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6 Working with Functions: Graphs and Graphing Calculators

6 Working with Functions: Graphs and Graphing Calculators

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SECTION 1.6

Working with Functions: Graphs and Graphing Calculators

65

(b) We make a scatter plot of the data and connect the points with a smooth curve as in

Figure 2(b). Notice that the height of the graph is the value of the function; in other

words, the height of the graph is the temperature of the water at the given time.

T (ºF)

T (ЊF)

100

100

80

60

40

70

5

0

40 x (min)

20

(a) Rough graph

10

20

30

40

50 x (min)

(b) Graph from data

f i g u r e 2 Graph of water temperature as a function of time

NOW TRY EXERCISE 49

It would be nice to have a precise rule (function) that gives the temperature of

the water at any time. Having such a rule would allow us to predict the temperature

of the water at any time and perhaps warn us about getting scalded. We don’t have

such a rule for this situation, but as we study more functions with different graphs,

we may be able to find a function that models this situation.

2

■ Graphs of Basic Functions

When the rule of a function is given by an equation, then to graph the function, we

first make a table of values. We then plot the points in the table and connect them by

a smooth curve. Let’s try graphing some basic functions.

In the next example we graph a constant function, that is, a function of the form

f 1x2 = c

where c is a fixed constant number. Notice that a constant function has the same output c for every value of the input.

e x a m p l e 2 Graph of a Constant Function

Graph the function f 1x 2 = 3.

Solution

We first make a table of values. Then we plot the points in the table and join them by

a line as in Figure 3. Notice that the graph is a horizontal line 3 units above the x-axis.

x

f(x) ‫ ؍‬3

-3

-2

-1

0

1

2

3

3

3

3

3

3

3

3

y

2

0

2

x

f i g u r e 3 Graph of f 1x2 = 3

NOW TRY EXERCISE 7

66

CHAPTER 1

Data, Functions, and Models

In the next example we graph the function f 1x2 = x. For each input, this function gives the same number as output. This function is called the identity function

(the output is identical to the input). We also graph f 1x2 = x 2, whose graph has the

shape of a parabola.

e x a m p l e 3 Graphs of Basic Functions

Graph the function.

(a) f 1x2 = x

(b) f 1x2 = x 2

(c) f 1x2 = x 3

Solution

We first make a table of values. Then we plot the points in the table and join them by

a line or smooth curve as in Figures 4, 5, and 6.

(a) f 1x2 = x

(b) f 1x2 = x 2

(c) f 1x2 = x 3

x

f 1x2 ‫ ؍‬x

-3

-2

-1

0

1

2

3

-3

-2

-1

0

1

2

3

f 1x2 ‫ ؍‬x2

x

-2

-1

- 12

0

4

1

1

2

1

4

1

2

1

4

1

4

0

x

f 1x2 ‫ ؍‬x3

- 32

-1

- 12

0

- 278

-1

- 18

0

1

2

1

8

1

1

3

2

27

8

y

y

y

2

2

2

0

2

figure 4

0

x

2

figure 5

0

x

2

x

figure 6

NOW TRY EXERCISES 15, 19, AND 23

Another basic function is the square root function f 1x2 = 1x, which we

graph in the next example.

e x a m p l e 4 Graph of the Square Root Function

Graph the function.

(a) f 1x2 = 1x

(b) g1x 2 = 1x + 3

SECTION 1.6

Working with Functions: Graphs and Graphing Calculators

67

Solution

We first make a table of values. Since the domain of f is 5x Έ x Ú 06 , we use only nonnegative values for x. Then we plot the points in the table and join them by a line or

smooth curve as in Figures 7 and 8.

(a) f 1x2 = 1x

(b) g1x 2 = 1x + 3

x

f 1x2 ‫ ؍‬1x

0

0

1

4

1

2

1

2

3

4

5

1

12

13

2

15

Decimal

x

g1x2 ‫ ؍‬1x ؉ 3

Decimal

0

0.5

1.0

1.4

1.7

2.0

2.2

0

0 +3

1

2 + 3

1 +3

12 + 3

13 + 3

2 +3

15 + 3

3.0

3.5

4.0

4.4

4.7

5.0

5.2

1

4

1

2

3

4

5

y

y

2

2

0

2

figure 7

0

x

2

x

figure 8

NOW TRY EXERCISE 33

In Example 4 notice how the graph of g1x2 = 1x + 3 has the same shape as the

graph of f 1x2 = 1x but is shifted up 3 units. We will study shifting of graphs more

systematically in Chapter 4.

2

■ Graphing with a Graphing Calculator

Algebra Toolkit D.3, page T80,

gives guidelines on using a

graphing calculator as well as

common graphing calculator

pitfalls.

A graphing calculator graphs a function in the same way you do: by making a table of

values and plotting points. Of course, the calculator is fast and accurate; it also frees us

from the many tedious calculations needed to get a good picture of the graph. But the

graph that is produced by a graphing calculator can be misleading. A graphing calculator must be used with care, and the graphs it produces must be interpreted appropriately.

e x a m p l e 5 Graphing a Function

Graph the function f 1x 2 = x 3 - 49x in an appropriate viewing rectangle.

Solution

To graph the function f 1x2 = x 3 - 49x, we first express it in equation form

y = x 3 - 49x

68

CHAPTER 1

Data, Functions, and Models

We now graph the equation using a graphing calculator. We experiment with different viewing rectangles. The viewing rectangle in Figure 9(a) gives an incomplete

graph—we need to see more of the graph in the vertical direction. So we choose the

larger viewing rectangle 3- 10, 104 by 3- 200, 2004 by choosing

Xmin = - 10

Ymin = - 200

Xmax = 10

Ymax = 200

The graph in this larger viewing rectangle is shown in Figure 9(b). This graph appears to show all the main features of this function. (We will confirm this when we

study polynomial functions in Chapter 5.)

100

200

_10

10

_10

10

_100

_200

(a)

(b)

f i g u r e 9 Graphing f 1x2 = x 3 - 49x

NOW TRY EXERCISE 35

e x a m p l e 6 Where Graphs Meet

Consider the functions f 1x 2 = x 2 - 1 and g1x2 = x 3 - 2x - 1.

(a) Graph the functions f and g in the viewing rectangle 3- 2, 34 by 3- 3, 64 .

(b) Find the points where the graphs intersect in this viewing rectangle.

6

Solution

3

_2

The graphs are shown in Figure 10. The graphs appear to meet at three different

points. Zooming in on each point, we find that the points of intersections are

1- 1, 02

_3

f i g u r e 10 Finding where

graphs meet

2

10, - 12

12, 32

You can check that each of these points satisfies both equations.

NOW TRY EXERCISE 41

■ Graphing Piecewise Defined Functions

Recall that a piecewise defined function is a function that is defined by different rules

on different parts of its domain. As you might expect, the graph of such a function

consists of separate “pieces,” as the following example shows.

SECTION 1.6

Working with Functions: Graphs and Graphing Calculators

69

e x a m p l e 7 Graphing a Piecewise Defined Function

Graph the function

f 1x2 = e

x

x+1

if x … 2

if x 7 2

Solution

The rule f is given by f 1x2 = x for x … 2 and f 1x2 = x + 1 for x 7 2. We use this

information to make a table of values and plot the graph in Figure 11.

On many graphing calculators the

graph in Figure 11 can be produced

by using the logical functions in the

calculator. For example, on the

TI-83 the following equation gives

the required graph:

x

f 1x 2

-2

-1

0

1

2

3

4

5

-2

-1

0

1

2

4

5

6

y

6

4

2

_2

0

_1

1

2

3

4

5 x

_2

xՅ2

2Ͻx

f i g u r e 11 Graph of the piecewise defined

Y1=(X◊2)*X+(X>2)*(X+1)

function f

(To avoid the extraneous vertical

line between the two parts of the

graph, put the calculator in Dot

mode.)

Notice the closed and open circles on the graph in Figure 11. The closed circle at the

point (2, 2) indicates that this point is on the graph. The open circle above it at (2, 3)

indicates that this point is not on the graph.

NOW TRY EXERCISE 45

e x a m p l e 8 Water Rates

A city charges its residents \$0.008 per gallon for households that use less than 4000

gallons a month and \$0.012 for households that use 4000 gallons or more a month

(see Example 9 in Section 1.5). The function

C 1x 2 = e

y

80

0.008x

0.012x

if x 6 4000

if x Ú 4000

gives the cost of using x gallons of water per month.

(a) Graph the piecewise defined function C.

(b) What does the break in the graph represent?

60

40

Solution

20

0

2000

4000

6000

f i g u r e 12 Graph of the cost

function C

x

(a) We graph the function C in two pieces. For x 6 4000 we graph the equation

y = 0.008x, and for x Ú 4000 we graph the equation y = 0.012x. The graph is

shown in Figure 12.

(b) From the graph we see that at 4000 gallons there is a jump in the cost of water.

This corresponds to the jump in the price from \$0.008 to \$0.012 per gallon.

NOW TRY EXERCISE 55

70

CHAPTER 1

Data, Functions, and Models

The absolute value function f 1x2 = 0 x 0 is a piecewise defined function:

f 1x2 = e

-x

x

if x … 0

if x 7 0

This function leaves positive inputs unchanged and reverses the sign of negative

inputs.

e x a m p l e 9 Graph of the Absolute Value Function

Sketch the graph of f 1x2 = 0 x 0 .

To graph f 1x 2 = 0 x 0 on the TI-83,

enter the function asY1=abs(X).

Solution

We make a table of values and then sketch the graph in Figure 13.

x

f 1x2 ‫ ؍‬0 x 0

-3

-2

-1

0

1

2

3

3

2

1

0

1

2

3

y

2

0

2

x

f i g u r e 13 Graph of f 1x2 = 0 x 0

NOW TRY EXERCISE 31

Test your skill in using your graphing calculator. Review the guidelines on graphing calculators in Algebra Toolkit D.3 on page T80.

1–4 Use a graphing calculator or computer to decide which viewing rectangle

(i)–(iv) produces the most appropriate graph of the equation.

1. y = x 4 + 2

(i)

(ii)

(iii)

(iv)

3- 2, 2 4 by 3- 2, 24 +

[0, 4] by [0, 4]

3- 8, 8 4 by 3- 4, 404

3- 40, 40 4 by 3- 80, 8004

3. y = 10 + 25x - x 3

(i)

(ii)

(iii)

(iv)

3- 4, 4 4 by 3- 4, 44

3- 10, 10 4 by 3- 10, 104

3- 20, 20 4 by 3- 100, 1004

3- 100, 100 4 by 3- 200, 2004

2. y = x 2 + 7x + 6

(i)

(ii)

(iii)

(iv)

3 - 5, 5 4 by 3- 5, 54

[0, 10] by 3- 20, 100 4

3- 15, 8 4 by 3- 20, 1004

3- 10, 3 4 by 3- 100, 204

4. y = 28x - x 2

(i)

(ii)

(iii)

(iv)

3- 4, 4 4 by 3- 4, 44

3- 5, 5 4 by 30, 100 4

3- 10, 10 4 by 3- 10, 404

3- 2, 10 4 by 3- 2, 64

5. Graph the equation in an appropriate viewing rectangle.

(a) y = 50x 2

(c) y = x 4 - 5x 2

(b) y = x 3 - 2x - 3

(d) y = 24 + x - x 2

SECTION 1.6

Working with Functions: Graphs and Graphing Calculators

71

1.6 Exercises

CONCEPTS

Fundamentals

1. To graph the function f, we plot the points (x, _______ ) in a coordinate plane. To

graph f 1x 2 = x 3 + 2, we plot the points (x, _______ ). So the point (2, _______ ) is

on the graph of f. The height of the graph of f above the x-axis when x = 2 is _______.

2. If f 122 = 3, then the point (2, _______ ) is on the graph of f.

3. If the point (1, 5) is on the graph of f, then f 112 = _______.

4. Match the function with its graph.

(a) f 1x2 = x 2

(b) f 1x2 = x 3

I

II

y

III

y

(d) f 1x2 = 0 x 0

(c) f 1x2 = 1x

IV

y

y

1

0

1

0

x

1

0

x

0

x

1

1

5. In what ways can a graph produced by a graphing calculator be misleading? Explain

using an example.

6. A student wishes to graph the following functions on the same screen:

y = x 1>3

y=

and

x

x +4

He enters the following information into the calculator:

Y1ϭX^1>3

and

Y2ϭX>Xϩ4

The calculator graphs two lines instead of the information he wanted. What went

wrong?

SKILLS

7–12

A function is given. Complete the table and then graph the function.

7. f 1x2 = 5

x

f(x)

8. g1x 2 = 2x - 4

x

g(x)

9. h 1x2 = 2x 2 - 3

x

-3

-3

-3

-2

-2

-2

-1

-1

-1

0

0

0

1

1

1

2

2

2

3

3

3

h(x)

x

72

CHAPTER 1

Data, Functions, and Models

10. k 1x 2 = x 3 + 8

x

11. F 1x 2 = 1x + 4

k(x)

x

12. G 1x2 = 2 0 x + 1 0

F(x)

x

G(x)

-3

-4

-4

-2

-2

-3

-1

0

-2

0

2

-1

1

4

0

2

8

1

3

13–28

Sketch the graph of the function by first making a table of values.

13. f 1x2 = 8

15. g1x 2 = x - 4

14. f 1x2 = - 1

16. g1x 2 = 3x - 7

17. h 1x2 = - x + 3, - 3 … x … 3

19. k 1x 2 = - x

20. k 1x 2 = 4 - x

24. G 1x2 = 1x + 22 3

23. G 1x 2 = x 3 - 8

26. A 1x 2 = 1x + 4

25. A 1x 2 = 1x + 1

27. r 1x 2 = 0 2x 0

2

22. F 1x 2 = 1x - 32 2

21. F 1x 2 = x - 4

2

29–34

18. h 1x2 = 3 - x, - 2 … x … 2

2

28. r 1x 2 = 0 x + 1 0

Graph the given functions on the same coordinate axes.

29. f 1x2 = x 2,

30. f 1x2 = x ,

2

g1x2 = x 2 - 3, h1x 2 = 14 x 2

g1x2 = 1x + 52 2,

h 1x 2 = 1x - 5 2 2

31. f 1x2 = 0 x 0 , g1x 2 = 2 0 x 0 , h 1x 2 = 0 x + 2 0

32. f 1x2 = 0 x 0 , g1x 2 = 0 3x 0 , h 1x 2 = 0 x 0 + 4

33. f 1x2 = 1x,

34. f 1x2 = 1x,

35–40

g1x2 = 14x, h 1x2 = 1x - 4

g1x2 = 1x - 9, h 1x 2 = 1x - 1 + 2

Draw a graph of the function in an appropriate viewing rectangle.

35. f 1x2 = 4 + 6x - x 2

37. f 1x2 = x - 4x

4

39. f 1x2 = `

41–42

36. f 1x2 = 212x - 17

38. f 1x2 = 0.1x 3 - x 2 + 1

3

x

+ 7`

2

40. f 1x2 = 2x - 0 x 2 - 5 0

Do the graphs of the two functions intersect in the given viewing rectangle? If

they do, how many points of intersection are there?

41. f 1x2 = 3x 2 + 6x - 12, g1x 2 = 27 42. f 1x2 = 6 - 4x - x ,

2

43–48

7

12

g1x 2 = 3x + 18;

x 2;

3- 4, 4 4 by 3- 1, 34

3- 6, 24 by 3- 5, 20 4

Sketch the graph of the piecewise defined function.

43. f 1x2 = e

0

1

if x 6 2

if x Ú 2

44. f 1x2 = e

1

x+1

if x … 1

if x 7 1

SECTION 1.6

CONTEXTS

Working with Functions: Graphs and Graphing Calculators

45. f 1x2 = e

1-x

5

if x 6 - 2

if x Ú - 2

46. f 1x2 = e

2x + 3

3-x

47. f 1x2 = e

1 - x2

x

if x … 2

if x 7 2

48. f 1x2 = e

2

x2

73

if x 6 - 1

if x Ú - 1

if x … - 1

if x 7 - 1

49. Filling a Bathtub A bathtub is being filled by a constant stream of water from the

faucet. Sketch a rough graph of the water level in the tub as a function of time.

50. Cooling Pie You place a frozen pie in an oven and bake it for an hour. Then you take

the pie out and let it cool before eating it. Sketch a rough graph of the temperature of

the pie as a function of time.

51. Christmas Card Sales The number of Christmas cards sold by a greeting card store

depends on the time of year. Sketch a rough graph of the number of Christmas cards

sold as a function of the time of year.

52. Height of Grass A home owner mows the lawn every Wednesday afternoon. Sketch a

rough graph of the height of the grass as a function of time over the course of a fourweek period beginning on a Sunday.

r

5

6

7

8

9

10

T(r)

53. Weather Balloon As a weather balloon is inflated, the thickness T of its latex skin is

related to the radius of the balloon by

T 1r 2 =

0.5

r2

where T and r are measured in centimeters. Complete the table in the margin and graph

the function T for values of r between 5 and 10.

54. Gravity near the Moon The gravitational force between the moon and an astronaut in

a space ship located a distance x above the center of the moon is given by the function

F 1x2 =

350

x2

where F is measured in newtons (N) and x is measured in megameters (Mm). Graph the

function F for values of x between 2 and 8.

55. Toll Road Rates The toll charged for driving on a certain stretch of a toll road

depends on the time of day. The amount of the toll charge is given by

5.00

7.00

T 1x2 = e 5.00

7.00

5.00

if 0 … x 6 7

if 7 … x … 10

if 10 6 x 6 16

if 16 … x … 19

if 19 6 x 6 24

where x is the number of hours since 12:00 A.M.

(a) Graph the function T.

(b) What do the breaks in the graph represent?

56. Postage Rates The domestic postage rate depends on the weight of the letter. In 2009,

the domestic postage rate for first-class letters weighing 3.5 oz or less was given by

0.44

0.61

P1x2 = d

0.78

0.95

if 0

if 1

if 2

if 3

6

6

6

x

x

x

x

where x is the weight of the letter measured in ounces.

(a) Graph the function P.

(b) What do the breaks in the graph represent?

1

2

3

3.5

74

CHAPTER 1

Data, Functions, and Models

1.7 Working with Functions:

Getting Information from the Graph

Reading the Graph of a Function

Domain and Range from a Graph

Increasing and Decreasing Functions

Local Maximum and Minimum Values

IN THIS SECTION... we use the graph of a function to get information about the

function, including where the values of the function increase or decrease and where the

maximum or minimum value(s) of the function occur.

GET READY... by reviewing interval notation in Algebra Toolkit A.2. Test your skill in

working with interval notation by doing the Algebra Checkpoint at the end of this

section.

The graph of a function allows us to “see” the behavior, or life history, of the function. For example, we can see from the graph of a function the highest or lowest value

of the function or whether the values of the function are rising or falling. So if a function represents cost, the lowest point on its graph tells where the minimum cost occurs. If a function represents profit, its graph can tell us where profit is increasing or

decreasing. In this section we examine how to obtain these and other types of information from the graph of a function.

2

■ Reading the Graph of a Function

If a function models a real-world situation, such as the weight of a person, its graph

is usually easy to interpret. For example, suppose the weight of Mr. Hector (in

pounds) is given by the function W, where the independent variable x is his age in

years. So

W 1x2 = “weight of Mr. Hector at age x”

The graph of the function W in Figure 1 gives a visual representation of how his

weight has changed over time. Note that Mr. Hector’s weight W 1x2 at age x is the

height of the graph above the point x.

W

200

180

160

140

120

100

80

60

40

20

0

W(30)=150

W(10)=80

10

20

30

40

50

60

70 x

f i g u r e 1 Graph of Mr. Hector’s weight

SECTION 1.7

Working with Functions: Getting Information from the Graph

75

e x a m p l e 1 Verbal Description from a Graph

Answer the following questions about the function W graphed in Figure 1.

(a) What was Mr. Hector’s weight at age 10? At age 30?

(b) Did his weight increase or decrease between the ages of 40 and 50? Between

the ages of 50 and 70?

(c) How did his weight change between the ages of 20 and 30?

(d) What was his minimum weight between the ages of 30 and 50?

(e) What was his maximum weight between the ages of 30 and 70?

(f) What is the net change in his weight from the age of 30 to 50?

Solution

(a) His weight at age 10 is W(10). The value of W(10) is the height of the graph

above the x-value 10. From the graph we see that W 1102 = 80. Similarly,

from the graph, W 1302 = 150.

(b) From the graph we see that the values of the function W were increasing

between the x-values 40 and 50, so Mr. Hector’s weight was increasing during

that period. However, the graph indicates that his weight was decreasing

between the ages of 50 and 70.

(c) From the graph we see that Mr. Hector’s weight was constant between the

ages of 20 and 30. He maintained his weight at 150 lb during that period.

(d) From the graph we see that the minimum value that W achieves between the

x-values of 30 and 50 is 130. So Mr. Hector’s minimum weight during that

period was 130 lb.

(e) From the graph we see that the maximum value that W achieves between the

ages of 30 and 70 is 200 lbs. So Mr. Hector’s maximum weight during that

period was 200 lb.

(f) From the graph we see that at age 30 Mr. Hector weighed 150 lb and at age 50

he weighed 200 lb. We have W1502 - W1302 = 200 - 150 = 50, so the net

change in his weight between those two ages is 50 lb.

NOW TRY EXERCISE 41

A complete graph of a function contains all the information about the function,

because the graph tells us which input values correspond to which output values. To

analyze the graph of a function, we must keep in mind that the height of the graph is

the value of the function. So we can read the values of a function from its graph.

e x a m p l e 2 Finding the Values of a Function from a Graph

T (*F)

40

30

20

10

0

1

figure 2

2

3

4

5

6

x

The function T graphed in Figure 2 gives the temperature between noon and 6:00 P.M.

at a certain weather station.

(a) Find T(1), T(3), and T(5).

(b) Which is larger, T(2) or T(4)?

(c) Find the value(s) of x for which T1x2 = 25.

(d) Find the values of x for which T1x2 Ú 25.

(e) Find the net change in temperature between 3:00 and 5:00 P.M.

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