. (l w)(5) = 3575. Identify the units of the domain and range.75h. x yd 60 yd 12.75. the cost in dollars of running the lawn mowers for 1 hour ⋅ ⋅ Activity 5 • Function Composition and Operations 87 . b. ( f ÷ g )(x ) = 5x + 2 . (c + h)(d) = 22d + 25 + 74d = 96d + 25 b. 13. d ≈ 5. (h − f)(x) = x − 3 − (5x + 2) = −4x − 5 5. Note any values that must be excluded from the domain. find (p − q)(3). Give an example that supports your answer. Sample example: Given that f(x) = 2x + 6 and g(x) = 4x + 4. Find (l w)(5). 26 B. For what value of d is (c + h)(d ) = 600? What does this value of d represent in this situation? Lesson 5-2 8. So. (f g)(x) = (5x + 2)(3 − x) = −5x2 + 13x + 6 6. (g h)(x) = (3 − x)(x − 3) = −x2 + 6x − 9 7. and what does it represent in this situation? 10. domain: gallons. 22.75 gives the total cost of a television set if the discount is applied before the sales tax is added on. The sales tax rate is 7. 88 SpringBoard® Mathematics Algebra 2. All rights reserved. Yes. r (t) = 2t + 1. Given that (r  s)( t ) = 2t + 11. and the rule for 9 9 (c o k)( f ) simplifies to 5 f + 1205. The range of the function d(t) is 0 ≤ d ≤ 4. a. Sample explanation: The rule for (k o c)( f ) simplifies to 5 f + 2297 . In Item 29. r (t) = 2t + 5. s(t) = t + 1 32. ( h  f )(x) 27. v(d(4)) = 324. Based on this information.075(p − 50) = 1. 31. Does it matter whether the sales clerk applies the sale discount first or adds on the sales tax first to find the total cost? Use compositions of the functions s and t to support your answer. (f o g)(x) = (x + 5)2 = x2 + 10x + 25 24. which could be the functions r and s? A. What is the composite function y = f ( g (x))? 35. ( g  h )(x) 28. /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . ( f  h )(x) 26. b. Write a function d(t) that gives the depth in feet of the water in the pool after t hours. (f o h)(x) = (4x − 6)2 = 16x2 − 48x + 36 26. r (t) = t + 5. ( h  g )(x) The function c( f ) = 5 ( f − 32) converts a temperature 9 f in degrees Fahrenheit to degrees Celsius. Yes.indd Page 88 05/12/13 4:07 AM s-059 ACTIVITY 5 Continued ADDITIONAL PRACTICE If students need more practice on the concepts in this activity. 33. 30. Sample explanation: The composite function s(t(p)) = 1. The function k(c) = c + 273 converts a temperature c in degrees Celsius to units called kelvins.5%. (g  f )(x) 25. 648 cubic feet 23.073-088_SB_AG2_SE_U01_A05. Write a composite function that can be used to convert a temperature in degrees Fahrenheit to kelvins. Unit 1 • Equations. f ( g (x)) = 4 − 6x2 C. g(f(−1)) = 6 34. and what does it represent in this situation? 22. does it matter whether you wrote (c  k)( f ) or (k  c)( f )? Explain..25 for the television set regularly priced at $800. What is v(d(4)). ( f  g )(x) 24. and h(x) = 4x − 6 to answer Items 23–28. f(g(−1)) = 22. After 4 hours. s(t) = t + 5 D.075p c. d(t) = 0. s(p) = p − 50 b. s(t) = 2t + 5 B. 23. Inequalities. A store is discounting all of its television sets by $50 for an after-Thanksgiving sale. Inequalities. 31. What is the composite function y = g( f (x))? MATHEMATICAL PRACTICES Model with Mathematics 36. r (t) = t + 1. 20. f ( g (x)) = 12 − 12x4 Use f (x) = 5x + 2 and g (x) = 3 − x to answer Items 33–35. v(d(t)) = 162(0. f ( g (x)) = 3(4 − 2x)2 D. . t(p) = 1. c. B 33. Functions SpringBoard® Mathematics Algebra 2. Find each function and simplify the function rule. This rule gives a total cost of $810 for the television set regularly priced at $800. Unit 1 • Equations. 29. A customer wants to buy a television regularly priced at $800. Write a function t(p) that gives the total cost including tax for a television priced at p dollars.5t 19. (g o f)(x) = x2 + 5 25.075p − 50 gives the total cost of a television set if the sales tax is added on before the discount is applied. The composite function t(s(p)) = 1. What is the composition f  g if f (x) = 4 − 2x and g (x) = 3x2? A. g(f(x)) = 1 − 5x 36. f(g(x)) = 17 − 5x 35. Once water begins to be pumped into the pool.. what is the greatest volume of water the pool can hold? Lesson 5-3 Use f (x) = x2. 18.075p − 53. so it matters which one you use. 88 Function Composition and Operations New from Old An empty swimming pool is shaped like a rectangular prism with a length of 18 feet and a width of 9 feet. and tell what this function represents in this situation. see the Teacher Resources for additional practice problems. Functions © 2015 College Board. ACTIVITY 5 continued © 2015 College Board. the depth of the water increases at a rate of 0. s(t) = 2t + 1 C. 19. Write the equation of the composite function v(d(t)). (h o g)(x) = 4(x + 5) − 6 = 4x + 14 29. All rights reserved. (g o h)(x) = 4x − 6 + 5 = 4x − 1 28. 21.5t) = 81t. the volume in cubic feet of water in the pool after t hours 21. What is the value of f ( g (−1)) and g ( f (−1))? 34. v(d) = 162d 20. Write a function s(p) that gives the sale price of a television regularly priced at p dollars. f ( g (x)) = 12x2 − 6x3 B. a. 18. (k o c)( f ) = 5 ( f − 32) + 273 9 30. C 32. Write a function v (d) that gives the volume in cubic feet of the water in the pool when the depth of the water is d feet. g (x) = x + 5. (h o f)(x) = 4x2 − 6 27. 9 9 These two functions are not the same. the volume of water in the pool will be 324 cubic feet.5 foot per hour. This rule gives a total cost of $806. Attend to precision. They also use composition of functions to determine if functions are inverses of one another.B. Write the answer equation from Item 3 using function notation. Activity Standards Focus My Notes Lesson 6-1 Green’s Grass Guaranteed charges businesses a flat fee of $30 plus $80 per acre for lawn mowing.B. $60 6 Pacing: 1 class period Chunking the Lesson #1 #2–6 #7–10 Example A Check Your Understanding Lesson Practice TEACH Bell-Ringer Activity As preparation for finding inverses of functions. Create Representations. Throughout this activity. Think-Pair-Share.4b(+) Verify by composition that one function is the inverse of another. range: dollars 1 Think-Pair-Share. HSF-BF. based on the amount of money a customer is willing to spend. Use the function to determine what part of an acre Jim could mow for each weekly fee. students form the inverse of the cost function by solving for A. Jim needs a function for calculating the maximum acreage that he can mow. [Note: HSF-BF. However. Work Backward. Debriefing Item 2 sets the stage for learning about the properties of inverse functions. Group Presentation. c. All rights reserved.089-098_SB_AG2_SE_U01_A06.B. the math concept and vocabulary term inverse function are not used at this point in the activity. What are the units of the domain and range of the cost function in Item 1? domain: acres. Quickwrite. SUGGESTED LEARNING STRATEGIES: Questioning the Text.4a Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. Solve the function equation from Item 1 for A in terms of C. $110 © 2015 College Board. 3 of an acre 8 b. Work Backward Item 1 gives students an opportunity to solve the cost equation repeatedly for a given dollar amount. Look for a Pattern.. It can be written C = 30 + 80A. PLAN Work on Items 1–10 with your group. G(C ) = C − 30 80 Activity 6 • Inverse Functions 89 Common Core State Standards for Activity 6 HSF-BF. Jim Green needs to determine the size of the yard he could mow for a particular weekly fee. Guided In Activity 6. emphasize how the domain of a function is related to the range of the inverse of the function and vice versa. Make use of structure. 2. In Items 3 and 4. Activity 6 • Inverse Functions 89 . students find inverse functions. y = 5x + 8 for x in terms of y  y −8 x =  5   2. ACTIVITY Inverse Functions ACTIVITY 6 Old from New Lesson 6-1 Finding Inverse Functions Learning Targets: • Find the inverse of a function.indd Page 89 16/01/15 3:24 PM ehi-6 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . C = 30 + 80A A = C − 30 80 4. The cost function F is C = F(A).B. All rights reserved. 1 acre © 2015 College Board. $80 5 of an acre 8 DISCUSSION GROUP TIP As you work in groups. For residential customers who may have a more limited budget. Debriefing. 3a + 2b = 7 for a in terms of b  a = 7 − 2b  3   3 3. 3.] 2–6 Work Backward. HSF-BF. students should be fluent in solving literal equations. • Write the inverse using the proper notation. as students will eventually learn that the domain of a function is the range of the inverse of the function and the range of a function is the domain of the inverse of the function.4 Find inverse functions. a. review the problem scenario carefully and explore together the information provided and how to use it to create a potential solution. Quickwrite. Students use this process later to write a rule for A acres as a function of C dollars. Discuss your understanding of the problem and ask peers or your teacher to clarify any areas that are not clear.. where G is the acreage function. where C is the cost to mow A acres. Note Taking 1.4b is introduced in this activity but is also addressed in higher level mathematics courses. c − 9d = 12 for c in terms of d 4 [c = 12d + 16] To make a profit and still charge a fair price. Have students solve each equation for the indicated variable: 1. . a. G(60) 3 8 b. Use the appropriate functions to evaluate each expression. Interpret the meaning of each expression and its corresponding value in Item 7. F(G(60)) Developing Math Language 60 Discuss that a function relates each element of a set with exactly one element of another set. range: acres 6. it will cost $190 to mow 2 acres. which are sometimes referred to as “input values. meaning that the domain of F is the range of G and that the domain of G is the range of F. Reason abstractly.089-098_SB_AG2_SE_U01_A06.” c.” The range is the set of values that actually come out of a function. it costs $60 to mow the number of acres you can mow for $60. Attend to precision. you can mow 2 acres for the cost of mowing 2 acres. 7. ACTIVITY 6 Continued . you can mow 38 of an acre for $60. The domains and ranges switch in these two functions. In part a. Lesson 6-1 Finding Inverse Functions ACTIVITY 6 continued My Notes 5. c. Later. The domain of the cost function C = F(A) is the range of the acreage function G. Students are first asked to recognize this relationship prior to determining that these are inverse functions.indd Page 90 05/12/13 1:51 PM user-s068a 123 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . Inequalities. G(F(2)) 2 8. Unit 1 • Equations. In part c. 90 90 SpringBoard® Mathematics Algebra 2. b. All rights reserved. Unit 1 • Equations. they will discover that the operations performed by the inner function in each composition are then undone by the operations performed by the outer function. 2–6 (continued) In Items 5 and 6. students are asked to name the domain and range of the function G and then to identify the relationship between the domains and ranges of the two functions. which are sometimes referred to as “output values. In part b. Functions SpringBoard® Mathematics Algebra 2. The domain is the set of values that can go into a function. Inequalities. All rights reserved. d. Sample explanation: The function C was rewritten to express A in terms of C. How are the domain and range of F(A) related to those of G(C)? Use your response to Item 3 to explain why the relationship exists. Be sure to include units in your explanation. Functions © 2015 College Board. The range of C is the domain of G. thereby forming the function G. a. This is the core concept of inverse functions that all students must understand after finishing this activity. © 2015 College Board. F(2) 190 d. What are the units of the domain and range of the function G? domain: dollars.. In part d. interchange the x and y variables and then solve for y. h(x) = 2 x − 5 3 h−1( x ) = 3 x + 15 2 Example A Note Taking Work through this example with your students step-by-step to demonstrate how to find the inverse of a function algebraically. See the Teacher Resources at SpringBoard Digital for a student page for this mini-lesson. Step 1: Let y = f(x). a mini-lesson is available to provide practice. and. Step 2: Interchange the x and y variables. Try These A Find the inverse of each function. a. Activity 6 • Inverse Functions 91 . Make sure to connect back to the work they did on the previous two pages of this activity. the range of a function is the domain of its inverse. undoing each other. this key concept of inverse functions is presented without actually mentioning the term inverse functions. WRITING MATH Example A © 2015 College Board.indd Page 91 15/04/14 11:43 AM user-g-w-728 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . What do the answers in Items 7–9 suggest about F and G? Sample answer: They undo each other. you can also write two equivalent composite functions: y = 2x − 4 x = 2y − 4 x + 4 = 2y y = x+4 2 x + 4 −1 f (x ) = 2 f g=x g f=x MATH TIP The –1 superscript in the function notation f −1 is not an exponent. MATH TERMS Functions f and g are inverse functions if and only if f(g(x)) = x for all x in the domain of g and g(f(x)) = x for all x in the domain of f.. in fact.. 10. g–1(x) = 4x − 12 d. The function notation f –1 denotes the inverse of function f and is read “f inverse. g (x ) = 1 (x + 12) 4 However. for any number n. and f −1 ≠ 1 when referring to f functions. b. These functions undo each other. Likewise. In general. what is the result when you evaluate F(G(x)) and G(F(x))? You get x. To find the inverse of a function algebraically.” Item 6 showed that the domain of a function is the range of its inverse. ACTIVITY 6 Continued Lesson 6-1 Finding Inverse Functions ACTIVITY 6 continued My Notes 9. Step 3: Solve for y. Debriefing Students need to interpret the meaning of F(G(60)) = 60 and G(F(2)) = 2 to understand the effect that composition of these functions has upon domain values. Find the inverse of the function f(x) = 2x − 4. Let y = f −1(x). © 2015 College Board. Item 8 asks students to attend to precision by using units to describe the answers. j(x ) = 3x − 2 6 j −1( x ) = 2 x + 2 3 Activity 6 • Inverse Functions 91 MINI-LESSON: Inverse vs. If f and g are inverse functions. the expression n–1 is the multiplicative inverse. While the verbal explanations of the composition may seem self-evident. and g( f(x)) = x for all x in the domain of f. Think-Pair-Share. Reciprocal Notation If students need additional help understanding the notation of f−1. of n. Solution: f −1 (x ) = x + 4 2 Step 4: f −1( x ) = −x + 8 3 c. they emphasize that the functions are. relating the domain and range of inverse functions. All rights reserved. All rights reserved. Two functions f and g are inverse functions if and only if: f(g(x)) = x for all x in the domain of g. or reciprocal.089-098_SB_AG2_SE_U01_A06. once again. Quickwrite. f(x) = −3x + 8 7–10 Create Representations. 18. Therefore. a. range: y ≥ 3. range: minutes. 20 15. h(x ) = x − 20 4 Students’ answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the lesson concepts and their ability to apply their learning. domain: miles. Demonstrate by evaluating a function for a given value. and then substituting the result into the inverse function. f (x) = 3x + 6 ASSESS 21. and the range of the function is y ≤ −2. the elevation is 500 m. range: miles. g−1(x) = −2x 21. Debrief students’ answers to these items to ensure that they understand concepts related to inverse functions. T ≥ 0}. Critique the reasoning of others.. explain how you can find f (g(20)) without knowing the equations for f and g. Use the function T = 50 − H . ADAPT Check students’ answers to the Lesson Practice to ensure that they understand inverse functions. If students struggle with the concept of inverse functions. Inequalities. What is G(25)? What does G(25) mean? 18. When the temperature is 25 degrees Celsius.5) = 25. {T | T ∈ . Unit 1 • Equations. j(x) = 5(x − 1) See the Activity Practice for additional problems for this lesson. a. d. 13. T ≥ 0} d. D ≥ 0} c. 10 T gives the number of miles Mariana can run in T minutes. ( ) c. The inverse function has “undone” the original function. Because f and g are inverse functions. 19. What is F(500)? What does F(500) mean? 16. b.indd Page 92 05/12/13 1:51 PM user-s068a 123 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . Just as inverse operations “undo” each other. D(T ) = 6 T = 1 T 60 10 b. {T | T ∈ . ∞). the answer is the original value. Is the student correct? Justify your answer. ∞). Label this function G. then h(j(x)) should equal x. which means that f(g(x)) = x. Unit 1 • Equations. Given that f and g are inverse functions. 25. The domain of a function is x ≥ 3. f −1 (x ) = 1 x − 2 3 20. 16. Functions © 2015 College Board. Check Your Understanding . The domain of a function is the range of its inverse. 500. 25 minutes. ACTIVITY 6 Continued continued My Notes Check Your Understanding Answers 11. j−1 (x ) = 1 x + 1 5 SpringBoard® Mathematics Algebra 2. What are the units of the domain and range of D(T)? Write the domain and range in both interval notation and set notation. 20. f(g(20)) = 20. {D | D ∈ . Find H in terms of T. Mariana’s average running speed is 6 miles per hour. ∞). h−1(x) = 4x + 20 22. 12. No. the temperature is 25 degrees Celsius. 92 92 SpringBoard® Mathematics Algebra 2. How many minutes will it take Mariana to run 2. All rights reserved. You may assign the problems here or use them as a culmination for the activity. 14. 12.5 miles? Explain how you can find the answer by using the inverse of D(T). 11. relate it to inverse operations. Evaluate T(D) for D = 2. Write a function D(T) that gives the distance in miles Mariana covers when running for T minutes. domain: [0. All rights reserved. Functions LESSON 6-1 PRACTICE 15. Because D(T ) = 1 T. 13. T(D) = 10D gives the number of minutes it takes Mariana to run D miles. {D | D ∈ . h(x) and j(x) are not inverse functions. Instead.5: T(2. Sample justification: If h(x) and j(x) are inverse functions. domain: minutes.089-098_SB_AG2_SE_U01_A06. Inequalities. Are the functions F and G inverses of each other? Explain. Find the inverse of each function. The function T = F(H) estimates the temperature (degrees Celsius) on a mountain given the height (in meters) above sea level. ∞). LESSON 6-1 PRACTICE Look for and make use of structure. G(T) = H = 1000 − 20T 17. G(F(x)) = F(G(x)) = x 19. What are the units of the domain and range of the inverse of D(T)? Write the domain and range of the inverse function in both interval notation and set notation. Domain: x ≤ −2. so do inverse functions. D ≥ 0}. Sample explanation: Let T(D) represent the inverse of D(T). When the elevation is 500 m. range: [0. they undo each other. domain: [0. What are the domain and range of the inverse function? Explain your answer. range: [0. h(j(x)) = 3(−3x) = −9x.. Yes.5) = 10(2. A student claims that h(x) = 3x and j(x) = −3x are inverse functions. 17. Lesson 6-1 Finding Inverse Functions ACTIVITY 6 © 2015 College Board. and the range of a function is the domain of its inverse. 14. g (x ) = − 1 x 2 22. a. Create Representations. © 2015 College Board. 2 Step 1: Compose f and f −1. 2 =x Solution: f(x) = 2x − 4 and f −1 (x ) = x + 4 are inverse functions. Discussion Groups. g (x ) = 1 x + 3 2 Activity 6 • Inverse Functions ( ) ( ) 93 Activity 6 • Inverse Functions 93 . Show your work.  f −1 ( x ) = x − 2  1. f(x) = 5x − 7  5   −1 1 3. Step 5: Substitute f into f −1. ACTIVITY 6 Continued Lesson 6-2 Graphs of Inverse Functions ACTIVITY 6 continued Lesson 6-2 PLAN Learning Targets: composition of functions to determine if functions are inverses of • Use each other. f(x) = 4x − 14 b. Try These A Answers a. f(x) = 3x + 2  3   f −1 ( x ) = x + 7  2. 2 Try These A Make use of structure. Debriefing Emphasize that both compositions are necessary to verify that two functions are inverses. you use logical reasoning to show that it is true. f(f −1(x)) = 2( f −1(x)) − 4 Step 2: Substitute f −1 into f. ACADEMIC VOCABULARY When you prove a statement. Example A Use the definition of inverse functions to prove that f(x) = 2x − 4 x+4 −1 and f (x ) = are inverse functions. Step 3: Simplify. All rights reserved. f ( x ) = x + 4 [f ( x ) = 2(x − 4)] 2 Example A Note Taking. Debriefing. Self Revision/Peer Revision You can use the definition of inverse functions to show that two functions are inverses of each other. = 2( x + 4 ) − 4 2 =x+4−4 =x f (x ) + 4 −1 f (f(x)) = 2 = 2x − 4 + 4 2 TEACH Bell-Ringer Activity Have students find the inverse of each function. g−1(x) = 2x − 6 g ( g −1 (x )) = 1 (2 x − 6) + 3 = x 2 g −1 ( g (x )) = 2 1 x + 3 − 6 = x 2 = 2x Step 6: Simplify. f −1 (x ) = x + 14 4 f −1 ( f (x )) = 4 x − 14 + 14 = x 4 −1 x + 14 − 14 = x f ( f (x )) = 4 4 b.. Step 4: Compose f −1 and f.089-098_SB_AG2_SE_U01_A06. Quickwrite.indd Page 93 15/04/14 11:45 AM user-g-w-728 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . © 2015 College Board. My Notes Pacing: 1 class period Chunking the Lesson Example A Check Your Understanding #4–6 #7–8 Check Your Understanding Lesson Practice SUGGESTED LEARNING STRATEGIES: Think-Pair-Share.. Then use the definition to prove the functions are inverses. • Graph inverse functions and identify the symmetry. Group Presentation. Find the inverse of the function. All rights reserved. Sample explanation: If the domain of f(x) = 2x − 4 is {x | x ∈ . You can use the relationship between the domain and range of a function and its inverse to graph the inverse of a function. If both compositions are equal to x. Functions SpringBoard® Mathematics Algebra 2.. 4. x ≥ 2}.indd Page 94 15/04/14 11:46 AM user-g-w-728 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . then (y. If (x. If students are not simply exchanging the ordered pairs from the first table when they graph the inverse. Domain: {x | x ∈ . y) values are reversed.089-098_SB_AG2_SE_U01_A06. Debriefing Students should recognize that switching the columns from Item 4 provides a table for the inverse function in Item 5 because the (x. Explain how to prove that two functions h(x) and j(x) are inverse functions. the functions are not inverses. x ≥ 0}. 3. could q(t) be the inverse of p(t)? Explain your answer.. Find h(j(x)) and j(h(x)). and the elements of the range of f(x) are the elements of the domain of f −1(x). Based on this information. y) is a point on the graph of a given function. Check Your Understanding . What would be the domain and range in set notation of f −1(x)? Explain your answer. See Item 4 for graph. y ≥ 2}. ∞). 94 94 y x x f −1(x) −8 −2 −2 0 7 3 10 4 SpringBoard® Mathematics Algebra 2. The range of q(t) is (−∞. ACTIVITY 6 Continued continued My Notes Check Your Understanding Answers 1. 4–6 Think-Pair-Share. y ≥ 0}. The domain of p(t) is [0. range: {y | y ∈ . Inequalities. Lesson 6-2 Graphs of Inverse Functions ACTIVITY 6 © 2015 College Board. Use the table in Item 4 to make a table of values for the inverse of f. –10 –8 –6 –4 –2 –2 f –1(x) = 2 4 6 (x + 2) 3 x 8 10 –4 –6 –8 –10 5. No. Inequalities. 1. All rights reserved. then its range is {y | y ∈ . x ≥ 2}. Then graph the inverse on the same coordinate axes. Debrief students’ answers to these items to ensure that they understand concepts related to proving two functions are inverse relations. Because the domain of p(t) is not the same as the range of q(t). Suppose that the domain of f(x) = 2x − 4 in Example A was restricted to {x | x ∈ R. bring this point out during debriefing. Complete the table of values for f(x) = 3x − 2. Functions © 2015 College Board. Students who find the inverse algebraically and then use that function to complete the table may not fully understand this property of inverse functions. If q(t) were the inverse of p(t). Unit 1 • Equations. Unit 1 • Equations. f(x) −2 −8 0 −2 6 3 7 4 4 10 2 10 f(x) = 3x – 2 8 Graph shows answers for Items 4–6. 2. then h(x) and j(x) are inverse functions. then the domain of p(t) would be the range of q(t). Use the values to graph the function on the coordinate axes below. x) is a point on the graph of its inverse. Construct viable arguments. Create Representations. 0]. The elements in the domain of f(x) are the elements of the range of f −1(x). 3. All rights reserved. 2. Inverse functions will be readdressed throughout Algebra 2 and will be in concepts in precalculus and calculus. which is not a function. The slopes are reciprocals or multiplicative inverses. Check Your Understanding Debrief students’ answers to these items to ensure that they understand concepts related to visual representations of inverse functions. or both. Graph the function f(x) = 2 and its inverse on the same coordinate plane. 0). Find the inverse of f(x) = x − 4. They should understand that parallel lines don’t intersect. What is the relationship between the slope of a nonhorizontal linear function and the slope of its inverse function? Explain your reasoning. students may find the inverse of any line with a slope of 1. Describe any symmetry among the three graphs. Debriefing In these items students will see that not all inverses intersect along the line y = x. so the slope of the x − x1 inverse function is 2 . therefore. f −1(x) = x + 4 My Notes CONNECT TO GEOMETRY Geometric figures in the coordinate plane can have symmetry about a point. y1) and (x2. students may simply use inverse operations. The lines are symmetric about the line y = x. then the graph of the function includes the point (c. Sample explanation: If c is an x-intercept of a function. Describe any symmetry that you see on the graph in Item 8a. Graph f(x) = x − 4. x2 − x1 The graph of the inverse of the function includes the points (y1.089-098_SB_AG2_SE_U01_A06. 10. The slopes xx22− −xx11 yy22− −yy11 must be reciprocals of each other. The two lines are symmetric about the line y = x. Model with mathematics. If students state the inverse of f(x) = 2x + 1 as f −1 (x ) = x − 1. The x-intercepts of a function are the same as the y-intercepts of its inverse. Ask students to generalize as to which functions intersect the line. x2). The inverse is the vertical line x = 2.indd Page 95 15/04/14 11:47 AM user-g-w-728 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . y1) and (x2. then y − y1 the slope of the function is 2 . Answers 9. x1) and (y2. ACTIVITY 6 Continued Lesson 6-2 Graphs of Inverse Functions ACTIVITY 6 continued 6. b is the x-intercept of the inverse of the function. MATH TIP Recall that the slope of a linear y − y1 function is equal to 2 . work 2 through additional examples as needed to reinforce the process.and y-intercepts of its inverse? Explain your reasoning. Likewise. For example. Activity 6 • Inverse Functions 95 10. 8. a. and the dotted line y = x on the coordinate axes. ⋅ Activity 6 • Inverse Functions 95 . its inverse f −1(x) from Item 7.. where x 2 − x1 (x1. Is the inverse of f(x) = 2 a function? Explain your answer. 7.and y-intercepts of a function and the x. then the graph of the function includes the point (0. any line with a slope of 1 will have an inverse parallel to the line y = x.. y f –1(x) = x + 4 10 6 Some students may have misconceptions about finding inverses. All rights reserved. c). c is a y-intercept of the inverse of the function. See Item 4 for graph. 0) and the graph of the inverse includes the point (0. 11. if b is the y-intercept of a function. Sample explanation: If (x1. © 2015 College Board. but they do not intersect because they are parallel. As an algebraic justification. No. Check Your Understanding 9. Show the graph of y = x as a dotted line on the coordinate axes in Item 4. they will find that f −1(x) = x − b. TEACHER to TEACHER 8 –10 –8 –6 –4 7–8 Create Representations. The y-intercept of a function is the same as the x-intercept of its inverse. y2) are two points on the function’s graph. In general. So. The y2 − y1 product of the slope of the function and the slope of its inverse is yy22− −yy11 xx22− −xx11 = =11. y  inverse of f(x) = 2 4 f(x) = 2 2 2 –2 4 x –2 11. y2) are 2 points on the graph of a nonhorizontal linear function. if f (x) = x + b. What is the relationship between the x. So. a line. 4 f(x) = x – 4 2 –2 –2 2 4 6 8 10 x –4 –6 –8 –10 b. All rights reserved. © 2015 College Board. b) and the graph of the inverse includes the point (b. 13. ) 13. A function and its inverse always intersect. Use reflective devices or paper-folding techniques to support students who have difficulty recognizing symmetry about y = x or reflecting graphs across y = x. −3 −1 f ( f ( x )) = 6 − 3 x − 6 = x −3 b. 4 2 –4 –2 –2 x 96 SpringBoard® Mathematics Algebra 2. a. In Items 16 and 17. ADAPT The composition of a function and Check students’ answers to the Lesson its inverse equals x. My Notes 12. Unit 1 • Equations. y  y  16. b. 17. state whether each statement is true or false. Reason abstractly and quantitatively. f −1 (x ) = x − 6 . ( LESSON 6-2 PRACTICE In Items 12–14. −3 f −1 ( f ( x )) = 6 − 3x − 6 = x . h −1(h(x)) = −(−x + 5) + 5 = x. Use the definition of inverse functions to verify that the two functions are inverses. Inequalities. 96 x © 2015 College Board. ACTIVITY 6 Continued Lesson 6-2 Graphs of Inverse Functions ACTIVITY 6 ASSESS continued Students’ answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the lesson concepts and their ability to apply their learning. All rights reserved. Explain your reasoning. y  16. Summarize the relationship between a function and its inverse by listing at least three statements that must be true if two functions are inverses of each other. Functions inverse 18. Functions © 2015 College Board.089-098_SB_AG2_SE_U01_A06. g −1(x) = x − 2. h −1(x) = −x + 5. g(x) = x + 2 14. graph the inverse of each function shown on the coordinate plane.. 2 –2 2 –2 –2 4 –4 –4 –2 SpringBoard® Mathematics Algebra 2. as shown in Item 14. False. .indd Page 96 05/12/13 1:51 PM user-s068a 123 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . False. h(x) = −x + 5 15. 2 4 18.. g −1(g(x)) = x + 2 − 2 = x. The rule for a function cannot equal the rule for its inverse. Inequalities. Unit 1 • Equations. Sample answer: The domain and range of a function and those of its –4 inverse are interchanged. Using your results in Items 12–14. –4 4 4 2 2 2 –2 4 x –4 –4 4 2 4 x –2 –4 inverse y  17. Item 13 provides a counterexample because the two lines are parallel. g(g −1(x)) = x − 2 + 2 = x 14. f(x) = 6 − 3x See the Activity Practice for additional problems for this lesson. The rule for a function can be the same as the rule for its inverse. Practice to ensure that they understand the properties of inverse functions and their graphs. h(h −1(x)) = −(−x + 5) + 5 = x 15. The graphs of a function and its inverse are symmetric about the line y = x. a. LESSON 6-2 PRACTICE 12. A function and its inverse may or may not intersect. You may assign the problems here or use them as a culmination for the activity. find the inverse of each function. All rights reserved. Express regularity in repeated reasoning. D 10. g −1(g(x)) = 4(0. What is the inverse of the function p(t) = 6t + 8? A. g(x) = −x − 3 8. ∞).indd Page 97 05/12/13 1:51 PM user-s068a 123 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . What are the units of the domain and range of G(c)? 5. range: acres 5. Use this function for Items 1–7. h −1(x) = 6x + 8 d. a. range: dollars b. f −1 (x ) = − 1 x + 1. 12. f(x) = 2(x − 4). f(x) = x + 3. 12. 4 8 12 16 20 x ) f −1 ( f (x )) = − 1 (−3x + 3) + 1 = x 3 b. g(x) = 2x − 6 2 c. Use the definition of inverse to determine whether or not each pair of functions are inverses. f ( g ( x )) = 2 x − 6 + 3 = x 2 x g ( f ( x )) = 2 + 3 − 6 = x 2 They are inverses. f −1(1) = −5 B. a = c − 50 . 4. Sample explanation: The functions are inverses because F (G(x )) = 60 x − 50 + 50 = x 60 60 and G(F (x )) = x + 50 − 50 = x. 8 Then I replaced a with G(m). f(x) = 2x − 10 b. h(x ) = 1 (x − 8) 6 d. 3 −1 f ( f ( x )) = −3 − 1 x + 1 + 3 = x. a. What is the inverse of F(a)? Label this function G. range: (50. f ( g ( x )) = 5 x + 3 − 3 = x + 12 5 They are not inverses. range: (0. domain: (50. g −1(x) = 4x − 5 c. Sample explanation: I evaluated G(c) for c = 200. G(3) = 8(3) = 24. f(x) = 5x − 3. a. j(x) = −5x + 2 9. F(40) is the cost in dollars of mowing 40 acres. f −1(5) = 1 C. What are the units of the domain and range of F(a)? b.6 = x. © 2015 College Board. j−1 (x ) = −x + 2 5 9. p−1(t) = −6t − 8 −t + 8 B. p−1 (t ) = t − 8 6 D. 10. and tell how you determined the rule for the inverse function. ( ) ( ) ( ) ( ) Activity 6 • Inverse Functions 97 . Label this function G. 4 11. G(170) = = 2. f −1(1) = 5 D.4..6) − 2. G(c) = c − 50 60 60 170 − 50 3. C 15. Find a in terms of c. 2 What is the actual distance in miles between the two towns? Explain how you determined your answer. 2 14. Are F(a) and G(c) inverse functions? Explain your answer.. How many acres will Mark mow for this amount? Explain how you determined your answer.4 = x 97 ACTIVITY PRACTICE 1. A customer has $200 to spend on mowing. 60 7. 3 16 –8 –4 Activity 6 • Inverse Functions 17. Two towns on the map are 4 1 inches apart. 36 miles. Given that f(1) = 5. Lesson 6-2 15. f(g(x)) = −x − 3 + 3 = −x They are not inverses. y 20 8 4 –4 f −1 4 (x) −2 −1 2 0 5 0 1 3 5 ( 12 –8 x 18.25(4x − 2. 13. Sample explanation: I evaluated G(m) for m = 4 1 . Mark’s cost-calculating function is c = 60a + 50. c.4) + 0. 7. Tell what the inverse function represents. ACTIVITY 6 Continued Inverse Functions Old from New ACTIVITY 6 continued The function m = F(a) = a gives the distance 8 in inches on a map between two points that are actually a miles apart. a. Lesson 6-1 Mark’s landscaping business Mowing Madness uses the function c = F(a) to find the cost c of mowing a acres of land. 8.089-098_SB_AG2_SE_U01_A06. Use this function for Items 10–13. a. domain: (0. f −1 (x ) = x + 10 2 b. What is G(3)? What does G(3) represent? 13. a. g (x ) = x + 5 4 c. 2. g (x ) = 1 x + 4 2 d. Show your work. domain: dollars. domain: acres. p−1 (t ) = 1 t − 8 6 ACTIVITY 6 PRACTICE Write your answers on notebook paper. Sample explanation: I solved the equation m = a for a. 2. F(40) = 60(40) + 50 = 2450. All rights reserved. What is F(40)? What does F(40) mean? 2. b. a.5 acres. b. b. What are the domain and range of G(c) in interval notation? 6. f −1(5) = −1 16. 1. a. f ( x ) = x + 3. a. a. The actual distance between 2 points that are 3 inches apart on the map is 24 miles. © 2015 College Board. ∞).25x + 0. ∞) b. ∞) 6. f ( g ( x )) = 2 1 x + 4 − 4 = x 2 1 g ( f ( x )) = [2( x − 4)] + 4 = x 2 They are inverses. g(g −1(x)) = 0. Yes. What is F(50)? What does F(50) represent? 11. g (x ) = x + 3 5 b. 3. F(50) = 50 = 6 1 . What is G(170)? What does G(170) mean? 4. p−1 (t ) = 6 C. All rights reserved. g −1(x) = 4x − 2. He charges a $50 fee plus $60 per acre. G(170) is 60 the number of acres that Mark will mow for $170. f and h are inverses because they are symmetric about the line y = x. a. d. G(m) = 8m. Find the inverse of each function. The inverse function represents the actual distance in miles between 2 points that are m inches apart on the map. which of the following statements must be true? A. 14. What are the domain and range of F(a) in interval notation? b. The distance on 8 4 a map between 2 points that are actually 50 miles apart is 6 1 inches. Find the inverse of each function. Use a graph to determine which two of the three functions listed below are inverses. Sample explanation: For each value of x greater than 2. Inequalities.. Unit 1 • Equations. 0). All rights reserved. c. Functions 22. 2 4 x 4 22. Graph the inverse of f(x) on the same coordinate plane.indd Page 98 15/04/14 11:48 AM user-g-w-728 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . A student says that the functions f(x) = 2x + 2 and g(x) = 2x − 2 are inverse functions because their graphs are parallel. graphed below. Graph each function and its inverse on the same coordinate plane. −3) 23.. b. range: {y | y ∈ . 26. Sample answer: (1) Let y = f(x). and (4) let y = f −1(x). No. d). (2) interchange the x and y variables. 5 5 20. x ≥ 2}. f(x) and g(x) are not inverse functions because f(g(x)) = 2(2x − 2) + 2 = 4x − 2 and g(f(x)) = 2(2x + 2) − 2 = 4x + 2.6 y  2 21. 4 98 2 –4 –2 –2 4 x 2 –4 g(x) = g-1(x) ADDITIONAL PRACTICE If students need more practice on the concepts in this activity. Is the student’s reasoning correct? Justify your answer. 3) B. Based on this information. −4) D. No. (3) solve for y. there is more than 1 output for each input. 98 x MATHEMATICAL PRACTICES 2 –2 –4 y  25. Graph the inverse of the function shown below. which point must lie on the graph of the function’s inverse? A. Graph the absolute value function f(x) = | x | + 2. 4 © 2015 College Board. inverse of f(x): domain: {x | x ∈ . 4 –4 16. . A function h(x) has two different x-intercepts. 6 –4 Inverse Functions Old from New ACTIVITY 6 19. Give the domain and range of f(x) and its inverse using set notation. f(x) = −3x + 3 b. c) and (0. –4 2 –2 4 x –2 y  f(x) g(x) 4 f(x) 2 –4 2 –2 4 –4 –2 –4 24. see the Teacher Resources at SpringBoard Digital for additional practice problems. are not inverse functions. g (x ) = 3 x − 6 2 3 c. 4). including the vertex. a. (−4. Sample explanation: List several ordered pairs from f(x). Sample justification: The graphs are parallel because both have a slope of 2. f(x) = 2x + 4 b. f (x ) = 2 x + 6 b. Then use the definition of inverse functions to verify that the two functions are inverses. d. Inequalities. c and d. range: {y | y ∈ } d. Explain why the functions f(x) and g(x). g(x) = −x − 2 18. Construct Viable Arguments and Critique the Reasoning of Others 26. SpringBoard® Mathematics Algebra 2. f(x): domain: {x | x ∈  }. 4 f -1(x) x 4 –4 –4 y  b. Switch the coordinates of each ordered pair to get new ordered pairs for the inverse function. there are 2 outputs. For the inverse of h(x). then the function is its own inverse. x f(x) −2 4 0 1 2 −1 3 0 19. Sample answer: The graphs of the functions are not symmetric about the line y = x. (−3. Describe a method for determining whether a function f(x) is its own inverse. If the rules for f(x) and f −1(x) are the same. y ≥ 2}. 24. 0) and (d.089-098_SB_AG2_SE_U01_A06. ACTIVITY 6 Continued continued y  f(x) inverse of f(x) 2 4 2 –2 x 6 –2 –4 b. g(x) = 0. a. (3. a–b. y  20. Is the inverse of f(x) a function? Explain your answer. Explain how you graphed the inverse. a. SpringBoard® Mathematics Algebra 2. c. The graph of a function passes through the point (−3. for the input 0. D 23. Is the inverse of h(x) a function? Explain your answer. Sample explanation: The graph of h(x) passes through the points (c. 25. a.25x + 0. Unit 1 • Equations. All rights reserved. Graph and connect the ordered pairs. h(x ) = 3 x − 9 2 17. However. No. inverse –2 21. Functions © 2015 College Board. a. and the graph of the inverse of h(x) must pass through the points (0. Write the inverse of the function defined by the table shown below. 4) C. (4. 30 3. The average cost of a plane ticket from Johannesburg. They identified a currency exchange service in Spain that will convert D dollars to euros with the function E(D) = 0.1(0. Write a piecewise-defined function that gives the cost C in South African rand for shipping a package with a mass of M grams. 35 28 21 14 7 200 400 600 800 1000 M Mass of Package (g) Unit 1 • Equations.1b Combine standard function types using arithmetic operations. {C|C ∈ . HSF-IF. Shipping Costs to U. All rights reserved. Explain how you can use the graph of g(x) to find the least and greatest ticket prices offered at the website. After converting USD to EUR in Spain. kf(x). Sample explanation: E(450) = 0. 0 < M ≤ 1000.1E − 10.00 b. 42 Cost to Ship (ZAR) © 2015 College Board. Write the range of the function using set notation.64D − 5) − 10 = 7. C a. how much will she have in South African rand? Explain the process you used to arrive at your answer. The function E(R) 12.C. and the range is ZAR (rand). all of the ticket prices are within $200 of the average price.64D − 5 and a currency exchange service in South Africa that will convert E euros to rand using R(E) = 12. All rights reserved.. then converts that amount in EUR to ZAR.S.B. a.A.64(450) − 5 = 283 R(283) = 12. Piecewise-Defined. a.64 and D(139) = 225. {M|M ∈ .. Show your work. and set notation. C(M) = 28 if 100 < M ≤ 200  35 if 200 < M ≤ 1000 1.099-100_SB_AG2_SE_U01_EA2. how many euros will she need? Explain how you determined your answer. Domain for E(D) is USD (dollars). including step functions and absolute value functions. find the value of k given the graphs.00 35. Graph g(x).5. The function g(x) = |x − 1300| gives the variation of a ticket costing x dollars from the average ticket price. Shipping Costs to U. 12. Inequalities. c. Write the composite function and identify the domain and range. South Africa. domain: USD. HSF-BF. For each function. 0.S. HSF-BF.5 = 3414. Domain for R(E) is EUR (euros). HSF-BF. Mass of Package (g) Assessment Focus Answer Key Use the information above to solve the following problems. b. and Inverse Functions Embedded Assessment 2 Use after Activity 6 CURRENCY CONVERSION Kathryn and Gaby are enrolled in a university program to study abroad in Spain and then in South Africa.744(450) − 70. D(E ) = E + 5 is the inverse function. 14 if 0 < M ≤ 100  5. Sample explanation: If the mass of the package is 283 grams.7 Graph functions expressed symbolically and show key features of the graph. She wants to ship it back to the United States. The package containing Kathryn’s bowl has a mass of 283 grams. Use an inverse function to find how much USD she converted. HSF-IF.indd Page 99 05/12/13 9:58 AM user-s068a 123 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ .1(283) − 10 = 3414. by hand in simple cases and using technology for more complicated cases.30 ZAR.B. They realize that they will have to convert US dollars (USD) to euros (EUR) in Spain. b. 1000]. The inverse of the function that converts euros to rand is E(R) = R +10 .00 28. give the units for the domain and range. and f(x + k) for specific values of k (both positive and negative). and R(E(450)) = 7. No more than 100 More than 100 and no more than 200 More than 200 and no more than 1000 Embedded Assessment 2 6. Inequalities. R(E(D)) = 12.4a Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. Functions 99 Common Core State Standards for Embedded Assessment 2 e. Gaby had 139 EUR. f(kx). Functions 99 .1 Write a function that describes a relationship between two quantities. 3. d. (0. the shipping will cost 35 rand.72. Kathryn and Gaby are shopping for plane tickets back to their home city of Chicago.1 HSF-BF. 2. 3. Graph the function. so Gaby had 225 USD converted. Unit 1 • Equations. range: ZAR 4. Cost to Ship (ZAR) • Piecewise-defined functions • Composition of functions • Inverse functions 1.744D − 70. Describe the graph of g(x) as a transformation of the graph of f(x) = |x|. Composite. to Chicago is $1300. and E(35) = 35 + 10 ≈ 3.30.A. C = 14 or C = 28 or C = 35} d. and the range is EUR (euros). Write the domain of the function using an inequality.3 Identify the effect on the graph of replacing f(x) by f(x) + k. Explain how to compose the functions E and R to answer Item 2. 2.C.7b Graph piecewise-defined functions. If Kathryn converts 450 USD in Spain to EUR.B. interval notation. HSF-BF. Kathryn buys a wooden bowl as a souvenir while in South Africa. 3414. and then convert EUR into South African rand (ZAR) for their time in South Africa. c.1 converts rand to euros. 5. If she needs to convert euros to rand to pay for the shipping. 4. 0 < M ≤ 1000} c. The table below shows the costs for one shipping service. © 2015 College Board. 14. At one travel website. e.4 Find inverse functions.72 euros. a. and function transformations (Items 1. composition. and set notation • Precise use of appropriate (Items 2. range. inverse. inverse. 6b) 50 800 1000 1200 1400 1600 1800 x Price of Ticket ($) b. and composite functions to model real-world scenarios Partial understanding of how to graph piecewisedefined functions and represent intervals using inequalities. Problem Solving (Items 3. Scoring Guide 250 © 2015 College Board. 3. and function transformations • Little or no understanding and inaccurate identification of function concepts including domain. inverse. inverse. composition. 6c) • SpringBoard® Mathematics Algebra 2 • A functional understanding • Partial understanding strategy that results in a correct answer math terms and language to describe function transformation and function composition Clear and accurate explanation of the steps to solve a problem based on a real-world scenario 100 SpringBoard® Mathematics Algebra 2 100 Emerging • An appropriate and efficient • A strategy that may include • A strategy that results in (Items 2. and composite functions to model real-world scenarios Inaccurate or incomplete understanding of how to graph piecewise-defined functions and represent intervals using inequalities. and Inverse Functions Embedded Assessment 2 Variation from Average Ticket Price . The intersection points are (1100. TEACHER to TEACHER You may wish to read through the scoring guide with students and discuss the differences in the expectations at each level. interval notation. Variation from Average Price ($) y Piecewise-Defined. inverse. inverse.. Embedded Assessment 2 6. All rights reserved. composition. 5e. interval notation. Check that students understand the terms used.. 3. inverse. 6c) Mathematical Modeling / Representations Proficient The solution demonstrates these characteristics: piecewise-defined. 5e. a translation of the graph of f(x) by 1300 units to the right c. and set notation • Misleading or confusing • Incomplete or inaccurate • • description of function transformation and function composition Misleading or confusing explanation of the steps to solve a problem based on a real-world scenario description of function transformation and function composition Incomplete or inadequate explanation of the steps to solve a problem based on a real-world scenario © 2015 College Board. Composite. so the least ticket price offered at the website is $1100 and the greatest ticket price is $1500. 6a) Reasoning and Communication Incomplete and accurate identification of function concepts including domain. range. 6b. Sample explanation: Find the points where the graph of the function intersects the line y = 200. interval notation. inverse. All rights reserved. 5c. and function transformations • No clear strategy when unnecessary steps but results in a correct answer some incorrect answers • Fluency in creating • Little difficulty in creating • Partial understanding of • Little or no understanding • • • • piecewise-defined. and set notation • Adequate description of • function transformation and function composition Adequate explanation of the steps to solve a problem based on a real-world scenario how to create piecewisedefined. 200). 5a-d. range. and composite functions to model real-world scenarios Clear and accurate understanding of how to graph piecewise-defined functions and represent intervals using inequalities. composition. 5b. range. 4. 200) and (1500. 4. and composite functions to model real-world scenarios Mostly accurate understanding of how to graph piecewise-defined functions and represent intervals using inequalities.099-100_SB_AG2_SE_U01_EA2.indd Page 100 05/12/13 9:58 AM user-s068a 123 /103/SB00001_DEL/work/indd/SE/M02_High_School/Algebra_02/Application_files/SE_A2_ . Use after Activity 6 CURRENCY CONVERSION 300 200 Mathematics Knowledge and Thinking 150 100 Exemplary • Clear and accurate identification of and understanding of function concepts including domain. and function transformations and partially accurate identification of function concepts including domain. interval notation. and set notation solving problems of how to create piecewisedefined.