ACTIVITY 11 Lesson 11-2 continued Shrinking, Stretching ...

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

My Notes ••Learning Targets:
Describe transformations of the parent function f(x) = x2.
MATH TIP Given a transformation of the function f(x) = x2, write the equation of
the function.
Unlike a rigid transformation, a SUGGESTED LEARNING STRATEGIES: Create Representations, Look
vertical stretch or vertical shrink will for a Pattern, Group Presentation, Quickwrite, Identify a Subtask
change the shape of the graph. 1. Graph the function f(x) = x2 as Y1 on a graphing calculator. Then graph
A vertical stretch stretches a graph each of the following functions as Y2. Describe the graph of each
away from the x-axis by a factor function as a transformation of the graph of f(x) = x2.
and a vertical shrink shrinks the a. g(x) = 2x2
graph toward the x-axis by a factor.
b. h(x) = 4x2

c. j(x) = 1 x2
2

d. k(x) = 1 x2
4

2. Express regularity in repeated reasoning. Describe any patterns © 2015 College Board. All rights reserved.
you observed in the graphs from Item 1.

MATH TIP 3. Graph the function f(x) = x2 as Y1 on a graphing calculator. Then graph
each of the following functions as Y2. Identify and describe the graph of
Reflections over axes do not each function as a transformation of the graph of f(x) = x2.
change the shape of the graph, so a. g(x) = −x2
they are also rigid transformations.

178 SpringBoard® Mathematics Algebra 2, Unit 2 • Quadratic Functions

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

b. h(x) = −4x2 My Notes

c. j(x) = − 1 x2
4

4. Describe any patterns you observed in the graphs from Item 3.

5. Make a conjecture about how the sign of k affects the graph of
g(x) = kx2 compared to the graph of f(x) = x2. Assume that k ≠ 0.

6. Make aofcog(nxj)ec=tukrxe2awbohuetnwchomethpearretdhetoabthsoelgurtaepvhaloufef(oxf)k=afxfe2c. tAsstshueme
graph
that k ≠ 0 and write your answer using absolute value notation. MATH TIP

© 2015 College Board. All rights reserved. In Item 6, consider the situation in
which |k| > 1 and the situation in
which |k| < 1.

7. Make use of structure. Without graphing, describe each function as
a transformation of f(x) = x2.
a. h(x) = 6x2

b. j(x) = − 1 x 2
10

Activity 11 • Transformations of y = x2 179

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

My Notes c. p(x) = −9x2

d. q(x) = 1 x 2
5

Check Your Understanding

8. The graph of g (x) is a vertical shrink of the graph of f(x) = x2 by a
1
factor of 6 . What is the equation of g(x)?

9. Reason quantitatively. The graph of h(x) is a vertical stretch of the
graph of f(x) = x2. If the graph of h(x) passes through the point (1, 7),
what is the equation of h(x)? Explain your answer.

10. The graph of j(x) = kx2 opens downward. Based on this information,
what can you conclude about the value of k? Justify your conclusion.

MATH TIP 11. Graph the function f(x) = x2 as Y1 on a graphing calculator. Then graph
each of the following functions as Y2. Identify and describe the graph of
A horizontal stretch stretches a each function as a horizontal stretch or shrink of the graph of f(x) = x2.
graph away from the y-axis by a a. g(x) = (2x)2
factor and a vertical shrink shrinks b. h(x) = (4x)2
the graph toward the y-axis by a
factor.

( )c. 1 2 © 2015 College Board. All rights reserved.
j(x) = 2 x

( )d. 1 2
k(x) = 4 x

180 SpringBoard® Mathematics Algebra 2, Unit 2 • Quadratic Functions

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

12. Describe any patterns you observed in the graphs from Item 11. My Notes

13. a. Use appropriate tools strategically. Graph the function
f(x) = x2 as Y1 on a graphing calculator. Then graph h(x) = (−x)2
as Y2. Describe the result.

b. Reason abstractly. Explain why this result makes sense.

14. Make a conjecture about how =thex2s.igAnssoufmk eaftfheacttskt≠he0g.raph of g(x) = (kx)2
compared to the graph of f(x)

15. Make aofcog(nxj)ec=tu(rkexa)b2 owuhtewnhceotmhepratrheedatbostohleutgervapalhueofoff(kx)a=ffexc2t.s the
graph
© 2015 College Board. All rights reserved. Assume that k ≠ 0.

16. Describe each function as a transformation of f(x) = x2.
a. p(x) = (6x)2

( )b. q(x) =1 2
10
x

Activity 11 • Transformations of y = x2 181

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

My Notes

Check Your Understanding

17. Describe how the graph of g(x) = 4x2 differs from the graph of
h(x) = (4x)2.

18. The graph of g(x) is a horizontal stretch of the graph of f(x) = x2 by a
factor of 5. What is the equation of g(x)?

19. Reason qu antitativel y. The graph of h(x) is a horizontal shrink
of the graph of f(x) = x2. If the graph of h(x) passes through the
point (1, 25), what is the equation of h(x)? Explain your answer.

20. Each function graphed below is a transformation of f(x) = x2. Describe
the transformation. Then write the equation of the transformed
function.
a. y  g(x)

8

6

4 (1, 3)
(–1, 3)

2

–4 –2 x
24

b. y 

8

(–6, 1) 4 (6, 1) h(x) © 2015 College Board. All rights reserved.
48 x
–8 –4
–4

–8

c. y 

2

–4 –2 24 x
(–2, –2) –2 (2, –2)

–4

–6

–8
j(x)

182 SpringBoard® Mathematics Algebra 2, Unit 2 • Quadratic Functions

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

d. y  k(x) My Notes
16
MATH TIP
12 (1, 9)
(–1, 9) 8 When graphing multiple
transformations of quadratic
4 x functions, follow this order:
24 1. horizontal translation
–4 –2 2. horizontal shrink or stretch
–4 3. reflection over the x-axis and/or

21. Model with mathematics. Multiple transformations can be vertical shrink or stretch
represented in the same function. Describe the transformations from 4. vertical translation
the parent function. Then graph the function, using your knowledge
of transformations only.
a. f(x) = −4(x + 3)2 + 2

© 2015 College Board. All rights reserved. y  x
10 5
5
x
–5 5
–5

–10

b. f(x) = 2(x − 4)2 − 3

y 
10
5

–5
–5

–10

Activity 11 • Transformations of y = x2 183

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

My Notes c. f(x) = 2(x + 1)2 − 4

y  x
10 5
5
x
–5 5
–5

–10

d. f(x) = −(x − 3)2 + 5

y 
10
5

–5
–5

–10

Check Your Understanding © 2015 College Board. All rights reserved.

22. Explain how you determined the equation of g(x) in Item 20a.
23. Without graphing, determine the vertex of the graph of

h(x) = 2(x − 3)2 + 4. Explain how you found your answer.
24. a. Start with the graph of f(x) = x2. Reflect it over the x-axis and then

translate it 1 unit down. Graph the result as the function p(x).
b. Start with the graph of f(x) = x2. Translate it 1 unit down and then

reflect it over the x-axis. Graph the result as the function q(x).
c. Construct viable arguments. Does the order in which the two

transformations are performed matter? Explain.
d. Write the equations of p(x) and q(x).

184 SpringBoard® Mathematics Algebra 2, Unit 2 • Quadratic Functions

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

LESSON 11-2 PRACTICE My Notes

Describe each function as a transformation of f(x) = x2.

25. g(x) = −5x2 26. h(x) = (8x)2

27. Make sense of problems. The graph of j(x) is a horizontal stretch of
the graph of f(x) = x2 by a factor of 7. What is the equation of j(x)?

Each function graphed below is a transformation of f(x) = x2. Describe the
transformation. Then write the equation of the transformed function.

28. y  k(x) 29. y 

6 4 m(x)

4 2
2
(–3, 3) (3, 3) (–9, 1) (9, 1) x
x 6 12
–12 –6
–4 –2 24 –2
–2
–4

Describe the transformations from the parent function. Then graph the
function, using your knowledge of transformations only.
1
30. n(x) = −3(x − 4)2 31. p(x) = 2 (x + 3) − 5

© 2015 College Board. All rights reserved.

Activity 11 • Transformations of y = x2 185