Lesson 11-2 Shrinking, Stretching, and Reflecting Parabolas ACTIVITY 11

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1 ACTIVITY 11 Lesson 11- M Notes Unlike a rigid transformation, a vertical stretch or vertical shrink will change the shape of the graph. A vertical stretch stretches a graph awa from the -ais b a factor and a vertical shrink shrinks the graph toward the -ais b a factor. Learning Targets: Describe transformations of the parent function f() =. Given a transformation of the function f() =, write the equation of the function. SUGGESTED LEARNING STRATEGIES: Create Representations, Look for a Pattern, Group Presentation, Quickwrite, Identif a Subtask 1. Graph the function f() = as Y1 on a graphing calculator. Then graph each of the following functions as Y. Describe the graph of each function as a transformation of the graph of f() =. a. g() = b. h() = c. j( ) = 1 d. k( ) = 1 Reflections over aes do not change the shape of the graph, so the are also rigid transformations.. Epress regularit in repeated reasoning. Describe an patterns ou observed in the graphs from Item Graph the function f() = as Y1 on a graphing calculator. Then graph each of the following functions as Y. Identif and describe the graph of each function as a transformation of the graph of f() =. a. g() = 01 College Board. All rights reserved. 178 SpringBoard Mathematics Algebra, Unit Quadratic Functions

2 Lesson 11- ACTIVITY 11 b. h() = M Notes c. j( ) = 1. Describe an patterns ou observed in the graphs from Item 3.. Make a conjecture about how the sign of k affects the graph of g() = k compared to the graph of f() =. Assume that k College Board. All rights reserved. 6. Make a conjecture about whether the absolute value of k affects the graph of g() = k when compared to the graph of f() =. Assume that k 0 and write our answer using absolute value notation. 7. Make use of structure. Without graphing, describe each function as a transformation of f() =. a. h() = 6 In Item 6, consider the situation in which k > 1 and the situation in which k < 1. b. j( ) = 1 Activit 11 Transformations of = 179

3 ACTIVITY 11 Lesson 11- M Notes c. p() = 9 d. q( ) = 1 Check Your Understanding 8. The graph of g () is a vertical shrink of the graph of f() = b a factor of 1. What is the equation of g()? 6 9. Reason quantitativel. The graph of h() is a vertical stretch of the graph of f() =. If the graph of h() passes through the point (1, 7), what is the equation of h()? Eplain our answer.. The graph of j() = k opens downward. Based on this information, what can ou conclude about the value of k? Justif our conclusion. A horizontal stretch stretches a graph awa from the -ais b a factor and a vertical shrink shrinks the graph toward the -ais b a factor. 11. Graph the function f() = as Y1 on a graphing calculator. Then graph each of the following functions as Y. Identif and describe the graph of each function as a horizontal stretch or shrink of the graph of f() =. a. g() = () b. h() = () c. j 1 ( ) = ( ) 01 College Board. All rights reserved. d. k 1 ( ) = ( ) 180 SpringBoard Mathematics Algebra, Unit Quadratic Functions

4 Lesson 11- ACTIVITY Describe an patterns ou observed in the graphs from Item 11. M Notes 13. a. Use appropriate tools strategicall. Graph the function f() = as Y1 on a graphing calculator. Then graph h() = ( ) as Y. Describe the result. b. Reason abstractl. Eplain wh this result makes sense. 1. Make a conjecture about how the sign of k affects the graph of g() = (k) compared to the graph of f() =. Assume that k College Board. All rights reserved. 1. Make a conjecture about whether the absolute value of k affects the graph of g() = (k) when compared to the graph of f() =. Assume that k Describe each function as a transformation of f() =. a. p() = (6) b. q 1 ( ) = ( ) Activit 11 Transformations of = 181

5 ACTIVITY 11 Lesson 11- M Notes Check Your Understanding 17. Describe how the graph of g() = differs from the graph of h() = (). 18. The graph of g() is a horizontal stretch of the graph of f() = b a factor of. What is the equation of g()? 19. Reason quantitativel. The graph of h() is a horizontal shrink of the graph of f() =. If the graph of h() passes through the point (1, ), what is the equation of h()? Eplain our answer. 0. Each function graphed below is a transformation of f() =. Describe the transformation. Then write the equation of the transformed function. a. 8 6 g() ( 1, 3) (1, 3) b. 8 c. ( 6, 1) (6, 1) (, ) (, ) h() 01 College Board. All rights reserved. 6 8 j() 18 SpringBoard Mathematics Algebra, Unit Quadratic Functions

6 Lesson 11- ACTIVITY 11 d. 16 k() M Notes 1 ( 1, 9) 8 (1, 9) 1. Model with mathematics. Multiple transformations can be represented in the same function. Describe the transformations from the parent function. Then graph the function, using our knowledge of transformations onl. a. f() = ( + 3) + When graphing multiple transformations of quadratic functions, follow this order: 1. horizontal translation. horizontal shrink or stretch 3. reflection over the -ais and/or vertical shrink or stretch. vertical translation b. f() = ( ) 3 01 College Board. All rights reserved. Activit 11 Transformations of = 183

7 ACTIVITY 11 Lesson 11- M Notes c. f() = ( + 1) d. f() = ( 3) + Check Your Understanding. Eplain how ou determined the equation of g() in Item 0a. 3. Without graphing, determine the verte of the graph of h() = ( 3) +. Eplain how ou found our answer.. a. Start with the graph of f() =. Reflect it over the -ais and then translate it 1 unit down. Graph the result as the function p(). b. Start with the graph of f() =. Translate it 1 unit down and then reflect it over the -ais. Graph the result as the function q(). c. Construct viable arguments. Does the order in which the two transformations are performed matter? Eplain. d. Write the equations of p() and q(). 01 College Board. All rights reserved. 18 SpringBoard Mathematics Algebra, Unit Quadratic Functions

8 Lesson 11- ACTIVITY 11 LESSON 11- PRACTICE Describe each function as a transformation of f() =.. g() = 6. h() = (8) 7. Make sense of problems. The graph of j() is a horizontal stretch of the graph of f() = b a factor of 7. What is the equation of j()? Each function graphed below is a transformation of f() =. Describe the transformation. Then write the equation of the transformed function k() ( 3, 3) (3, 3) ( 9, 1) (9, 1) m() M Notes Describe the transformations from the parent function. Then graph the function, using our knowledge of transformations onl. 30. n() = 3( ) 31. p( ) = 1 ( + 3) 01 College Board. All rights reserved. Activit 11 Transformations of = 18