Stewart J. College Algebra: Concepts and Contexts

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x 1223 4 6 8

578710 12 14

224 CHAPTER 2

■

Linear Functions and Models

39–40

■ Suppose that y is directly proportional to x. Find the constant of proportionality k

under the given condition, and use it to solve the rest of the problem.

39. If x  3, then y  15. Find y when x  7.

40. If x  6, then y  14. Find x when y  35.

41–42

■ Find a proportionality equation of the form y  kx that relates y to x.

41. The number y of millimeters in x inches. (1 inch equals 25.4 millimeters.)

42. The number y of legs on x horses (assuming that none of the horses has lost a leg).

43–44 ■ A table of values is given.

(a) Make a scatter plot of the data.

(b) Find the regression line for the data, and graph it on your scatter plot.

43.

x 023 3 568

10 3 6 12 14 5 4

44.

45–50

■ Linear functions f and are given.

(a) Sketch graphs of f and .

(b) From your graph, estimate the coordinates of the points where the graphs of f and

intersect.

45. 46.

47. 48.

49. 50. f 1x 2= 1 + 5x;

g1x 2= 2x - 7f 1x 2= 6 + 2x;g1x 2= 15 - 3x

f 1x 2= 13;

g1x 2= 7x - 8f 1x 2= 5 - 3x;g1x 2= 12 + 4x

f 1x 2= 10 - 3x;

g1x 2= 5xf 1x 2= 2x + 4;g1x 2= 5x - 8

These exercises test your understanding by combining ideas from several sections in a

single problem.

51. Points and Lines The table lists the coordinates of six points, P

through P

. Answer

the following questions about these points and the lines that they determine.

Point

Coordinates

(-1, 1) (-1, -2)

(1, 0) (2, 1) (0, 3) (3, 4)

(a) Plot the points on a coordinate plane.

(b) Which two points lie on the same vertical line? What is the equation of that line?

(d) Find an equation in slope-intercept form for the line containing and .

(e) Which two points lie on a line perpendicular to the line you found in part (c)? What

is the equation of the line containing these two points?

(f) Let f be the function whose graph contains and . What is the rate of change of f ?

(g) Find equations in slope-intercept form for the line that contains and , and the

line that contains and . Find the coordinates of the point of intersection of these

two lines.

CONNECTING

THE CONCEPTS

CHAPTER 2

■

Review Exercises 225

(h) Find the regression line for the six points in the table, and graph it on your scatter

plot from part (a). Do most of the points lie close to the line, or are they scattered

far away from it?

52. Supply and Demand for Milk The price of fresh milk varies widely from state to

state. The Ackamee grocery store in Dalewood, Maryland, determines from sales data

that when milk is priced at $1.10 per quart, they sell 4000 quarts every week, but when

it is priced at $1.25 per quart, they sell only 3700 quarts in a week.

(a) Use the given price and quantity data to find a linear demand equation of the form

that gives the weekly demand y (in quarts) when the price of milk is

p dollars per quart. Sketch a graph of the equation.

(b) What does the slope of the graph of the demand equation represent? Why is the

slope negative?

(d) At what price will the demand for milk drop to 3000 quarts?

(e) A survey of local dairies indicates that the supply equation for milk sold at

Ackamee is . Sketch this line on your graph from part (a).

(f) What does the p-intercept of the supply equation represent?

(g) From your graph, estimate the coordinates of the equilibrium point (the point at

which the graphs of the supply and demand equations meet).

(h) Find the coordinates of the equilibrium point algebraically. What is the equilibrium

price, and how many quarts of milk are sold at this price?

y = 7800p - 5070

y = b + mp

53. U.S. Population The Constitution of the United States requires a national census

every 10 years. The census data for 1800–2000 are given in the table. (Express all rates

in millions/yr.)

(a) Draw a partial scatter plot of the data, using only the data for years divisible by 20

(that is, 1800, 1820, 1840, and so on).

(b) From your scatter plot in part (a), do you detect a general trend in the rate of

change of population over the past two centuries?

(d) Find the average rate of change of population in the 19th century and in the 20th

century.

(e) Find the average rate of change of population for the years 1800–2000. Compare

this to the rates you found in part (d).

(f) In which decade of the 20th century was the rate of change of population the

greatest? What was that rate of change?

CONTEXTS

Year

Population

(millions)

1800 5.3

1810 7.2

1820 9.6

1830 12.9

1840 17.1

1850 23.2

1860 31.4

Year

Population

(millions)

1870 38.6

1880 50.2

1890 63.0

1900 76.2

1910 92.2

1920 106.0

1930 123.2

Year

Population

(millions)

1940 132.2

1950 151.3

1960 179.3

1970 203.3

1980 226.5

1990 248.7

2000 281.4

54. Price of Gasoline The price of gasoline in the United States has fluctuated considerably

in recent years. A function that models the average price per gallon of gas in Montgomery

County, Pennsylvania, for the period from July 1, 2008, to March 31, 2009, is

f 1x 2= 4 - 0.08x - 0.002x

+ 0.00008x

226 CHAPTER 2

■

Linear Functions and Models

where x is the number of weeks since July 1, 2008, and is the average price in

dollars of a gallon of gas in week x.

(a) Draw a graph of f with a graphing calculator, using the viewing rectangle by

(b) What does the graph indicate about the general trend of the rate of change of the

price of gasoline over this period?

July 1 to December 31, 2008 (i.e., from x  0 to x  26).

(d) Find the average rate of change in the price of gasoline from January 1 to March 31,

2009 (i.e., from x  26 to x  39).

55. Growing Corn During its initial growth phase, a corn plant’s height increases

linearly. It then stops growing as the ears of corn mature. The official start of corn

season in a certain Pennsylvania farming community is May 15, a couple of weeks after

most of the farmers have planted their corn. Let be the height (in inches) of a

particular cornstalk x days after May 15. The table in the margin shows the height of

this cornstalk on various days of the corn season.

(a) Find the rate at which the cornstalk is growing.

(b) Find the linear function h that models the growth of the cornstalk.

(d) The cornstalk stops growing on day 40 of the season. What height does it reach on

this day?

(e) What was the height of the cornstalk on May 15 (that is, when x is 0)?

(f) How many days before May 15 did the seedling break the surface of the soil (that

is, start at height 0)?

(g) When did the cornstalk reach a height of 35 inches?

56. Length of a Coho Salmon Coho salmon inhabit the coastal waters of the Pacific

Northwest, living a fixed lifespan of four years before they die as they spawn in the

same riverbed in which they were born. They grow steadily throughout their lives, and

their length is proportional to their age. A coho two years old is 16 inches long.

(a) Find the equation of proportionality that relates a coho’s length in inches to its age

in months.

(b) How long is a coho 30 months old?

(d) What length are cohoes when they die?

(e) A few cohoes do survive spawning, returning to the ocean for another four-year

cycle. A fisherman catches such a coho that is 40 inches long. How old is this coho?

57. Super-Size Portions A study published in the American Journal of Public Health

found that the portion sizes in many fast-food restaurants far exceed the sizes of those

offered in the past. For instance, in the mid-1950s, McDonald’s had only one size of

french fries, and now that size is labeled “Small” and is one-third the weight of the

largest size available. The problem with larger portions is that we tend to eat more when

we are offered more food, contributing to the national obesity epidemic. The table at left

shows the number of ounces in a “large” drink at a particular convenience store chain

since 1975.

(a) Find the regression line for the number of ounces in the “large” drink as a function

of the number of years since 1975.

(b) Graph the regression line along with a scatter plot of the data.

in 2010. Does this prediction make sense?

(d) If the model is accurate, in what year will a “large” drink contain 60 oz?

h 1x 2

31,

4.54

30, 404

f 1x 2

Year Large size (oz)

1975 12

1980 15

1985 16

1990 20

1995 24

2000 32

2005 36

Day x Height h(x)

521

10 28

20 42

35 63

CHAPTER 2

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Review Exercises 227

x ⴝ Beginning weight

212 161 165 142 170 181 165 172 312 302

y ⴝ Weight after 12 months

188 151 166 131 165 156 156 152 276 289

58. Weight Loss A weight-loss clinic keeps records of its clients’ weights when they first

joined and their weight 12 months later. The data for ten clients are shown in the table.

(See Chapter 1 Review, Exercise 71, page 123.)

(a) Find the regression line for the weight after 12 months in terms of the beginning

weight.

(b) Make a scatter plot of the weights, together with the regression line you found in

part (b). Does the line seem to fit the data well?

12 months after entering the program?

59. Distance, Time, and Speed Annette leaves her home at 8:00

A.M. and cycles at a

constant speed of 7 mi/h to the gardening supply store where she works. Fifteen

minutes later, her husband Theo notices that she has forgotten her keys, so he leaves the

house and follows her in his car, driving at 24 mi/h, so that he can give her the keys.

(a) Find linear functions f and that model the distance Annette and Theo have

traveled, as a function of the time t (in hours) that has elapsed since 8:00

A.M.

(b) At what time will Theo catch up to Annette? How far will he have driven when he

catches up?

60. Cost and Revenue Amanda silkscreens artistic designs on tee shirts, which she sells

to souvenir shops in her beachside town. She has fixed costs of $60 every week, and

each tee shirt costs her $3.50, while dyes and other supplies cost 50 cents per shirt. She

sells each completed tee shirt for $15.

(a) Find a linear function C that models Amanda’s total cost when she produces

x tee shirts in a week.

(b) Find a linear function R that models Amanda’s total revenue when she sells

x tee shirts.

(d) How many shirts must Amanda sell to break even?

P 1x 2

R 1x 2

C 1x 2

228 CHAPTER 2

■

Linear Functions and Models

Weight of

guinea pig (oz)

Food per

week (oz)

18 15

22 12

23 14

23 15

25 17

27 18

4. A description of a line is given. Find the equation of the line in slope-intercept form.

(a) The line contains the points (2, 5) and (4, 13).

(b) The line contains the point and is parallel to the line .

5. Find equations for the vertical line and the horizontal line that pass through the point .

6. A linear equation is given. Put the equation in general form and in slope-intercept form.

Then graph the equation.

(a) (b)

7. The distance by which a rubber band stretches when it is pulled is proportional to the

force used in pulling it. If a rubber band stretches by 6 inches when a 2-pound weight is

hung from it, by how much does it stretch when a weight of 3.5 pounds is hung from it?

8. The data in the table give the weight of six guinea pigs and the amount of food each one

eats in one week (both measured in ounces).

(a) Make a scatter plot of the data.

(b) Find the regression line for the data, and graph it on your scatter plot.

would eat.

9. A pair of supply and demand equations is given.

(a) Find the coordinates of the point at which the graphs of the lines intersect (the

equilibrium point).

(b) What is the equilibrium price of the commodity being modeled?

Demand:

y =-2p + 350

Supply:

y = 4p - 100

2x + 3y = 6 + yx =

y - 1

16, - 82

2x + 5y = 20

2x + 5y = 201- 3, 32

TEST

1. Find the average rate of change of the function f between the given values of x.

(a) between x  1 and x  4

(b) between x  2 and x  2  h

2. Let , and let .

(a) Only one of the two functions f and is linear. Which one is linear, and why is the

other one not linear?

(b) Sketch a graph of the linear function.

3. The figure shows the graph of a linear function. Express the function in the form

.f 1x 2= b + mx

g1x 2= 6 - 3xf 1x 2= x

- 5

f 1x 2= 3 + 5x

f 1x 2= x

- 3x

CHAPTER

EXPLORATIONS 229

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EXPLORATIONS

When Rates of Change Change

OBJECTIVE To recognize changes in the average rate of change of a function and to

see how such changes affect the graph of the function.

In an episode of the popular television show The Simpsons, Homer reads from his

US of A Today newspaper and says, “Here’s good news! According to this eye-

catching article, SAT scores are declining at a slower rate.”

In this statement Homer is talking about the rate of change of a rate of change: SAT

scores are changing (declining), but the rate at which they are declining is itself chang-

ing (slowing down).* In the real world, rates of change are changing all the time. When

you drive, your speed (rate of change of distance) increases when you accelerate and

decreases when you decelerate. As teenagers grow, the rate at which their height in-

creases slows down and eventually stops as they become adults. In this Exploration we

investigate how a changing rate of change affects the graph of a function.

I. Changes in Tuition Fees

1. The table gives the average annual cost of tuition at public 4-year colleges in

the United States. The first section in the table gives the tuition in actual

current dollars; the second section adjusts these numbers for inflation to

constant 2006 dollars.

(a) Fill in the “Rate of change” columns by finding the change in tuition and

fees in dollars per year, over the preceding year.

(b) Fill in the “Annual percentage change” columns by expressing the rate of

change as a percentage of the preceding year’s tuition (to the nearest

percent).

Academic

year

Tuition

Rate of

change

Annual

% change

99–00 $3362

——

00–01 $3508

$146 4.3%

01–02 $3766

02–03 $4098

03–04 $4645

04–05 $5126

05–06 $5492

06–07 $5836

Tuition

Rate of

change

Annual

% change

$4102

——

$4139

$37 0.9%

$4326

$4624

$5131

$5516

$5702

$5836

Source: The College Board, New York, NY.

Current dollars Inflation-adjusted dollars

*References to mathematical ideas are frequent on The Simpsons. Professor Sarah Greenwald of

Appalachian State University and Professor Andrew Nestler of Santa Monica College maintain a website

devoted to “the mathematics of The Simpsons”: www.mathsci.appstate.edu/~sjg/simpsonsmath/.

EXPLORATIONS

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2. From the table we see that the cost of tuition has changed from year to year

over this ten-year period, in both actual and inflation-adjusted dollars.

(a) Did tuition increase every year over this period?

(b) Did the rate of change increase every year over this period? If not,

describe how the rate of change of tuition changed over this period.

3. Consider the actual current dollar data shown in the table.

(a) Over what period was the rate of change of tuition increasing?

(b) Over what period was the rate of change of tuition decreasing?

4. Repeat Question 3 for the inflation-adjusted data.

5. (a) Which do you think is a better way of measuring the change in tuition,

actual dollars or inflation-adjusted dollars?

(b) Which do you think is a better way of expressing the rate of change of

tuition, dollars per year or the percentage change per year?

II. Rates of Change and the Shapes of Graphs

The graphs in Figure 1 show the temperatures in Springfield over a 12-hour period

on two different days, starting at midnight. From the graphs we see that a warm front

was moving in overnight, causing the temperature to rise.

230 CHAPTER 2

1. Complete the following tables by finding the average rates of change of

temperature over consecutive 2-hour intervals. Read the temperature from

each graph as accurately as you can.

Temperature

(°C)

Temperature

(°C)

2468

Day 1

10 12

Time

2468

Day 2

10 12

figure 1

Time interval (h) [0, 2] [2, 4] [4, 6] [6, 8] [8, 10] [10, 12]

Average rate of

change on interval

11 - 5

2 - 0

= 3.0

Temperature on Day 1

EXPLORATIONS

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EXPLORATIONS 231

Time interval (h) [0, 2] [2, 4] [4, 6] [6, 8] [8, 10] [10, 12]

Average rate of

change on interval

5.5 - 5

2 - 0

= 0.25

Temperature on Day 2

2. It is obvious from the graphs that the temperature increases on both days. Is

the average rate of change of temperature increasing or decreasing on Day 1?

On Day 2?

3. Which of the basic shapes in Figure 2 below best describe each of the graphs

in Figure 1?

figure 2 Basic shapes

4. Use your answers to Questions 2 and 3 to complete the following table.

Day 1 Day 2

Temperature

Increasing

Average rate of change

Decreasing

Shape of graph

5. On Day 3 a cold front moves into Springfield. The following table shows the

temperatures on that day.

(a) Plot the data and connect the points with a smooth curve.

Time

Temperature

(ⴗC)

024

223

421

618

814

10 9

12 2

24681012

Time

Day 3

Temperature

(°C)

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Day 3 Day 4

Temperature

Average rate of change

Shape of graph

change of temperature increasing or decreasing?

(d) Which of the basic shapes in Figure 2 best describes the graph for Day 3?

6. On Day 4 the temperature also decreases from to between midnight

and noon. But this time, the rate of change of temperature is increasing.

Sketch a rough graph that describes this situation. (You may find it helpful to

consider the basic shapes in Figure 2.)

7. Use your answers to Questions 5 and 6 to complete the following table.

2°C24°C

8. Six different functions are graphed. For each function, determine whether the

function is increasing or decreasing and whether the average rate of change is

increasing or decreasing.

232 CHAPTER 2

Function: Increasing

Rate of change: Decreasing

(b) Complete the following table by finding the average rates of change of

temperature over consecutive 2-hour intervals.

Interval time (h) [0, 2] [2, 4] [4, 6] [6, 8] [8, 10] [10, 12]

Average rate of

change on interval

23 - 24

2 - 0

=-0.5

Temperature on Day 3

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EXPLORATIONS 233

Function:

_____________ _____________ _____________

Rate of change:

_____________ _____________ _____________

III. SAT Scores

We began this exploration with a discussion of the change in the rate of change of

SAT scores. Let’s examine what really happened to SAT scores. The table shows the

combined verbal and mathematical SAT scores between 1988 and 2002.

Year 1988 1990 1992 1994 1996 1998 2000 2002

SAT score 1006 1001 1001 1003 1013 1017 1019 1020

Linear Patterns

OBJECTIVE To recognize linear data and find linear functions that fit the data

exactly.

Finding patterns is a basic human urge. Recognizing patterns in our environment

helps us to avoid the unexpected and so feel more secure in our daily lives. In fact,

psychologists tell us that our need to find patterns is so strong that sometimes we

make up patterns that aren’t really there!

But finding real patterns in large data sets can be very profitable for those who

know how to do it. For example, Business Week reported that the Sunnyvale campus

of Yahoo employs a team of more than 100 very highly paid mathematicians and

computer scientists to sift through Yahoo’s immense pool of data. Their goal is to

find patterns in the online activity of the over 200 million registered Yahoo users.

Yahoo’s employees “mine” the data for useful information using methods from a

new branch of mathematics called data mining. The patterns they find help the com-

pany to be more responsive to its customers’ needs.

In this exploration we consider the simplest type of pattern: linear patterns. In

later explorations (in Chapters 3–5) we build on what we learned here to find more

complex patterns.

Let’s examine SAT scores from 1994 to 2002.

1. Did SAT scores decrease or increase in this period?

2. Did the rate of change in SAT scores increase or decrease in this period?

3. Did Homer Simpson’s US of A Today newspaper report the facts accurately?

Data mining