: Now : Why? You analyzed graphs

of functions. Lessons

1-2 through 1-4)

Identify, graph, and

describe parent

functions.

Identify and graph

transformations of

parent functions.

The path of a 60-yard punt can be modeled

by the function at the right. This function is

related to the basic quadratic function

fix) = x2.

NewVocabulary parent function constant function zero function identity function quadratic function cubic function square root function reciprocal function absolute value function step function greatest integer function transformation translation reflection dilation

.2 40 Q

r 2 0 >

f y h{x) = - ^ - x 2 + Ax + i H

\ \ X

20 40 60 80

Horizontal Distance (yd)

1 Parent Functions A family of functions is a group of functions w i t h graphs that display one or more similar characteristics. A parent funct ion is the simplest of the functions in a family.

This is the function that is transformed to create other members i n a family of functions.

In this lesson, you w i l l study eight of the most commonly used parent functions. You should already be familiar w i t h the graphs of the fo l lowing linear and polynomial parent functions.

Key Concept Linear and Polynomial Parent Functions A constant function has the form f(x) = c, where c is any real number. Its graph is a horizontal line. When c = 0, f(x) is the zero function.

y

Ax) = c

6 X

The quadratic function f(x) = x2 has a U-shaped graph.

The identity function f(x) = x passes through all points with coordinates (a, a).

i \( Y ) -

0 X

The cubic function f(x) = x3 is symmetric about the origin.

1 • - f

- fix) = X2 / / 0 X

1 1

f

You should also be familiar w i t h the graphs of both the square root and reciprocal functions.

Concept Square Root and Reciprocal Parent Functions

The square root function has the form fix) = yfx.

y

fix) =

0 X

The reciprocal function has the form fix)

fix)

I r" connectED.mcgraw-hill.com I 45 a

Another parent function is the piecewise-defined absolute value function.

KeyConcept Absolute Value Parent Function

Words The absolute value function, denoted f(x) = \x\, is a V-shaped function defined as

Model

f(x) = -x if x < 0 x if x > 0

Examples |-5| = 5,|0| = o,|4| = 4

y

// 0 X

- / f _ 1 1 / - /

i

StudyTip Floor Function The greatest integer function is also known as the floor function.

y

I f(x) = Vx I

0 X

Figure 1.5.1

A piecewise-defined function i n which the graph resembles a set of stairs is called a step funct ion. The most wel l -known step function is the greatest integer function.

Concept Greatest Integer Parent Function

Words The greatest integer function, denoted f{x) = [AJ, is defined as the greatest integer less than or equal to x.

Model

Examples [ - 4 1 = - 4 , [ - 1 . 5 ] = - 2 ,

y

-- )

[o X

)—

Using the tools you learned i n Lessons 1-1 through 1-4, you can describe characteristics of each parent function. Knowing the characteristics of a parent function can help you analyze the shapes of more complicated graphs i n that family.

Describe Characteristics of a Parent Function

Describe the f o l l o w i n g characteristics of the graph of the parent funct ion f(x) = yfx: domain, range, intercepts, symmetry, continuity, end behavior, and intervals on w h i c h the graph is increasing/decreasing.

The graph of the square root function (Figure 1.5.1) has the fol lowing characteristics.

• The domain of the function is [0, oo), and the range is [0, oo).

• The graph has one intercept at (0, 0).

• The graph has no symmetry. Therefore,/(x) is neither odd nor even.

• The graph is continuous for all values in its domain.

• The graph begins at x = 0 and l i m /(x) = oo.

• The graph is increasing on the interval (0, oo).

p GuidedPractice

1. Describe the fol lowing characteristics of the graph of the parent function/(x) = \x\: domain, range, intercepts, symmetry, continuity, end behavior, and intervals on which the graph is increasing/ decreasing.

2Transformations Transformations of a parent function can affect the appearance of the parent graph. Rigid transformations change only the position of the graph, leaving the size and

shape unchanged. Nonrigid transformations distort the shape of the graph.

46 I L e s s o n 1-5 I P a r e n t F u n c t i o n s a n d T r a n s f o r m a t i o n s

A translation is a r i g i d transformation that has the effect of shifting the graph of a function. A vertical translation of a function/shifts the graph o f / u p or d o w n , whi le a horizontal translation shifts the graph left or right. Horizontal and vertical translations are examples of r ig id transformations.

Concept Vertical and Horizontal Translations Vertical Translations Horizontal Translations

The graph of g[x) = f(x) + k is the graph of f{x) translated The graph of g(x) = f(x - h) is the graph of f(x) translated

• k units up when Ar > 0, and • h units right when h > 0, and

• * units down when k<0. • h units left when h<0.

TechnologyTip Translations You can translate a :raph using a graphing calculator. Under IY— L place an equation in v 1 . Move to the Y2 line, and then

oress VARS • E N T E R

ENTER |. This will place Y1 in the Y2 line. Enter a number to Translate the function. Press

G r a p h |. The two equations ••/ill be graphed in the same window.

Graph Translation;

Use the graph of fix) = \x\o graph each function.

a. gix) = \x\ 4 This function is of the form g(x) =f(x) + 4. So, the graph of g(x) is the graph of f (x) = \x\ translated 4 units up , as shown i n Figure 1.5.2.

b. gix) = \x + 3\

This function is of the form g(x) =f{x + 3) ov g(x) =f[x - ( -3 ) ] . So, the graph of g(x) is the graph of/(x) = \x\d 3 units left, as shown i n Figure 1.5.3.

c. gix) = \x - 2| - 1

This function is of the form g(x) =f{x — 2) — 1 . So, the graph of g(x) is the graph of f(x) = \x\ translated 2 units right and 1 unit down, as shown i n Figure 1.5.4.

fix) = \x\ y fix) = \x\

J. 9

u X

y <(x) = \x\

V

0 X

gix)-. = \x-2\1

Figure 1.5.2 Figure 1.5.3 Figure 1.5.4

W GllidedPraCtlCe Use the graph of fix) = x 3 to graph each funct ion.

2A. h(x) 2B. h(x) = (x-3): 2C. h{x) = (x + If + 4

Another type of r ig id transformation is a reflection, which produces a mirror image of the graph of a function w i t h respect to a specific line.

KeyConcept Reflections in the Coordinate Axes Reflection in x-axis

g(x) = -f(x) is the graph of f(x) reflected in the x-axis.

Reflection in y-axis

g(x) = f{-x) is the graph of f(x) reflected in the y-axis.

y y y=f(x) g(x) = f(-x) y=f(x)

0 Vs. X

g(x) = -f(x)

0 X

When w r i t i n g an equation for a transformed function, be careful to indicate the transformations correctly. The graph of g(x) = —\Jx — 1 + 2 is different from the graph of g(x) = — (Vx — 1 -I- 2).

y

- gix) = - V x - 1 + 2 f —

P — X 0 X

g(x) = _ ( N /73T + 2)

reflection of f(x) = Vx in the x-axis, then translated 1 unit to the right and 2 units up

translation of f(x) = Vx 1 unit to the right and 2 units up, then reflected in the x-axis

y

\ y \ fi \ j r l\M — A

0 X

Figure 1.5.5

Exampl Write Equations for Transformations

Describe how the graphs oifix) = xz andg(x) are related. Then wri te an equation tor g(x).

b. y

y= gix)

7̂ 0 f v X / \ / \ / \

• J /

y = gix)

f '

/ \ 0 \ X

1 \ t \ The graph of g(x) is the graph of f(x) = x2

translated 5 units to the right and reflected in the x-axis. So, g(x) = — (x — 5) 2 .

The graph of g(x) is the graph of/(x) = x 2

reflected in the x-axis and translated 2 units up. So, g(x) = - x 2 + 2.

f GuidedPractice I • •

Describe how the graphs of fix) = — and g(x) are related. Then wr i te an equation for g(x).

3A.

V

X X)

3B. 1 y

1

/ y = gix — - y = gix

X

48 | L e s s o n 1-5 | P a r e n t F u n c t i o n s a n d T r a n s f o r m a t i o n s

A di la t ion is a nonrigid transformation that has the effect of compressing (shrinking) or expanding (enlarging) the graph of a function vertically or horizontally.

'Tip Sometimes pairs of

look similar such as a jxpansion and a il compression. It is not to tell which dilation a nation is from the graph, t compare the equation of iformed function to the inction.

Concept Vertical and Horizontal Translations

Vertical Dilations

If a is a positive real number, then g(x) = a • f(x), is

• the graph of f{x) expanded vertically, if a > 1. • the graph of f(x) compressed vertically, if 0 < a < 1.

gix) = a* fix), a > 1

y=f(x)

|£fU) = a . f U ) , 0 < a < 1 - f

Horizontal Dilations

If a is a positive real number, then g(x) = f{ax), is

• the graph of f{x) compressed horizontally, if a > 1. • the graph of f{x) expanded horizontally, if 0 < a < 1.

g(x) = f(ax), a > 1

y=f(x)

y=f(x)

g(x) = fiax), 0 < a < 1

escribe and Graph Transformations

Ident i fy the parent function/(x) of g(x), and describe h o w the graphs of g(x) and/(x) are related. Then graph fix) andg(x) on the same axes.

a. gix) = ± x 3

The graph of g(x) is the graph of

f{x) = x3 compressed vertically because

£(x) = ± x 3 = i / ( x ) a n d O < l < l .

b. gix) = -i0.2x)2

The graph of gix) is the graph of f{x) = x2 expanded horizontally and then reflected i n the x-axis because g{x) = - (0 .2x) 2 = -/(0.2x) and 0 < 0.2 < 1.

• Guided Practice

4A. gix) = M-4:

f(x) = x2

gix) = -i02x?

4B. g{x) 15

You can use what you have learned about transformations of functions to graph a piecewise-defined function.

Real-WorldLink The record for the longest punt in NFL history is 98 yards, kicked by Steve O'Neal on September 21, 1969.

Source: National Football League

Graph a Piecewise-Defined Function

{ 3x 2 i f x < - 1 - 1 i f - 1 < x < 4.

(x - 5) 3 + 2 i f x > 4

O n the interval (—oo, —1), graph y = 3x2. O n the interval [—1,4), graph the constant function y = — 1 . O n the interval [4, oo), graph y = (x — 5 ) 3 + 2.

Draw circles at (—1,3) and (4, —1) and dots at (—1, —1) and (4,1) because/(-l) = - 1 and/(4) = 1 .

w GuidedPractice

i

Graph each funct ion. r x - 5 i f x < 0

5A. 0 < x < 2

x > 2 5B. /z(x)

(x + 6 ) 2 if x < - 5 7 if - 5 < x < 2

|4-x| i f x > 2

You can also use what you have learned about transformations to transform functions that model real-world data or phenomena.

UiiJOiy D Transformations of Functions

FOOTBALL The path of a 60-yard p u n t can be modeled by g(x) = —jgx2 + 4x + 1 , where g(x) is the vertical distance i n yards of the footbal l f r o m the ground and x is the horizontal distance i n yards such that x = 0 corresponds to the k i c k i n g team's 20-yard l ine .

a. Describe the transformations of the parent function/(x) = x 2 used to graph g(x).

Rewrite the function so that it is i n the form g(x) = a(x — h)2 + k by completing the square.

^ - x 2 + 4x + 1

- 60x) + 1

(x 2 - 60x

(x - 30) 2 + 61

- U 2 + 4x. 15

900) + 1 + Y5 (900)

Original function

Factor

Complete the square.

Write x 2 — 60x + 900 as a perfect square and simplify.

So, g(x) is the graph of/(x) translated 30 units r ight, compressed vertically, reflected i n the x-axis, and then translated 61 units up.

b. Suppose the punt was f r o m the k i c k i n g team's 30-yard l ine . Rewrite g(x) to reflect this change. Graph both functions on the same graphing calculator screen.

A change of position f rom the kicking team's 20- to 30-yard line is a horizontal translation of 10 yards to the right, so subtract an additional 10 yards f rom inside the squared

expression. [0,100] scl: 1 by [0,100] scl: 20

3 0 - 1 0 ) 2 + 61 or£(x) 15

(x - 40) 2 + 61

w GuidedPracti c e

6. ELECTRICITY The current i n amps f lowing through a D V D player is described by I(x) -where x is the power in watts and 11 is the resistance i n ohms. A. Describe the transformations of the parent function/(x) = V x used to graph I(x).

B. The resistance of a lamp is 15 ohms. Write a function to describe the current f lowing through the lamp.

C. Graph the resistance for the D V D player and the lamp on the same graphing calculator screen.

1 1 '

50 | L e s s o n 1-5 l P a r e n t F u n c t i o n s a n d T r a n s f o r m a t i o n s

Another nonridgid transformation involves absolute value.

HrtnologfTip Bolute Value Transformations

ran check your graph of an so Jte value transformation by sng your graphing calculator. L ran also graph both functions

TB same coordinate axes.

Concept Transformations with Absolute Value

9(x)=\f(x)\

This transformation reflects any portion of the graph of f(x) that is below the x-axis so that it is above the x-axis.

\" L

, \ \y=M

J '

V y // o X

gix) = \f(x)\

m=n\x\)

This transformation results in the portion of the graph of f(x) that is to the left of the y-axis being replaced by a reflection of the portion to the right of the y-axis.

\V V V 0 X

/< q X

g(x) = rthfl)] X X

y

/ \ / \ f 0 X

—, 1 \ i — i 0 = --X3- - 4 x

Figure 1.5.6

Describe and Graph Transformations Use the graph of f(x) = x 3 — 4x i n Figure 1.5.6 to graph each funct ion.

a. £ ( x ) = |/(x)|

The graph of f(x) is below the x-axis on the intervals (—oo, —2) and (0,2), so reflect those portions of the graph in the x-axis and leave the rest unchanged.

b. h(x) = f(\x\)

Replace the graph of f(x) to the left of the y-axis w i t h a reflection of the graph to the right of the y-axis.

V /

0 / X

g(x) = \f(x)\ g(x) = \f(x)\

— -Mx) = f(lxl)l

• GuidedPractice

Use the graph o f / ( x ) shown to graph ^ ( x ) = |/(x)| and h(x) =/(|x|).

7A. 7B. y

I_ 4 4 f (x) [2 x] f (x) [2 x]

—( >— 7 - 2 0 I- 4 X

-2 , I -2 i—(

connectED.mcgraw-hill.com 51

Exercises = Step-by-Step Solutions begin on page R29.

Describe the f o l l o w i n g characteristics of the graph of each parent funct ion: domain, range, intercepts, symmetry, continuity, end behavior, and intervals on w h i c h the graph is increasing/decreasing. (Example 1)

.3 1. / ( * ) = [ * ]

4. f{x) = x 4

2. f(x)

5. f(x)

3. /(x) = ; r

6. / ( * ) = *

Use the graph of f(x) = Vx to graph each funct ion. (Example 2}

7. g(x) = V x - 4 8. g(x) = \Jx + 3

9. g(x) = \/x + 6 - 4 10. g{x) = V x - 7 + 3

Use the graph of/(x) = j to graph each funct ion. (Example 2)

1 1 . gix)

13. g(x)

x) = i + 4 7 X

12. £(x) = ^ - 6

x - 6 14. g(x) = j ^ - 4

Describe h o w the graphs of/(x) = [x] andg(x) are related. Then wri te an equation forg(x) . (Example 3)

15. y

0 9(x)\

I T

16. n : : •©—I

r

T v

0 X

17. 18. y

•OHh /-» i — (>

• -ih

M l PROFIT A n automobile company experienced an unexpected two-month delay on manufacturing of a new car. The projected profit of the car sales before the delay p(x) is shown below. Describe how the graph of p(x) and the graph of a projection including the delay d(x) are related. Then wri te an equation for d(x). (Example 3)

Projected First-Quarter Profit

s re o = - E O

1 ° 40 p{x) = 10x3 - 70x2 + 1 5 0 x - 2

1 2 3

Months After January

Describe how the graphs of/(x) = |x| andg(x) are related. Then wr i te an equation for gix). (Example 3)

20. y u

4 9W

- 4 0 4 8 1 2x

-4 -4

- - 8

i - - 8

i

21, 0 y U

A tix) 4 tix)

I y i fc

- 8 _S 0 8 x

I

-8; -8;

22. 0 y u

A g(x) H g(x)

- 8 - 4 0 /A 8 x

-8 -8

23. n y U

A (x) Q (x)

-I ] - 4 ' A 8 x

-4 -4

P - o

Ident i fy the parent funct ion fix) of gix), and describe h o w the graphs of g(x) and/(x) are related. Then graph fix) and gix) on the same axes. (Example 4)

24. gix) = 3|x| - 4

26. ^ ) = ^ T

28. gix) = - 5 [ x - 2]

so. m=fx+7

25. g(x) = 3Vx + 8

27. gix) = 2[x - 6]

29. g(x) = -2|x + 5|

3 1 . gix)=^±*

Graph each funct ion . (Example 5)

[ - x 2 if x < - 2 32. fix) =\ i f - 2 < x < 7

( x - 5 ) 2 + 2 i f x > 7

33. gix)

34. /(x)

35. h(x)

36. #(x)

37. fix) =

x + 4 if x < - 6 1 X

i f - 6 < x < 4

6 if x > 4

4 i f x < - 5 X 3 i f - 2 < x < 2

V x + 3 if x > 3

| x - 5 | if x < - 3 4x - 3 i f - 1 < x < 3

V ^ if x > 4

2 if x < - 4 - 3x 3 + 5 if - 1 < x < 1

[x] + l i f x > 3

- 3 x - 1 if x < - 1 0.5x + 5 if - 1 < x < 3

| x - 5 | + 3 if x > 3

52 I L e s s o n 1-5 I P a r e n t F u n c t i o n s a n d T r a n s f o r m a t i o n s

38. POSTAGE The cost of a first-class postage stamp i n the U.S. from 1988 to 2008 is shown i n the table below. Use the data to graph a step function. (Example 5)

Year Price (0)

1988 25

1991 29

1995 32

1999 33

2001 34

2002 37

2006 39

2007 41

2008 42

38. BUSINESS A no-contract cell phone company charges a flat rate for daily access and $0.10 for each minute. The cost of the plan can be modeled by c(x) = 1.99 + 0.1 [xfl, where x is the number of minutes used. (Example 6)

a. Describe the transformation(s) of the parent function f(x) = [x] used to graph c(x).

b. The company offers another plan in which the daily access rate is $2.49, and the per-minute rate is $0.05. What function c(x) can be used to describe the second plan?

c. Graph both functions on the same graphing calculator screen.

d. Would the cost of the plans ever equal each other? If so, at how many minutes?

40. GOLF The path of a drive can be modeled by the function shown, where g(x) is the vertical distance in feet of the ball from the ground and x is the horizontal distance i n feet such that x = 0 corresponds to the init ial point. (Example 6)

Drive Path

g y u

I ' 6

Q "TO 8 w r

> 0

-g( x) = = 0. 76, < — 0.0( )04 g y u

I ' 6

Q "TO 8 w r

> 0

g y u

I ' 6

Q "TO 8 w r

> 0

g y u

I ' 6

Q "TO 8 w r

> 0

g y u

I ' 6

Q "TO 8 w r

> 0

g y u

I ' 6

Q "TO 8 w r

> 0

g y u

I ' 6

Q "TO 8 w r

> 0 100 200 300 400 x

Horizontal Distance (ft)

a. Describe the transformation(s) of the parent function f(x) = x2 used to graph g(x).

b. If a second golfer hits a similar shot 30 feet farther d o w n the fairway from the first player, what function h(x) can be used to describe the second golfer's shot?

C. Graph both golfers' shots on the same graphing calculator screen.

d. If both golfers hit their shots at the same time, at what horizontal and vertical distances w i l l the shots cross paths?

Use the graph of/(x) to graphg(x) = |/(x)| and h(x) = /(|x|). (Example 7)

41- /(*) = !

43. f{x) = x 4 - x 4x'

+ 5

42. f{x) = V - T - 4

44. fix) = | x 3 + 2x2 - 8x - 2

46. fix) = V * + 2 - 6

47. TRANSPORTATION I n New York City, the standard cost for taxi fare is shown. One unit is equal to a distance of 0.2 mile or a time of 60 seconds when the car is not i n motion. $2.50 per trip plus

$0.40 per unit

a. Write a greatest integer function fix) that w o u l d represent the cost for units of cab fare, where x > 0. Round to the nearest unit.

b. Graph the function.

c. H o w w o u l d the graph of fix) change if the fare for the first unit increased to $3.70? Graph the new function.

48. PHYSICS The potential energy i n joules of a spring that

has been stretched or compressed is given by pix) =

where c is the spring constant and x is the distance from the equil ibrium position. When x is negative, the spring is compressed, and when x is positive, the spring is stretched.

mmmw mnm QJUUUU Compressed Equilibrium Stretched

a. Describe the transformation(s) of the parent function f[x) = x2 used to graph pix).

b. The graph of the potential energy for a second spring passes through the point (3,315). Find the spring constant for the spring and wri te the function for the potential energy.

Write and graph the funct ion w i t h the given parent funct ion and characteristics.

(49) fix) = j ; expanded vertically by a factor of 2; translated

7 units to the left and 5 units up

50. fix) = f x ] ; expanded vertically by a factor of 3; reflected i n the x-axis; translated 4 units d o w n

PHYSICS The distance an object travels as a funct ion of t ime

is given by fit) = -^at2 + v0t + XQ, where a is the

acceleration, v0 is the i n i t i a l velocity, and x 0 is the i n i t i a l posit ion of the object. Describe the transformations of the parent funct ion fit) = t2 used to graph fit) for each of the f o l l o w i n g .

51. a

53. a

2, v< 2, x0 = 0

4, y n = 8, x.

M

52. a = 2, v0 = 0, x0 = 10

54. a = 3, v0 = 5, x0 = 3

m e c t E ^ i T c g r a ^ i i l ^ o n J 53

Write an equation for each g(x).

55. p\ y * u \ s J

(4, 6) s

8 x A

-H

0

i

56.

57. 58. > i I fix)

i o i I fix)

-o

M Mr-

-©--

J

•-o -o

59. SHOPPING The management of a new shopping mall originally predicted that attendance in thousands w o u l d follow f(x)= \[7x for the first 60 days of operation, where x is the number of days after opening and x = 1 corresponds w i t h opening day. Write g(x) i n terms of f(x) for each situation below.

a. Attendance was consistently 12% higher than expected.

b. The opening was delayed 30 days due to construction.

c. Attendance was consistently 450 less than expected.

Ident i fy the parent function/(x) of gix), and describe the transformation oif(x) used to graph gix).

60. y

- 8 - 4 0 4 8 x

s v4 I f 1 v4 f_ .9 i j)

/ \ I

/ 12 \y=gix)

12 vr

6 1 . 0 y 0 A 4(3, 5)

- 8 - 4 0 4 8 x

-4 K = = n(x) -4 J 7 /

-8 -8

62. 63. 8 y 8

• \

4 \ >) 4 V 17, 3)

6 20 60 80x

-4 1

-4 y = g\x) P

-o

Use fix) to graph g(x).

64. gix) = 0.25/(*) + 4

65. = 3/(x) - 6

66. g ( . r ) = / ( . v - 5 ) + 3

67. = -2fix) + 1

0 y (J

V /I

\ 4

- 8 - 4 \ O * 8 x > /i 1

-8 -8

Use fix) = — 2 = -y/x + 6

68. gix) = 2fix) + 5

70. gix)=fi4x)-5

4 to graph each funct ion.

69. gix) = -3fix) + 6

7 1 . g{x) =/(2x + 1) + 8

72. J MULTIPLE REPRESENT/: In this problem, you w i l l investigate operations w i t h functions. Consider

• fix) = x2 + 2x + 7,

• = 4x + 3, and

• hix) = x2 + 6x + 10.

a. TABULAR Copy and complete the table below for three values for a.

9ia) m + M m

b. VERBAL H o w are fix), gix), and hix) related?

c. ALGEBRAIC Prove the relationship from part b algebraically.

H.O.T. Problems Use Higher-Order Thinking Skills

73. ERROR ANALYSIS Danielle and Miranda are describing the transformation gix) = \x + 4 ] . Danielle says that the graph is shifted 4 units to the left, whi le Miranda says that the graph is shifted 4 units up. Is either of them correct? Explain.

74. REASONING Let fix) be an odd function. If gix) is a reflection of fix) in the x-axis and hix) is a reflection of gix) i n the y-axis, what is the relationship between fix) and hix)? Explain.

75. WRITING IN MATH Explain w h y order is important when transforming a function w i t h reflections and translations.

REASONING Determine whether the f o l l o w i n g statements are sometimes, always, or never true. Explain your reasoning.

76. If/(x) is an even function, then/(x) = |/(x)|.

77. If/(x) is an odd function, then/(—x) = — |/(x)|.

78. If/(x) is an even function, then/(—x) = — |/(x)|.

79) CHALLENGE Describe the transformation of fix) = V x if (—2,-6) lies on the curve.

80. REASONING Suppose (fl, b) is a point on the graph of fix). Describe the difference between the transformations of ia, b) when the graph of fix) is expanded vertically by a factor of 4 and when the graph of fix) is compressed horizontally by a factor of 4.

8 1 . WRITING IN MATH Use words, graphs, tables, and equations to relate parent functions and transformations. Show this relationship through a specific example.

54 | L e s s o n 1-5 | P a r e n t F u n c t i o n s a n d T r a n s f o r m a t i o n s

Spiral Review

d the average rate of change of each funct ion on the given interval . (Lesson 1 -4)

C g(x) = -2x2 + x - 3; [ - 1 , 3 ] 83. g(x) = x2 - 6x + 1; [4, 8] 84. f{x) = -2x3 - x2 + x - 4; [ - 2 , 3 ]

fee the graph of each funct ion to describe its end behavior. Support the conjecture umerically. (Lesson 1-3)

86. f{x) = ^ f -x xl

85- q(x) 87. pix) x + 2 x-3

Use the graph of each funct ion to estimate its y-intercept and zero(s). Then f i n d these values algebraically. (Lesson 1 -2)

y8 j y8 J -4 -4 j

-I } - 4 0 UJ I ]x A

(x 3 y - (x 3

I I I

90.

\Jx-2 - 1

-8 -4 I S

8x

9 1 . GOVERNMENT The number of times each of the first 42 presidents vetoed bills are listed below. What is the standard deviation of the data? (Lesson 0-8)

2, 0, 0, 7, 1 , 0, 12, 1, 0, 10, 3, 0, 0, 9, 7, 6, 29, 93, 13, 0, 12, 414, 44, 170, 42, 82, 39, 44, 6, 50, 37, 635, 250, 181, 21, 30, 43, 66, 31, 78, 44, 25

92. LOTTERIES In a multi-state lottery, the player must guess which five of the white balls numbered from 1 to 49 w i l l be drawn. The order i n which the balls are drawn does not matter. The player must also guess which one of the red balls numbered f rom 1 to 42 w i l l be drawn. H o w many ways can the player complete a lottery ticket? (Lesson 0-7)

Skills Review for Standardized Tests

93. SAT/ACT The figure shows the graph of y = gix), which has a m i n i m u m located at (1, —2). What is the maximum value of hix) = —3gix) — 1?

\ y / \ A / \ 4 / \ n /

/ \ / - 2 4 X

-2 \ -2 (x) L (x)

A 0

B 1

C 2

D 3

E It cannot be determined f rom the information given.

94. REVIEW What is the simplified form of 4 x 3 y 2 2 _ 1 ,

{x-Vzf

95. What is the range of y = —-—?

F {y\y± +2V2} G { y | y > 4 }

H { y | y > 0 }

J {y I y < 0}

96. REVIEW What is the effect on the graph of y = kx2 as k decreases from 3 to 2?

A The graph of y = 2x2 is a reflection of the graph of y = 3x2 across the y-axis.

B The graph is rotated 90° about the origin.

C The graph becomes narrower.

D The graph becomes wider.

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