CHAPTER 8 Transformations Content Summar In Chapter 8, students continue their work with functions, especiall nonlinear functions, through further stud of function graphs. In particular, the consider three was of changing the location, orientation, and size of those graphs. (Note: You might skip the material on matrices if our student s teacher is not covering Lesson 8.7; matrices are often covered as part of an advanced algebra curriculum.) Translations A translation shifts points or graphs on the plane. If a point (, ) is shifted to the right h units and up k units, the resulting point is ( h, k). A function graph can be shifted in the same wa, b replacing with h in each occurrence of in the graph s equation, and b replacing with k in each occurrence of. As an eample, consider the rational function f(), which the book introduces in this chapter. If ou shift the graph of to the left units and up units, the result has the equation. You can also think of the resulting equation as the graph of function g(). Although f() is undefined for 0 and has 0 as an asmptote (a line that the graph approaches but never touches), g() is undefined units to the left, where, and has an asmptote units higher, at. 4 It is possible to represent transformations with matrices. In particular, a translation can be represented b matri addition. To do so, ou can represent the point (, ) b the matri and the translation b h. Then the image point is k represented b the sum of these matrices: h h. k k In fact, this matri approach can represent the translation of more than one point. Discovering Algebra shows how a matri can represent the vertices (corners) of a polgon, with each column being the coordinates of a verte. For eample, the matri represents the pentagon pictured on the net page. 0 4 A shift to the right units and up units can be represented b the matri. (continued) 008 Ke Curriculum Press Discovering Algebra: A Guide for Parents 7
Chapter 8 Transformations (continued) The resulting polgon is represented b the sum of the matrices: 4 0 4 4 4 It is shown as the dashed polgon on the graph. Reflections The book eamines reflections (or flips) of points and graphs across the aes. When ou reflect a point (, ) across the -ais, the result is the point (, ). Reflecting the point (, ) across the -ais ields an image of (, ). To reflect a graph across the -ais, ou replace each occurrence of in its equation with. To reflect a graph across the -ais, ou replace with. For eample, the reflection of the graph of across the -ais has equation f(). The reflection of across the -ais has equation,or. Reflections can also be represented b matri multiplication. For eample, to reflect the pentagon represented b 0 4 across the -ais, multipl b the matri 0 to get 0 0 0 0 4 0 4 Stretches and Shrinks A vertical stretch b positive factor a changes (, ) to (, a). If a is less than (but still positive), the stretch is a shrink. To stretch or shrink a graph verticall b positive factor a, replace each occurrence of with a in the equation of the graph. For eample, replacing with in the equation f() creates a vertical stretch b a factor of. This is the equation of the function g(). (continued) 8 Discovering Algebra: A Guide for Parents 008 Ke Curriculum Press
Chapter 8 Transformations (continued) Stretches and shrinks can also be represented b matri multiplication. You can find the image of pentagon after it is stretched verticall b a factor of 0 4 if ou multipl b the matri 0 0 0 4 0 0 0 4 to get. Summar Problem. 0.7 0. 0.0. 0.8 0.7 0. 0. Use transformations of the function f() to fit the data in the preceding table as well as ou can. Questions ou might ask in our role as student to our student include: Wh do different translations give the same result? Wh do different reflections give the same result? Wh do different combinations of stretches and shrinks give the same result? Does the order in which ou do the transformations matter? Could a starting function other than f() be transformed to fit the data points better? Sample Answers For a line with equation a b, replacing with h gives a b bh, or bh a b. So a horizontal translation b h is the same as a vertical translation b bh. Algebra also shows that a reflection of a line across the -ais can be accomplished b a reflection across the -ais, combined with a translation, and that stretching a line in one direction is equivalent to shrinking it in the perpendicular direction. A nonlinear function will probabl fit the data better. 008 Ke Curriculum Press Discovering Algebra: A Guide for Parents 9
Chapter 8 Review Eercises Name Period Date. (Lessons 8., 8., 8.4) Draw this triangle on graph paper or on our calculator. Then draw the image under each of the following transformations. Describe each transformation. a. (, ) b. (, ) c. (, ) d. (0., ) 7 7. (Lessons 8. 8.4) The graph of the function is shown below. Name the functions that give the following transformations of the graph. Check each answer b graphing it on our calculator. f() 0 a. Translate right units. b. Reflect across the -ais and translate up unit. c. Reflect across the -ais, shrink verticall b a factor of 0., translate left unit, and translate up units.. (Lesson 8.) Reduce each epression to lowest terms. State an restrictions on the variable. a. 0 4 4 b. 4 c. 4 ( ) ( ) d. 008 Ke Curriculum Press Discovering Algebra: A Guide for Parents 4
SOLUTIONS TO CHAPTER 8 REVIEW EXERCISES. a. Translate left units and up unit. b. Reflect across the -ais. c. Reflect across the -ais. d. Shrink horizontall b a factor of 0.; stretch verticall b a factor of. d 7 0. Solve for. 0. ( ) Replace with to translate the graph left unit. 0. Replace with to translate the graph up units. 0. 7 a c b. a. Original function. ( ) Replace with to translate the graph right units. [9.4, 9.4,,.,., ] b. Original function. () Replace with to reflect the graph across the -ais. Replace with to translate the graph up unit. [4.4, 4.4,,.,., ]. a. 0 4 4,where 0 The restriction 0 is necessar because the -value 0 would make the denominator of the original epression zero. b. 4 ( ),where 0 The restriction 0 is necessar because the -value 0 would make the denominator of the original epression zero. c. 4 ( ) ( ),where. d.,where 0. c. Original function. Replace with to reflect across the -ais. Solve for. Replace with 0.,or, to shrink the graph verticall b a factor of 0.. 4 Discovering Algebra: A Guide for Parents 008 Ke Curriculum Press