Stewart J. College Algebra: Concepts and Contexts

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14 CHAPTER 1

■

Data, Functions, and Models

example

Domain and Range of a Relation

The data in the table give the number of house finches seen at a bird feeder on vari-

ous days in February 2008.

(a) Express the data as a relation.

(b) What does the pair (12, 15) represent?

Solution

(a) The set of ordered pairs that defines this relation is

(b) The pair (12, 15) tells us that 15 house finches were seen at the bird feeder on

February 12.

set .

■ NOW TRY EXERCISE 37 ■

53, 10, 15, 20, 216

52, 5, 12, 15, 21, 256

512, 212, 15, 202, 112, 152, 115, 102, 121, 152, 125, 326

■ Graphing Two-Variable Data in a Coordinate Plane

Two-variable data consist of ordered pairs of numbers, so to graph such data, we

graph the ordered pairs in a coordinate plane, with one variable plotted on the x-axis

and the other on the y-axis. For instance, to plot the depth-pressure data tabulated in

Example 6 of Section 1.1, we would plot the points (0, 14.7), (10, 19.2), (20, 23.7),

and so on, where the first coordinate represents depth and the second represents the

corresponding pressure. Such a graph of data is called a scatter plot. Graphs and

scatter plots are so common in everyday life (you see them in magazines, in news-

papers, on television news, and in advertising) that you may already have thought of

making a graph of the data in the preceding examples.

example

Graphing Two-Variable Data

Draw a scatter plot of the time-temperature data and the depth-pressure data from

Examples 5 and 6 of Section 1.1. Comment on any trends or patterns you observe

from the graphs.

Day

Number of

house finches

2 21

5 20

12 15

15 10

21 15

25 3

Hours since

6:00

A.M.

Temperature

(°F)

0 59

2 62

4 68

6 65

8 58

10 60

12 62

Depth

(ft)

Pressure

(lb/in

)

0 14.7

10 19.2

20 23.7

30 28.2

40 32.7

50 37.2

60 41.7

To review how we plot points in

a coordinate plane, see Algebra

Toolkit D.1, page T67.

Solution

We plot the points given by the ordered pairs listed in the tables. The graphs are

shown in Figures 1 and 2.

SECTION 1.2

■

Visualizing Relationships in Data 15

In the time-temperature data we see that as time increases, the temperature first

increases, then decreases, then increases again. In the depth-pressure data there ap-

pears to be a very precise trend: As the depth increases, so does the pressure. In fact,

pressure appears to increase in proportion to the increase in depth.

■ NOW TRY EXERCISES 27 AND 39 ■

Finding relationships in two-variable data is a fundamental activity in every

branch of science. For example, scientists test different chemicals in the blood of ex-

pectant mothers to determine whether there is a relationship between the levels of

these chemicals and the chances of the baby having birth defects. Knowing the po-

tential for birth defects can sometimes give doctors a chance to intervene to repair

the defect. One way to discover such relationships is to collect data on the levels of

these chemicals for different expectant mothers. Graphing the data helps scientists

visually discover any relationships that may exist. The next example illustrates this

method.

IN CONTEXT ➤

(*F)

2468

Hours since 6:00

A.M.

10 12

figure 1 Time and temperature

(lb/in™)

10 20 30 40

x (ft)

50 60

figure 2 Depth and pressure

example

Levels of Enzymes in Expectant Mothers

A biologist measures the levels of three different enzymes (call them A, B, and C) in 20

blood samples taken from expectant mothers. The data she obtains are given in the table,

where enzyme levels are in milligrams per deciliter (mg/dL). The biologist wishes to

determine whether there is a relationship between the levels of the different enzymes.

(a) Make a scatter plot of the levels of the pairs of enzymes A and B.

(b) Make a scatter plot of the levels of the pairs of enzymes A and C.

Sample A B C

11 2.2 0.6 25

12 1.5 4.8 32

13 3.1 1.9 20

14 4.1 3.1 10

15 1.8 7.5 31

16 2.9 5.8 18

17 2.1 5.1 30

18 2.7 2.5 20

19 1.4 2.0 39

20 0.8 2.3 56

Sample A B C

1 1.3 1.7 49

2 2.6 6.8 22

3 0.9 0.6 53

4 3.5 2.4 15

5 2.4 3.8 25

6 1.7 3.3 30

7 4.0 6.7 12

8 3.2 4.3 17

9 1.3 8.4 45

10 1.4 5.8 47

Lev Dolgachov/Shutterstock.com 2009

16 CHAPTER 1

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Data, Functions, and Models

Solution

Scatter plots for parts (a) and (b) are shown in Figures 3 and 4. Note that each point

plotted represents the results for one sample; for instance, Sample 1 had 1.3 mg/dL

of enzyme A and 1.7 mg/dL of enzyme B, so we plot the point (1.3, 1.7) to represent

this data pair.

2413

figure 3 Enzymes A and B

2143

figure 4 Enzymes A and C

of enzymes A and B. From Figure 4 we see that when the level of enzyme A

goes up, the level of enzyme C tends to goes down.

■ NOW TRY EXERCISES 29 AND 41 ■

■ Reading a Graph

Data are often presented in the form of a graph. In the next example we read the data

from a graph.

example

Reading a Graph

The average annual precipitation in Medford, Oregon is 21 inches. The graph in

Figure 5 shows the total annual precipitation for the ten-year period 1996–2005. In

the graph Year 1 corresponds to 1996, Year 2 to 1997, and so on.

(a) What was the precipitation in 2002?

(b) Which year(s) had a total precipitation of 29 in?

(in.)

21436587109

Year

figure 5 Precipitation in Medford, Oregon

SECTION 1.2

■

Visualizing Relationships in Data 17

Solution

(a) The year 2002 is Year 7 on the graph. We need to find the height of the point

in the graph above Year 7. From the graph in Figure 6(a) we see that the total

rainfall was 18 in.

(b) Precipitation of 29 in. corresponds to the horizontal line in Figure 6(b). From

the graph we see that only Year 3 had this level of precipitation, that is, the

year 1998.

21436587109

Year

(a) Rainfall in Year 7

21436587109

Year

(b) Rainfall of 29 in.

(in.) y (in.)

figure 6

horizontal line as in Figure 7. From Figure 7 we see that Years 1, 3, and

10 had more than 21 in. of precipitation. These years correspond to 1996,

1998, and 2005.

21436587109

Year

(in.)

figure 7 Average rainfall is 21 in.

■ NOW TRY EXERCISES 15 AND 45 ■

18 CHAPTER 1

■

Data, Functions, and Models

Test your knowledge of plotting points in a coordinate plane. You can review this

topic in

Algebra Toolkit D.1

on page T67.

1. Plot the ordered pair in a coordinate plane.

(a) (3, 4) (b) (5, 1) (c) (d)

(e) (f) (g) (h) (2, 1)

2. Find the coordinates of the points shown in the figure to the left.

3–4 A set of points is given.

(a) Give a verbal description of the set.

(b) Graph the set in a coordinate plane.

3. 4.

51x, y2冨 y =-1651x, y2冨 x = 26

1- 2, 5215, - 321- 2,- 42

1- 1, 121- 2, 32

1.2 Exercises

100

200

300

Wage

1052015 3025

Hours

CONCEPTS

Fundamentals

1. The domain of a relation is the set of all _______, and the range of a relation is the set

of all

_______.

2. To express two-variable data as a relation, we represent the data as a set of

_______

pairs.

3. To graph a relation, we graph each ordered pair by plotting the input on the

____-axis

and the output on the

____-axis.

4. The scatter plot in the margin gives the relation between a worker’s wages (in dollars)

and the number of hours he works.

(a) What does the ordered pair (10, 100) represent?

(b) How many dollars does he earn when he works 20 hours?

Think About It

5. Give examples of two-variable data where some inputs have more than one output.

6. Give examples of two-variable data where several inputs give the same output.

7–10

■ A collection of ordered pairs defining a relation is given.

(a) Find the domain and range of the relation.

(b) Sketch a diagram of the relation.

10. 514, 12, 14, 32, 15, 22, 16, 126

515, 12, 13, 22, 1- 2, 52, 15, 5 26

511, 02, 12, - 12, 13, 0 2, 13, - 126

511, 12, 12, 22, 13, 42, 14, 626

SKILLS

SECTION 1.2

■

Visualizing Relationships in Data 19

11–12 ■ Two-variable data are given.

(a) Express the data as a set of ordered pairs where x is the input and y is the

output.

(b) Find the domain and range of the relation.

(d) Find the output(s) corresponding to the input 5.

(e) Find the inputs(s) corresponding to the output 3.

11. 12.

13–14

■ Two-variable data are given.

(a) Express the data as a relation where x is the input and y is the output.

(b) Find the domain and range of the relation.

(d) Find the output(s) corresponding to the input 70.

(e) Find the inputs(s) corresponding to the output 20.

13. 14.

15–18

■ The graph of a relation is given.

(a) List three ordered pairs in the relation.

(b) Find the output(s) corresponding to the input 4.

15. 16.

x 10 20 40 50 70

y 40 50 60 50 40

x 10 20 30 40 70

y 20 20 20 70 20

x 1 2 3 5 5 6

y 7 2 8 3 10 10

x 1 2 3 4 5 6

y 6 9 3 2 8 3

100

42861210

17. 18.

200

150

100

4286

100

428106

100

428106

23–26 ■ For each scatter plot, decide whether there is a relationship between the variables.

If there is, describe the relationship.

23. 24.

20 CHAPTER 1

■

Data, Functions, and Models

19–22

■ The graph of a relation is given. Match the description of the relation to the

appropriate graph.

19. The y values get larger as the x values get larger.

20. The y values get smaller as the x values get larger.

21. The y values get larger and then get smaller as the x values get larger.

22. There is no obvious relationship between x and y.

400

200

428

Graph A

10 12 146

428

Gra

h B

10 12 146

428

Gra

h C

10 12 146

428

Graph D

10 12 146

25. 26.

SECTION 1.2

■

Visualizing Relationships in Data 21

27–28 ■ A table of data is given.

(a) Make a scatter plot of the data in the given table.

(b) Determine whether there is a relationship between the variables. If there is,

describe the relationship.

27.

28.

29–30

■ A table of data is given.

(a) Make three scatter plots: one for A and B, one for B and C, and one for A and C.

(b) From each scatter plot, determine whether there is a relationship between the

variables. If there is, describe the relationship.

29. 30.

A B C

1 5 12

3 2 4

2 3 5

6 6 14

8 1 2

4 5 11

2 4 8

5 9 16

A B C

6 8 40

4 8 50

5 3 45

7 2 20

8 5 10

6 4 25

6 7 30

5 3 40

x 2 4 3 1 3 5 2 1

y 10 6 7 10 9 2 1 5

x 2 4 6 8 10 12 14

y 1 3 6 10 12 20 21

31–34 ■ Answer the following questions for the data in the indicated exercise from

Section 1.1.

(a) Express the data as a relation.

(b) Draw a diagram of the data.

31. Exercise 25 32. Exercise 26

33. Exercise 27 34. Exercise 28

35. Income and Home Prices The median incomes and median home prices in several

neighborhoods across the United States are shown in the table.

(a) Express the data as a relation.

(b) Draw a diagram of the relation.

(d) Find the output(s) corresponding to the input 80,000.

(e) Find the domain and range of the relation.

36. Student Directory The student directory at a university lists the last four digits of

each student’s ID number and the number of years the student has completed. The table

on the next page contains some of this information.

(a) Express the data as a relation.

(b) Draw a diagram of the relation.

CONTEXTS

Median

income

Median

home price

80,000 400,000

30,000 250,000

70,000 300,000

55,000 250,000

80,000 450,000

60,000 300,000

55,000 300,000

150,000 1,500,000

22 CHAPTER 1

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Data, Functions, and Models

37. Hybrid Car Sales With rising gas prices, there has been increasing interest in hybrid

cars. The table below shows the number of hybrid cars sold in some recent years.

(a) Express the data as a relation.

(b) What does the pair (2005, 213) represent?

(d) Find the output(s) corresponding to the input 6731.

(e) Find the domain and range of the relation.

ID number 8271 4357 6731 3642 3371 5291 2273 1942

Year completed 1 2 1 3 5 1 4 2

Year 2000 2004 2005 2006 2007

Hybrid cars sold (ⴛ 1000)

78 83 213 244 350

38. Women in Government Women did not have the right to vote in the United States until

1920. Since then, women have played an increasing role in government. The table shows

the number of women who served in the U.S. Senate in a given year.

(a) Express the data as a relation.

(b) What does the pair (2003, 14) represent?

Year

Women in the

U.S. Senate

1975 0

1977 3

1979 2

1981 2

1983 2

1985 2

1987 2

1989 2

1991 4

Year

Women in the

U.S. Senate

1993 7

1995 9

1997 9

1999 9

2001 14

2003 14

2005 14

2007 16

2009 17

39. Christmas Bird Count For the past 100 years ornithologists have traditionally made a

worldwide bird count at Christmas time. The table below shows the number of house

finches observed in a Christmas bird count in California.

(a) Make a scatter plot of the data in the table.

(b) What trends do you detect in the house finch population of California?

Year Years since 1960 Bird count

1960 0 33,621

1965 5 44,787

1970 10 53,838

1975 15 86,507

1980 20 73,767

1985 25 91,659

1990 30 87,632

1995 35 107,190

2000 40 69,733

2005 45 61,053

Steve Byland/Shutterstock.com 2009

SECTION 1.2

■

Visualizing Relationships in Data 23

Year

Average

litter size

Coyotes

killed

1 3 310

2 4 250

3 5 360

4 4 280

5 8 570

6 9 640

7 7 550

Year 12345678910

Weight (lb) 150 152 156 162 180 183 169 166 165 162

Systolic pressure (mm Hg) 110 114 125 132 151 160 124 124 122 125

42. Life Expectancy and Demographics Over the course of the past century both

population and average life expectancy have increased in the United States. It is

interesting to see how the age distribution of the population (demographics) is affected

by these changes. The table below lists the percentage of the U.S. population that is

between the ages of 30 and 39 years, the median age of the U.S. population, and the life

expectancy at birth for a given year.

(a) Make a scatter plot of the data for the percentage of the U.S. population that is 30–39

and the life expectancy at birth. Describe any trends you detect from the graph.

(b) Make a scatter plot of median age and life expectancy of U.S. residents. Describe

any trends you detect from the graph.

43. Women in the Work Force The disparity between the median incomes of men and

women has narrowed significantly in the last two decades. The table on the next page

shows the median income of men and women over a 40-year period.

(a) On the same graph, make scatter plots of the data for the yearly median income of

men and the yearly median income of women.

Year

Years

since 1980

Percentage

age 30–39

Median

age

Life

expectancy

1980 0 6.1 34.2 73.7

1990 10 7.9 34.9 75.4

1995 15 8.4 35.2 75.8

1998 18 8.3 35.3 76.7

1999 19 8.2 34.0 76.7

2000 20 8.0 36.5 76.7

2001 21 7.8 35.6 77.2

2002 22 7.6 35.7 77.3

2003 23 7.4 35.9 77.6

2004 24 7.1 36.0 77.9

40. Coyote Reproduction Coyotes live throughout North America. They have become

a nuisance for pet owners and ranchers, and often trapping and killing them seems to

be the only solution. The table in the margin gives data in a certain county for the

average number of coyote pups in a litter in a given year and the number of coyotes

killed in that year.

(a) Make a scatter plot of the relation between litter size and number of coyotes killed.

(b) What trend do you detect from your graph?

41. Blood Pressure Data A man has his blood pressure and weight measured at his

annual physical exam. The table below shows these data for a 10-year period.

(a) Make a scatter plot of the year and the systolic pressure. Describe any trends you

detect from the graph.

(b) Make a scatter plot of the weight and systolic pressure. Describe any trends you

detect from the graph.