Stewart J. College Algebra: Concepts and Contexts

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f 1x 2 f 1x ⴙ 22

- 6

105

- 4

- 2

0 53

2 20

4 6

6 2 —

f 1x 2 f 1x2ⴙ 1

- 3

- 2

- 1

0 42

1 39

2 70

3 21

424 CHAPTER 5

■

Quadratic Functions and Models

19–24

■ A function f is given by the table. Complete the table for the given transformation.

19. 20.

21. 22.

f 1x 2 f 1x2ⴚ 1

- 3

- 2

- 1

0 20

1 31

2 27

3 19

f 1x 2 f 1x ⴚ 22

- 6

13 —

- 4

- 2

0 49

2 55

4 62

6 75

f 1x 2 f 1ⴚx2

- 6

105

- 4

- 2

0 53

2 20

4 6

6 2

f 1x 2 1 ⴚ f 1x2

- 6

105

- 104

- 4

- 2

0 53

2 20

4 6

6 2

25–28 ■ Sketch the graph of the function, not by plotting points, but by starting with the

graph of a basic function and applying a vertical shift.

25. 26.

27. 28.

29–32

■ Sketch the graph of the function, not by plotting points, but by starting with the

graph of a basic function and applying a horizontal shift.

29. 30.

31. 32. f 1x 2= 1x + 4

f 1x 2= 0x - 3 0

f 1x 2= 1x + 12

f 1x 2= 1x - 52

f 1x 2= 0x 0- 1f 1x2= 1x + 1

f 1x 2= x

+ 5f 1x 2= x

- 1

23. 24.

SECTION 5.1

■

Working with Functions: Shifting and Stretching 425

33–36 ■ Sketch the graph of the function, not by plotting points, but by starting

with the graph of a basic function and applying vertical stretching or

shrinking.

33. 34.

35. 36.

37–52

■ Sketch the graph of the function, not by plotting points, but by starting with the

graph of a basic function and applying transformations.

37. 38.

39. 40.

41. 42.

43. 44.

45. 46.

47. 48.

49. 50.

51. 52.

53–60

■ A function f is given, and the indicated transformations are applied to its graph (in

the given order). Write the equation for the final transformed graph.

53. ; shift upward 3 units

54. ; shift downward 1 unit

55. ; shift 2 units to the left

56. ; shift 1 unit to the right

57. ; shift 3 units to the right and shift upward 1 unit

58. ; shift 2 units to the left and reflect in the x-axis

59. ; shift downward 2 units

60. ; shift 2 units to the right and reflect in the y-axis

61–64

■ The graphs of f and are given. Find a formula for the function .

61. 62.

f 1x 2= log

f 1x 2= 2

f 1x 2= x

f 1x 2= 0 x 0

f 1x 2= 1x

f 1x 2= x

f 1x 2=-log

1x + 22f 1x 2=-log

1- x2

f 1x 2= log

1x + 22f 1x 2= log

1x - 42

f 1x 2= 1 + 2

-x

f 1x 2=-3

f 1x 2= 3

+ 5f 1x 2= 2

- 3

f 1x 2= 3 - 0

x - 2 0f 1x2= 6 - 1x + 4

f 1x 2=-1x + 42

- 3f 1x 2=-1x - 32

+ 5

f 1x 2=-1x

f 1x 2=-0 x 0

f 1x 2=-2 - x

f 1x 2= 1 - x

f 1x 2=

f 1x 2= 2x

f 1x 2= 3 0 x 0f 1x2=

f(x) = x

f(x) =

log¤ x

426 CHAPTER 5

■

Quadratic Functions and Models

65. Bungee Jumping Luisa goes bungee jumping from a 500-foot-high bridge. The

graph shows Luisa’s height (in feet) after t seconds.

(a) Describe in words what the graph indicates about Luisa’s bungee jump.

(b) Suppose Luisa goes bungee jumping from a 400-foot-high bridge. Sketch a new

graph that shows Luisa’s height after t seconds.

Express the function H in terms of h.

H1t 2

h1t 2

CONTEXTS

t (s)

(ft)

500

f(x) = 2

12_2 _1

f(x) = x

63. 64.

66. Train Ride Liam is riding a train that is traveling at a constant speed of 20 m/s. Liam

is inside the train walking in the direction opposite to that in which the train is moving.

Liam’s speed relative to the train at time t (in seconds) is /s. Find a

function that gives Liam’s speed relative to the ground. How are the graphs of and

related?

67. Distance Alessandra leaves home 30 minutes after her brother Alberto and drives at a

constant speed of 90 km/h on a straight road.

(a) Find a function d that models Alessandra’s distance from home t hours after she

started her trip.

(b) Find a function D that models Alessandra’s distance from home t hours after

Alberto started his trip.

68. Sales Growth The annual sales of a certain company can be modeled by the function

, where t represents years since 1990 and is measured in millions

of dollars.

(a) What shifting and shrinking operations must be performed on the function to

obtain the function ?

(b) Suppose you want t to represent years since 2000 instead of 1990. What

transformation would you have to apply to the function to accomplish

this? Write the new function that results from this transformation.y = g1t2

y = f 1t 2

y = t

f 1t 2f 1t 2= 4 + 0.01t

V√V

√1t 2= 415t

SECTION 5.1

■

Working with Functions: Shifting and Stretching 427

69. Swimming Laps Miyuki practices swimming laps with her team. The function

graphed below gives her distance (in meters) from the starting edge of the pool

t seconds after she starts her laps.

(a) Describe in words Miyuki’s swim practice. What is her average speed for the first

30 seconds?

(b) Graph the function . How is the graph of the new function related to the

graph of the original function?

y = 1.2f 1t 2

y = f 1t 2

t (s)

d (m)

t (min

)

d (ft)

200

70. Field Trip A class of fourth graders walks to a park on a field trip. The function

graphed in the margin gives their distance from school (in feet) t minutes after

they left school.

(a) What is the average speed going to the park? How long was the class at the park?

How far away is the park?

(b) Graph the function . How is the graph of the new function related to

the graph of the original function? What is the average speed going to the new

park? How far away is the new park?

the graph of the original function? How does the field trip described by this

function differ from the original trip?

71. Bacterial Infection A culture of the bacteria Streptococcus pneumoniae

(S. pneumoniae) initially has 10 bacteria and increases at a rate of 5% per minute.

(a) Find a function that represents the number of bacteria in the culture after

t minutes.

(b) Suppose the culture initially has 50 bacteria and is the number of bacteria after

t minutes. Describe the stretching or shrinking operations that must be performed

on the function to obtain the function . Write the new function

that results from this transformation.

72. Rabbit Population A population of 20 rabbits is introduced to a small island. The

population increases at a rate of 60% per year.

(a) Find a function that represents the number of rabbits on the island after t years.

(b) Suppose 100 rabbits were introduced to the island and is the number of rabbits

after t years. Describe the stretching or shrinking operations that must be performed

on the function to obtain the function . Write the new function

that results from this transformation.

73. Newton’s Law of Cooling A cup of coffee has a temperature of and is placed

in a room that has a temperature of . The temperature of the coffee after t minutes

is modeled by the function

(a) Describe the transformations you would apply to the function to obtain

the function .

(b) Graph the function T by applying transformations to the graph of .y = 10.95 2

y = T1t2

y = 10.95 2

T1t 2= 75 + 11510.95 2

75°F

190°F

y = g1t2

y = g1t2y = f 1t 2

g1t 2

f 1t 2

y = g1t2

y = g1t2y = f 1t 2

g1t 2

f 1t 2

y = f 1t - 102

y = 0.50f 1t 2

y = f 1t 2

428 CHAPTER 5

■

Quadratic Functions and Models

■ The Squaring Function

A squaring function is a function of the form

where C is a number different from zero. We’ll see in this section that the squaring

function is the prototype quadratic function.

We all know that when we drop an object from a high building, the object falls

faster and faster. In fact, the distance d (in feet) it falls in t seconds is given by the

function . But suppose that you make a mistake in your physics class and

you write this function as (not squared). This apparently small error

makes a huge difference in reality. If objects fell according to this rule, we would

have quite a different world. For one thing, we would be able to leap over tall trees

with ease, and we wouldn’t need a parachute to go skydiving. We compare these two

different functions in the next example.

d1t 2= 16t

f 1x2= Cx

Every time you throw a ball, it

travels according to a quadratic

function.

example

Comparing Linear and Squaring Functions

Dave and his friend Harriet (who has superpowers) go skydiving together. In t sec-

onds Dave falls , but Harriet falls feet.

(a) Make a table of values for the functions D and H, and find their average rate

of change on intervals of length 1 second, for t between 0 and 5 seconds.

(b) What do the tables in part (a) indicate about how fast Dave and Harriet are

falling? Which skydiver do you think will have the softer landing?

H1t 2= 16tD1t 2= 16t

5.2 Quadratic Functions and Their Graphs

■

The Squaring Function

■

Quadratic Functions in General Form

■

Quadratic Functions in Standard Form

■

Graphing Using the Standard Form

IN THIS SECTION… we study quadratic functions and compare them to linear

functions. We learn how to express quadratic functions in standard form and then use the

standard form to graph such functions.

GET READY… by reviewing how to expand expressions and how to complete the square

in Algebra Toolkits B.1 and B.2. Test your understanding by doing the Algebra

Checkpoint at the end of this section.

Some of the most important functions in modeling the real world are functions that

involve squaring a quantity. Such functions are called quadratic functions. For ex-

ample, you are familiar with the function that gives the area of a square, ,

or the area of a circle, . One of the most fundamental functions in our

everyday experience is the quadratic function . This function gives the

distance d that a falling object travels in t seconds under the force of gravity. Every

time you throw a ball, drop your keys, or ride a roller coaster, you are experiencing

this function in action. In this section we begin our study of this important family of

functions.

d1t 2= 16t

A1r 2= pr

A1x 2= x

SECTION 5.2

■

Quadratic Functions and Their Graphs 429

H(t)  16 t Harriet

DaveD(t)  16 t

The average rate of change of

distance with respect to time is the

average speed. Average speed is

studied in Section 2.1.

D1t 2

Rate of

change

0 0 —

1 16 16

2 64 48

3 144 96

4 256 128

5 400 160

H1t 2

Rate of

change

0 0 —

1 16 16

2 32 16

3 48 16

4 64 16

5 80 16

these graphs?

Solution

(a) The tables below give the values and average rates of change of the functions

D and H.

d (ft)

t (s)

12345

100

200

300

400

figure 1 Graphs of D and H

(b) Since H is a linear function, its rate of change is constant (Section 2.2). So

Harriet falls at a constant speed of 16 ft/s. Dave falls a lot faster; from the

table we see that his average speed increases on each succeeding one-second

interval. Clearly, Harriet will have the softer landing.

and how far Harriet has fallen keeps widening with passing time.

■ NOW TRY EXERCISE 43 ■

Example 1 shows that a squaring function grows much faster than a linear function.

On the other hand, exponential functions grow much faster than squaring functions, as

we will see in Example 1 of Section 6.2, page 494.

■ Quadratic Functions in General Form

A quadratic function is a combination of a squaring function and a linear function.

Quadratic Functions

A quadratic function is a function of the form

where a, b, c are real numbers and . This form of a quadratic function

is called the general form.

a  0

f 1x2= ax

+ bx + c

430 CHAPTER 5

■

Quadratic Functions and Models

example

Identifying Quadratic Functions

Determine whether the given function is a quadratic function.

(a) (b) (c)

Solution

(a) Notice that , so f is a quadratic function with ,

, and .

(b) The function is not a quadratic function, since has a term involving .

term involving .

■ NOW TRY EXERCISE 7 ■

c = 3b = 0

a = 0.1f 1x2= 0.1x

+ 0

x + 3

h1x 2= 3

x + 1g1x 2= x

- 2x + 1f 1x2= 0.1x

+ 3

example

Expressing a Quadratic Function in General Form

Express the function in the general form of a quadratic

function.

Solution

We use the distributive property to multiply the factors of f:

Given function

Distributive property

Combine like terms

The general form is .

■ NOW TRY EXERCISE 9 ■

f 1x2= 2x

- 16x - 96

= 2x

- 16x - 96

= 2x

+ 8x - 24x - 96

f 1x 2= 12x + 8 21x - 122

f 1x2= 12x + 821x - 122

We study how to multiply

algebraic expressions using the

distributive property in Algebra

Toolkit B.1, page T25.

example

Expressing a Quadratic Function in General Form

Express the function in the general form of a quadratic

function.

Solution

We multiply out the squared term of f using Special Product Formula 1 (see

Algebra

Toolkit B.1

Given function

Special Product Formula 1

Multiply

Combine constant terms

The general form is .

■ NOW TRY EXERCISE 13 ■

f 1x2= 3x

- 24x + 49

= 3x

- 24x + 49

= 3x

- 24x + 48 + 1

= 31x

- 8x + 162+ 1

f 1x 2= 31x - 42

+ 1

f 1x2= 31x - 4 2

+ 1

SECTION 5.2

■

Quadratic Functions and Their Graphs 431

■ Quadratic Functions in Standard Form

In Example 4 we saw that the function can be expressed in the

general form of a quadratic function. This is an example of a quadratic function in

standard form according to the following definition.

Standard Form of a Quadratic Function

f 1x2= 31x - 4 2

- 1

example

Expressing Quadratic Functions in Standard Form

Express the quadratic function in standard form.

Solution

To get f in standard form we “complete the square” by first taking half of the coeffi-

cient of x and squaring it: . We then add and subtract the result.

The standard form is .

■ NOW TRY EXERCISE 21 ■

f 1x2= 1x + 82

- 40

= 1x + 82

- 40

= 1x

+ 16x + 642- 64 + 24

perfect square

f 1x 2= x

+ 16x + 24

= 64

f 1x2= x

+ 16x + 24

A quadratic function can be expressed in the standard

form

by completing the square.

f 1x2= a1x - h 2

+ k

f 1x2= ax

+ bx + c

The next example shows how “completing the square” allows us to express a

quadratic function in standard form.

We study how to factor perfect

squares in Algebra Toolkit B.2,

page T33.

Given function

Complete the square: Add 64 inside

parentheses, and subtract 64 outside

Factor and simplify

example

Expressing a Quadratic Function in Standard Form

Express the quadratic function in standard form.

Solution

Since the coefficient of is not 1, we must factor this coefficient from the terms in-

volving x before completing the square.

The standard form is .

■ NOW TRY EXERCISE 23 ■

f 1x2= 21x - 3 2

+ 5

= 21x - 32

+ 5

= 21x

- 6x + 92- 2

9 + 23

perfect square

= 21x

- 6x2+ 23

f 1x 2= 2x

- 12x + 23

f 1x2= 2x

- 12x + 23

Given function

Factor 2 from x-terms

Complete the square: Add 9 inside

parentheses, and subtract outside

Factor and simplify

432 CHAPTER 5

■

Quadratic Functions and Models

■ Graphing Using the Standard Form

Recall that the graph of the squaring function is a U-shaped curve called

a parabola. The vertex of this parabola is at the point (0, 0) (see Example 3 in

Section 1.6). The graph of f is shown in Figure 2(a).

f 1x2= x

Vertex (0, 0)

(a) Graph of f(x)=x

(b) Graph of f(x)=2(x-1)

Vertex (1, 3)

figure 2

If we transform f by shifting 1 unit to the right, stretching by a factor of 2, and

then shifting upward 3 units, we get the function

We recognize this as a quadratic function in standard form (see Figure 2(b)). Since

any quadratic function can be put into standard form, we see that the graph of any

quadratic function is a parabola obtained by transforming the basic squaring func-

tion . Notice that the standard form of a quadratic function allows us to

quickly identify the location of the vertex. In this example the vertex is at (1, 3)

because the original vertex (0, 0) has been shifted 1 unit to the right and 3 units

upward.

Graphs of Quadratic Functions

f 1x2= x

f 1x2= 21x - 1 2

+ 3

To graph a quadratic function, we first express it in standard form.

The graph of the quadratic function in standard form

is a parabola with vertex ; the parabola opens upward if or

downward if .a 6 0

a 7 01h, k 2

f 1x2= a1x - h 2

+ k

Ï=a(x-h)™+k, a>0

Ï=a(x-h)™+k, a<0

Vertex (h, k)

SECTION 5.2

■

Quadratic Functions and Their Graphs 433

example

Graphing a Quadratic Function

Let .

(a) Express f in standard form.

(b) Sketch a graph of f.

Solution

(a) In Example 6 we found that the standard form of f is

(b) The standard form tells us that we get the graph of f by taking the parabola

and performing the following transformations: Shift to the right 3 units,

stretch by a factor of 2, then shift upward 5 units. The vertex of the parabola is at

(3, 5). The parabola opens upward because the coefficient 2 is positive.

Vertex (3, 5)

Opens upward

The y-intercept is obtained by evaluating f at 0. This is more easily done by

using the general form of a quadratic function. In this case

So the y-intercept is 23. The graph is shown in Figure 3.

■ NOW TRY EXERCISE 29 ■

f 102= 210 2

- 1210 2+ 23 = 23

f 1x2= 21x - 3 2

+ 5

y = x

f 1x2= 21x - 3 2

+ 5

f 1x2= 2x

- 12x + 23

Ï=2(x-3)™+5

Vertex (3, 5)

figure 3

example

Graphing a Quadratic Function

Let .

(a) Express f in standard form.

(b) Sketch the graph of f.

Solution

(a) To express this quadratic function in standard form, we complete the square.

(b) From the standard form we see that the graph is a parabola with vertex .

Vertex

Opens downward

f 1x2=-Ax -

=-Ax -

=-Ax

- x +

B+ 2 - 1- 12

=-1x

- x2+ 2

f 1x 2=-x

+ x + 2

f 1x2=-x

+ x + 2

perfect square

Given function

Factor from the x-terms

Complete the square: Add inside

parentheses, and subtract outside

Factor and simplify

A- 1

- 1