Introduction to Transformation Geometry



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Welcome to 2006!A little course philosophy before we jump into the next topic Is mathematics a creative discipline? Do you need to be creative to be good at math? Does creativity help you or hurt you when you are working on math problems? What is the difference between an exercise and a problem? Spend a few minutes answering these questions on paper. Put your response in the journal section of your notebook. It might make for a good introduction for your January journal entries. Transformation Geometry A transformation of the plane can be viewed as a function. Definition: A one-to-one function with the set of all points in the plane as the domain and the range is called a transformation. What is a one-to-one function? A function is one-to-one if the elements of the range are not repeated. Each element of the range corresponds to only on element in the domain. For transformations, this means that no two points can get mapped to the same point. The inverse of a One-to-One function is also a function. Let's play...Name that Transformation (With a little introduction to some new notation) 1. T(x,y) = (-x, y) 2. R(x,y) = (-x, -y) 3. U(x,y) = (x, -y) 4. (x,y) --> (y, x) 5. (x,y) -> (-y, -x) 6. (x,y) = (3x, 3y) 7. R(x,y) = (x2, y) 8. S(x,y) = (x+3, y) 9. T(x,y) = (3, -y) 1. ry-axis = reflection in the y axis. 2. r0 = reflection in the origin. 3. rx-axis = reflection in the x axis. 4. ry=x = reflection in line y=x. 5. ry=-x =reflection in line y = -x. 6. D3 = Dilation by a factor of 3. 7. Not a transformation! 8. T3,0= Translation 3 units right. 9. Not a transformation! (not 1-to-1) Transformations: Translations, Dilations & Notations A translation is a transformation that shifts all points by a fixed distance in a particular direction. Ta,b= Translation a units right and b units up. Ta,b(x, y) = (x+a, y+b) [of course, if a<0 we shift left. If b<0 we shift down] Dk = Dilation by a factor of k. If k>1, the dilation increases the size of [dilates] images. Dk(x,y) = (kx, ky). If k<0, the the dilation changes the size and also reflects the image through the origin. The center of dilation is the origin, unless otherwise specified. Transformations: Rotation Notation A rotation is denoted by a capital R. The center of rotation is the origin unless otherwise indicated. The ones you should know are: R90 (x,y) = (-y, x) R180 (x,y) = (-x,-y) R270 (x,y) = ( y,-x) Do you remember the formulas? The Geometer's Sketchpad does a great job demonstrating these. If you can visualize these transformations, you can always derive the formula if you forget! Transformations: Sample Problems You need to remember all the formulas. If you get stuck, draw a diagram. 1. What is the image of P(4,-2) after R90 ? R90(x,y) = (-y,x) R90(4,-2) = (2,4) 2. The image of Q(3,5) after a translation is Q'(0,-1). What is the translation? Find the image of P(4,-2) under this same translation. The translation is T-3,-6 (3 left and 6 down). T-3,-6 (4,-2) = (1,-8) 3. What is the image of (-3,5) after RO RO(x,y) = (-x,-y) (reflection in the origin)? RO(-3,5) = (3,-5) 4. Give a simpler way of describing the transformation R-300 . R60 Transformations: More Problems 5. What is the image of P(4,-2) after R-90 ? R-90(x,y) = R270(x,y) = (y,-x) R270(4,-2) = (-2, -4) 6. The domain of f is {x | -5 < x < 5}. What is the domain of g if g is the image of f after D2? D2(x,y) = (2x,2y). domain of g is {x | -10 < x < 10}. 7. The domain of f is {x | 1 < x < 100} and the range is {y | 0 < y < 2}. What is the domain of g if g is the image of f after T2,6? Can you think of a function like f? It smells like the log function! T2,6(x,y) = (x+2,y+6). domain of g is {x | 3 < x < 102}. Range of g is {y | 6 < y < 8}. 8. The image of Q(-7,5) after a line reflection is (-5,7). What line was Q releflected in? (x,y)->(-y,-x) The line y= -x Transformations: Examples of compositions (8.3 in RB) Determine the image of the point specified. 1. T2,-1(3,0) = (5,-1) 2. rx-axis (T5,4(2,1)) = rx-axis(7,5)=(-7,5) 3. ry=x (T4,0(-2,1)) = r (2,1) = (1,2) y=x 4. T3,2 D3(3,0) = (5,-1) 5. ry-axis T1,-4(-2,3) = (5,-1) 6. ry=-x T4,0(0,1)) = (5,-1) 7. T3,2 D3(x,y) = (5,-1) 8. ry-axis T1,-4(x,y) = (5,-1)
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