Stewart J. College Algebra: Concepts and Contexts

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184 CHAPTER 2

■

Linear Functions and Models

Now that we know k is 0.2, we can write the equation of proportionality:

(b) To find how much electricity is produced by 16 panels, we replace x by 16 in

the equation

Equation of proportionality

Replace x by 16

Calculate

So 16 panels produce 3.2 kilowatts.

equation of a line with slope 0.2 and y-intercept 0.

y = 0.2x

y = 3.2

y = 0.2 ⴢ 16

y = 0.2x

2 4 6 8 10 12 14 16 18 20 22 24

figure 10 Graph of y = 0.2x

■ NOW TRY EXERCISE 53 ■

2.4 Exercises

CONCEPTS

Fundamentals

1. The graph of the line and the graph of the line have the same

_______, so they are ___________ lines.

2. The graph of the line and the graph of the line have the same

___________. Are these two lines parallel, perpendicular, or neither?

3. A line has the equation .

(a) This line has slope

________.

(b) Any line parallel to this line has slope

_______.

4. (a) If the quantities x and y are related by the equation , then we say that y is

________ _________ to x and the constant of proportionality is ______.

(b) If y is directly proportional to x and if the constant of proportionality is 3, then x

and y are related by the equation .y =

ⵧ

y = 3x

y = 3x + 2

y = 4 + 3xy = 4 + 2x

y = 6 + 3xy = 5 + 3x

SECTION 2.4

■

Varying the Coefficients: Direct Proportionality 185

Think About It

5. Compare the slopes and y-intercepts of the lines l

, l

, and l

. Which lines have the

greatest slope? Which lines are parallel to each other? Which lines are perpendicular to

each other?

(a) (b)

l⁄

l¤

l›

l‹

l⁄

l¤

l›

l‹

6. Suppose you are given the coordinates of three points in the plane and you want to find

out whether they form a right angle. How can you do this using slopes? Can you think

of another method?

7. True or false?

(a) If electricity costs $0.20 per kilowatt-hour, then the cost of electricity is directly

proportional to the number of kilowatt-hours used.

(b) If a phone plan costs $5.95 per month plus $0.22 per minute, then the monthly

phone cost is directly proportional to the number of minutes used.

proportional to the weight.

(d) If the price of tomatoes is $2.99 per pound, then the cost of a box of tomatoes is

directly proportional to the number of tomatoes in the box.

8. Use the following tables to determine which variable(s) (r, w, or t) could be directly

proportional to x.

x r

0 3.200

1 3.400

2 3.440

3 3.448

4 3.450

x w

0 0.0

1 5.7

2 11.4

3 17.1

4 22.8

x t

0 0

1>3

1>9

1>27

1>81

SKILLS

9–12 ■ Use a graphing device to graph the given family of linear equations in the same

viewing rectangle. What do the lines have in common?

9. (a) for

(b) for

10. (a) for

(b) for

11. (a) for

(b) for

12. (a) for

(b) for m = 0, - 0.5, - 1, - 2y = 2 + m 1x + 32

m = 0, 0.5, 1, 2y = 2 + m 1x + 32

m = 0, - 0.25, - 0.75, - 1.5y = m 1x - 32

m = 0, 0.25, 0.75, 1.5y = m 1x - 32

m = 0, - 0.25, - 0.75, - 1.5y = mx - 3

m = 0, 0.25, 0.75, 1.5y = mx - 3

= 0, - 1, - 2, - 3y = 21x + b2

b = 0, 1, 2, 3y = 21x + b2

186 CHAPTER 2

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Linear Functions and Models

l›

l‹

l⁄

l¤

_2 2

13–16 ■ An equation of a line l and the coordinates of a point P are given.

(a) Find an equation for the line that is parallel to l and passes through P.

(b) Sketch a graph of both lines on the same coordinate axes.

13. ; (2, 4)

14. ;

15. ; (4, 0)

16. ;

17–18

■ Graphs of lines l

, l

, and l

are shown in the figure in the margin. Are the given

pairs of lines perpendicular?

17.

18.

19–22

■ An equation of a line l and the coordinates of a point P are given.

(a) Find an equation for the line that is parallel to l and passes through P.

(b) Find an equation for the line that is perpendicular to l and passes through P.

19. ;

20. ;

21. ; (0, 3)

22. ; (2, 1)

23–26

■ The graph of a line l is sketched.

(a) Find an equation for l.

(b) Find an equation for the line parallel to l and with x-intercept 3.

23. 24.

5x + 4y = 2

4x - 3y + 1 = 0

1- 6, 122y + 1 = 21x + 32

1- 1, 42y =-

x + 1

and l

13, - 22x - 2y = 3

2x + 3y + 6 = 0

1- 1, 22y - 3 = 51x - 22

y = 3x - 4

1_1 2 3

25. 26.

1_1 2 3

_3 _1_2 1

SECTION 2.4

■

Varying the Coefficients: Direct Proportionality 187

27–28 ■ Match the equation of the line with one of the lines l

, l

, or l

in the graph.

27. (a) 28. (a)

(b) (b)

y =

x - 2 y =-x + 1

y = 4x - 2 y =-

x + 1

29–40

■ Find an equation of the line that satisfies the given conditions.

29. Through ; parallel to the line

30. Through (2, 3); parallel to the line

31. Through (2, 2); parallel to the line y ⫽ 5

32. Through (4, 5); parallel to the y-axis

33. Through ; parallel to the line

34. Through (4, 5); parallel to the x-axis

35. Through (2, 6); perpendicular to the line

36. Through ; perpendicular to the line

37. Through ; perpendicular to the line

38. Through ; perpendicular to the line

39. Through (1, 7); parallel to the line passing through

40. Through ; perpendicular to the line passing through

41–42

■ Write an equation that expresses the statement.

41. T is directly proportional to x.

42. P is directly proportional to w.

43–44

■ Express the statement as an equation. Use the given information to find the

constant of proportionality.

43. y is directly proportional to x. If x ⫽ 6, then y ⫽ 42.

44. is directly proportional to t. If t ⫽ 3, then .

45–46

■ Use the given information to solve the problem.

45. y is directly proportional to x, with constant of proportionality . Find y when

x ⫽ 5.

46. is directly proportional to t, with constant of proportionality . Find when

t ⫽ 25.

47–50

■ Find the equation of proportionality that relates y to x.

47. The number of feet y in x miles. (One mile is equal to 5280 feet.)

48. The number of seconds y in x hours.

zk = 0.25z

k = 2.4

z = 5z

11, 1 2 and 15, - 121- 2, - 112

12, 5 2 and 1- 2, 12

4x - 8y = 11

, -

4y = 5x - 315, -

y =

x - 21- 6, 62

y =

x + 5

x =-113, - 32

y =-2x - 5

y = 4x + 71- 1, 22

_2 2

l‹

l⁄

l¤

_2 2

l‹

l⁄

l¤

188 CHAPTER 2

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Linear Functions and Models

49. The number of fluid ounces y in x barrels of crude oil. (One barrel of crude oil contains

42 gallons of oil, and each gallon contains 128 fluid ounces.)

50. The number of ounces y in x tons. (One ton is equal to 2000 pounds and one pound is

equal to 16 ounces.)

CONTEXTS

51. Kayaking Mauricio and Thanh are kayaking south down a river heading toward some

rapids. Mauricio leaves 45 miles north of the rapids at 6:00

A.M., and Thanh leave

24 miles north of the rapids at 8:00

A.M. Both boys maintain a constant speed of 5 mi/h.

(a) For each boy, find a linear equation that relates their distance y from the rapids at

time x. (Take time x ⫽ 0 to be 6:00

A.M., so 8:00 A.M. would be time x ⫽ 2.)

(b) Sketch a graph of the linear equations you found in part (a).

52. Catching Up Kathie and her friend Tia are on their motorcycles heading north to

Springfield on the same straight highway. Kathie leaves from a point 120 miles south of

Springfield at 10:00

A.M., and Tia leaves from a point 35 miles south of Springfield at

11:00

A.M. Both girls maintain a constant speed of 75 mi/h.

(a) For each girl, find a linear equation that relates her distance y from Springfield at

time x. (Take time x ⫽ 0 to be 10:00

A.M., so 11:00 A.M. would be time x ⫽ 1.)

(b) Sketch a graph of the linear equations you found in part (a).

53. Crude Oil in Plastic A small amount of crude oil is used for manufacturing plastic.

Scientists estimate that about 3 fluid ounces of crude oil is used to manufacture

1 million plastic bottles. So the number of fluid ounces of crude oil y used to manufacture

plastic bottles is directly proportional to the number of bottles x that are manufactured.

(a) Find the equation of proportionality that relates y to x.

(b) In 2006, about 29 billion plastic bottles were used in the United States. Use the

equation found in part (a) to determine the number of barrels of crude oil that were

used to manufacture these plastic bottles. (Use the equation found in Exercise 49 to

convert fluid ounces to barrels of oil.)

54. Power from Windmills A power company installs 100 wind turbines in a large field.

When all the windmills are operating the electrical capacity is 55,000 kilowatts (kW) of

electricity.

(a) Find the equation of proportionality that relates the number of operating windmills

to the total electrical capacity (in kW).

(b) What is the electrical capacity (in kW) when 56 windmills are operating?

55. Interest The amount of interest i earned from a CD is directly proportional to the

amount of money P invested in the CD. Hiam invests $1500 in a 12-month CD and

earns $90.00 in interest at maturity. Find the equation of proportionality that relates

i to P. What does the constant of proportionality represent?

56. Interest Perry invests in a high-yield money market account that has an APY of

6.17%. This means that when the effects of compounding are included, Perry’s

investment yields 6.17% each year. Find the equation of proportionality that relates the

amount of interest i earned in one year to the amount of the investment P. If Perry

invests $2500, what is the amount of interest that the investment earns after one year?

57. Hooke’s Law Hooke’s Law states that the force F needed to keep a spring stretched

x units beyond its natural length is directly proportional to x. The constant of

proportionality is called the spring constant.

SECTION 2.5

■

Linear Regression: Fitting Lines to Data 189

(a) Write Hooke’s Law as an equation.

(b) If a spring has a natural length of 10 cm and a force of 40 N is required to maintain

the spring stretched to a length of 15 cm, find the spring constant.

58. Weighing Fish An Alaskan fisherman uses a spring scale to weigh the fish he

catches. He observes that a salmon weighing 10 pounds stretches the spring of the scale

1.5 inches beyond its natural length.

(a) Find the spring constant for the scale.

(b) If a halibut stretches the scale by 4 in. what is its weight?

59. Calorie Calculator The number of calories a person’s body burns is directly

proportional to the amount of energy the body uses. Search the Internet to find a site

that calculates the number of calories burned for a given level of activity. Choose a

weight and activity, and then calculate the projected number of calories burned for one

hour spent on that activity. Use the calculation to find the equation of proportionality

that relates the number of calories burned y to the duration of time x spent performing

the chosen activity for the chosen weight.

60. Blood Alcohol Level The blood alcohol level in your body is directly proportional to

the number of alcoholic drinks you have had. Search the Internet to find a site that

calculates blood alcohol level (there are many of them). Choose a weight, type of drink,

and time elapsed since drinking. Then calculate the projected blood alcohol level for

one drink. Use the calculation to find the equation of proportionality that relates the

blood alcohol level y to the number of alcoholic drinks x for your chosen parameters.

5 cm

2.5 Linear Regression: Fitting Lines to Data

■

The Line That Best Fits the Data

■

Using the Line of Best Fit for Prediction

■

How Good Is the Fit? The Correlation Coefficient

IN THIS SECTION… we model real-life data whose scatter plot doesn’t lie exactly on a

line but only appears to exhibit a “linear trend.” We can model such data by a line that best

fits the data—called the line of best fit or the regression line.

GET READY… by reviewing Section 1.6 on graphing with a graphing calculator. We need

a graphing calculator or computer spreadsheet program for this section.

So far in this chapter we have used linear functions to model data whose scatter plots

lie exactly on a line. But in most real-life situations data seldom fall into a precise

line. Because of measurement errors or other random factors, a scatter plot of real-

world data may appear to lie more or less on a line, but not exactly. For example, the

scatter plot in Figure 1(b) on page 190 shows the results of a study on childhood obe-

sity; the graph plots the body mass index (BMI) versus the number of hours of tele-

vision watched per day for 25 adolescent subjects. Of course, we would not expect

an exact relationship between these variables, as in Figure 1(a), but the scatter plot

clearly indicates a linear trend: The more hours a subject spends watching TV, the

190 CHAPTER 2

■

Linear Functions and Models

Fitting lines to data is one of the most important tools available to researchers

who need to analyze numerical data. In this section we learn how to find and use lines

that best fit almost-linear data.

■ The Line That Best Fits the Data

Until recently, infant mortality in the United States was declining steadily. Table 1

gives the nationwide infant mortality rate for the period from 1950 to 2000; the rate

is the number of infants who died before reaching their first birthday out of every

1000 live births. Over this half century the mortality rate was reduced by over 75%,

a remarkable achievement in neonatal care.

The scatter plot in Figure 2 shows that the data lie roughly on a straight line. We

can try to fit a line visually to approximate the data points, but since the data aren’t

exactly linear, there are many lines that might seem to work. Figure 3 shows two at-

tempts at “eyeballing” a line to fit the data.

Hours Hours

BMI

12345

BMI

123

(a) Line fits data exactly (b) Line of best fit

figure 1

10 20 30 40 50

Years since 1950

Infant mortality rate

figure 3 Attempts to visually fit

line to data

Year Rate

1950 29.2

1960 26.0

1970 20.0

1980 12.6

1990 9.2

2000 6.9

table 1

U.S. infant mortality

higher the BMI tends to be. So although we cannot fit a line exactly through the data

points, a line like the one in Figure 1(b) shows the general trend of the data.

Of all the lines that run through these data points, there is one that “best” fits the

data, in the sense that it provides the most accurate linear model for the data. We now

describe how to find this line.

10 20

Years since 1

950

Infant mortality rate

30 40 50

figure 2 U.S. infant mortality

rate

SECTION 2.5

■

Linear Regression: Fitting Lines to Data 191

It seems reasonable that the line of best fit is the line that is as close as possible

to all the data points. This is the line for which the sum of the vertical distances from

the data points to the line is as small as possible (see Figure 4).

For technical reasons it is best to use the line for which the sum of the squares

of these distances is smallest. This is called the regression line. The formula for the

regression line is found by using calculus, but fortunately, the formula is pro-

grammed into most graphing calculators. In Example 1 we see that we can use a

TI-83 calculator to find the regression line for the infant mortality data just de-

scribed. (The process for other calculator models is similar.)

figure 4 The line of best fit

example

Finding the Regression Line for the U.S. Infant Mortality Data

Find the regression line for the infant mortality data in Table 1. Graph the regression

line on a scatter plot of the data.

Solution

To find the regression line using a TI-83 calculator, we must first enter the data into

the lists L1 and L2, which are accessed by pressing the key and selecting

Edit. Figure 5 shows the calculator screen after the data have been entered. (Note

that we are letting x ⫽ 0 correspond to the year 1950 so that x ⫽ 50 corresponds to

2000. This makes the equations easier to work with.)

We then press the key again and select CALC, then 4:LinReg(ax+b),

which provides the output shown in Figure 6(a). This tells us that the regression line is

Here, x represents the number of years since 1950, and y represents the corresponding

infant mortality rate.

y =-0.48x + 29.4

STAT

The scatter plot and the regression line are plotted on a graphing calculator screen in

Figure 6(b).

■ NOW TRY EXERCISE 13 ■

0 29.2

10 26

20 20

30 12.6

40 9.2

50 6.9

-------

L2(7)=

L3 1

figure 5 Entering the data

y=ax+b

a=-.4837142857

b=29.40952381

LinReg

(b) Scatter plot and regression line(a) Output of the

LinReg

command

figure 6

■ Using the Line of Best Fit for Prediction

The main use of the regression line is to help us make predictions about new data

points that are outside the domain of the collected data. In general, extrapolation

means using a model to make predictions about data points that are beyond the

192 CHAPTER 2

■

Linear Functions and Models

example

Using the Regression Line for Prediction

Use the regression line from Example 1 to do the following:

(a) Estimate the infant mortality rate in 1995.

(b) Predict the infant mortality rate in 2006. Search the Internet for the actual rate

for 2006. Compare it with your prediction.

Solution

(a) The year 1995 is 45 years after 1950, so substituting 45 for x, we find that

So the infant mortality rate in 1995 was about 7.8.

(b) To predict the infant mortality rate in 2006, we replace x by 56 in the

regression equation to get

So we predict that the infant mortality rate in 2006 would be about 2.5.

■ NOW TRY EXERCISE 15 ■

An Internet search shows that the actual infant mortality rate was 7.6 in 1995

and 6.4 in 2006. So the regression line is fairly accurate for 1995 (the actual rate was

slightly lower than the predicted rate), but it is considerably off for 2006 (the actual

rate was more than twice the predicted rate). The reason is that infant mortality in the

United States stopped declining and actually started rising in 2002, for the first time

in more than a century. This shows that we have to be very careful about extrapolat-

ing linear models outside the domain over which the data are spread.

Since the modern Olympic Games began in 1896, achievements in track and

field events have been improving steadily. One example in which the winning

records have shown an upward linear trend is the pole vault. Pole vaulting began in

the northern Netherlands as a practical activity; when traveling from village to vil-

lage, people would vault across the many canals that crisscrossed the area to avoid

having to go out of their way to find a bridge. Households maintained a supply of

wooden poles of lengths appropriate for each member of the family. Pole vaulting

for height rather than distance became a collegiate track and field event in the mid-

1800s and was one of the events in the first modern Olympics. In the next example

we find a linear model for the gold-medal-winning records in the men’s Olympic

pole vault.

y =-0.48156 2+ 29.4 = 2.5

y =-0.48145 2+ 29.4 = 7.8

IN CONTEXT ➤

domain of the given data, and interpolation means using a model to make predic-

tion about new data points that are between the given data points. In the next ex-

ample we use the regression line of Example 1 to predict the infant mortality rate

for a future year (extrapolation) and for a year between those given in the data

points (interpolation).

Tim Mack, 2004 Olympic gold

medal winner

AP Images

SECTION 2.5

■

Linear Regression: Fitting Lines to Data 193

example

Regression Line for Olympic Pole Vault Records

Table 2 gives the men’s Olympic pole vault records up to 2004.

(a) Find the regression line for the data.

(b) Make a scatter plot of the data and graph the regression line. Does the regres-

sion line appear to be a suitable model for the data?

(d) Use the model to predict the winning pole vault height for the 2008 Olympics.

table

Men’s Olympic pole vault records

Solution

(a) Let so that 1896 corresponds to , 1900 to x ⫽ 0, and

so on. Using a calculator, we find the regression line:

(b) The scatter plot and the regression line are shown in Figure 7. The regression

line appears to be a good model for the data.

y = 0.0266x + 3.40

x =-4x = year - 1900

y=ax+b

a=.0265652857

b=3.400989881

LinReg

Output of the LinReg

function on the TI-83

20 40 60 80 100

Height

(m)

Years since 1900

figure 7 Scatter plot and regression line for pole vault data

Year Gold medalist Height (m)

1956 Robert Richards, USA 4.56

1960 Don Bragg, USA 4.70

1964 Fred Hansen, USA 5.10

1968 Bob Seagren, USA 5.40

1972 W. Nordwig, E. Germany 5.64

1976 T. Slusarski, Poland 5.64

1980 W. Kozakiewicz, Poland 5.78

1984 Pierre Quinon, France 5.75

1988 Sergei Bubka, USSR 5.90

1992 M. Tarassob, Unified Team 5.87

1996 Jean Jaffione, France 5.92

2000 Nick Hysong, USA 5.90

2004 Timothy Mack, USA 5.95

Year Gold medalist Height (m)

1896 William Hoyt, USA 3.30

1900 Irving Baxter, USA 3.30

1904 Charles Dvorak, USA 3.50

1906 Fernand Gonder, France 3.50

1908 A. Gilbert, E. Cook, USA 3.71

1912 Harry Babcock, USA 3.95

1920 Frank Foss, USA 4.09

1924 Lee Barnes, USA 3.95

1928 Sabin Can, USA 4.20

1932 William Miller, USA 4.31

1936 Earle Meadows, USA 4.35

1948 Guinn Smith, USA 4.30

1952 Robert Richards, USA 4.55