Chapter 1 Section 3 Linear Functions 29 - Lamar.edu calculus/hoffman/ch0… · B2 4A2 B 2A Chapter 1 Section 3 Linear Functions 29 EXAMPLE 3.1EXAMPLE 3.1 Linear Functions 3. Solution

41. Show that the vertex of the parabola y � Ax2 � Bx � C (A � 0) occurs at the

point where x � . [Hint: First verify that

Then note that the largest or smallest value of

f(x) � Ax2 � Bx � C

must occur where x � .

42. Graph . Determine the values of x for which the function

is defined.

43. Graph . Determine the values of x for which the function is

defined.

In many practical situations, the rate at which one quantity changes with respect toanother is constant. Here is a simple example from economics.

A manufacturer’s total cost consists of a fixed overhead of $200 plus production costsof $50 per unit. Express the total cost as a function of the number of units producedand draw the graph.

f(x) �8x2 � 9x � 3

x2 � x � 1

f(x) ��9x2 � 3x � 4

4x2 � x � 1

Vertex(low point)

Vertex(high point)

2A–B

2A–B

x

y

x

y

�B

2A �

Ax2 � Bx � C � A��x �B

2A�2

� �C

A�

B2

4A2��

�B

2A

Chapter 1 � Section 3 Linear Functions 29

EXAMPLE 3 .1EXAMPLE 3 .1

LinearFunctions

3

SolutionLet x denote the number of units produced and C(x) the corresponding total cost.Then,

Total cost = (cost per unit)(number of units) + overhead

where

Hence,

C(x) � 50x � 200

The graph of this cost function is sketched in Figure 1.15.

The total cost in Example 3.1 increases at a constant rate of $50 per unit. As aresult, its graph in Figure 1.15 is a straight line that increases in height by 50 unitsfor each 1-unit increase in x.

In general, a function whose value changes at a constant rate with respect to itsindependent variable is said to be a linear function. This is because the graph of sucha function is a straight line. In algebraic terms, a linear function is a function of theform

f(x) � a0 � a1x

FIGURE 1.15 The cost function C(x) � 50x � 200.

700

600

500

400

300

200

100

(2,300)

(0, 200)

(3,350)

1 2 3 4 5

C(x)

x

Overhead � 200

Number of units � x

Cost per unit � 50

30 Chapter 1 Functions, Graphs, and Limits

E x p l o r e !E x p l o r e !Input the cost function, Y1 �

50x � {200, 300, 400} into the

equation editor of your graphing

calculator. Use the WINDOW

menu to set the viewing window

to [1, 5]1 by [�100, 700]100.

Explain the effect of varying

the overhead value.

where a0 and a1 are constants. For example, the functions ,

and f(x) � 12 are all linear. Linear functions are traditionally written in the form

y � mx � b

where m and b are constants. This standard notation will be used in the discussionthat follows.

A surveyor might say that a hill with a “rise” of 2 feet for every foot of “run” has aslope of

The steepness of a line can be measured by slope in much the same way. In particu-lar, suppose (x1, y1) and (x2, y2) lie on a line as indicated in Figure 1.16. Betweenthese points, x changes by the amount x2 � x1 and y by the amount y2 � y1. Theslope is the ratio

It is sometimes convenient to use the symbol y instead of y2 � y1 to denote thechange in y. The symbol y is read “delta y.” Similarly, the symbol x is used todenote the change x2 � x1.

The Slope of a Line � The slope of the nonvertical line passingthrough the points (x1, y1) and (x2, y2) is given by the formula

Slope �change in y

change in x�

y2 � y1

x2 � x1

m �rise

run�

2

1� 2

THE SLOPE OF A LINE

Linear Functions � A linear function is a function that changes at aconstant rate with respect to its independent variable.

The graph of a linear function is a straight line.

The equation of a linear function can be written in the form

y � mx � bwhere m and b are constants.

f(x) �3

2� 2x, f(x) � �5x


Slope � y

x�

y2 � y1

x2 � x1

The use of this formula is illustrated in the following example.

Find the slope of the line joining the points (�2, 5) and (3, �1).

Solution

The situation is illustrated in Figure 1.17.

FIGURE 1.17 The line joining (�2, 5) and (3, �1).

∆y = –1 – 5 = –6

∆ x = 3 – (–2) = 5

(–2, 5)

(3, –1)

x

y

Slope � y

x�

�1 � 5

3 � (�2)�

�6

5

FIGURE 1.16 .

y2 – y1 = ∆y

x

y

x2 – x1 = ∆ x

(x2, y2)

(x1, y1)

(Run)

(Rise)


EXAMPLE 3 .2EXAMPLE 3 .2E x p l o r e !E x p l o r e !Construct a family of straight

lines through the origin by stor-

ing Y1 � L1*X in the equation

editor and entering the values 2,

1, .5, �.5, �1, �2 into L1 using

the STAT menu and the Edit op-

tion. Graph using the Decimal

Window and trace the values of

the different lines for x � 1.

Slope �y2 � y1

x2 � x1�

y x

The sign and magnitude of the slope of a line indicate the line’s direction andsteepness, respectively. The slope is positive if the height of the line increases as xincreases and is negative if the height decreases as x increases. The absolute value ofthe slope is large if the slant of the line is severe and small if the slant of the line isgradual. The situation is illustrated in Figure 1.18.

Horizontal and vertical lines (Figures 1.19a and 1.19b) have particularly simple equa-tions. The y coordinates of all of the points on a horizontal line are the same. Hence,a horizontal line is the graph of a linear function of the form y � b, where b is aconstant. The slope of a horizontal line is zero, since changes in x produce no changesin y.

FIGURE 1.19 Horizontal and vertical lines.

y = b(0, b)

x = c

(c, 0)

(a) (b)

x

y

x

y

HORIZONTALAND VERTICAL LINES

FIGURE 1.18 The direction and steepness of a line.

m = 2

m = 1

m =12

m = –2

m = –1

m = –12

x

y


The x coordinates of all the points on a vertical line are equal. Hence, verticallines are characterized by equations of the form x � c, where c is a constant. Theslope of a vertical line is undefined. This is because only the y coordinates of points

on a vertical line can change, and so the denominator of the quotient is

zero.

The constants m and b in the equation y � mx � b of a nonvertical line have geo-metric interpretations. The coefficient m is the slope of the line. To see this, supposethat (x1, y1) and (x2, y2) are two points on the line y � mx � b. Then, y1 � mx1 �b and y2 � mx2 � b, and so

The constant b in the equation y � mx � b is the value of y corresponding to x � 0. Hence, b is the height at which the line y � mx � b crosses the y axis, andthe corresponding point (0, b) is the y intercept of the line. The situation is illustratedin Figure 1.20.

Because the constants m and b in the equation y � mx � b correspond to theslope and y intercept, respectively, this form of the equation of a line is known as theslope-intercept form.

The slope-intercept form of the equation of a line is particularly useful when geo-metric information about a line (such as its slope or y intercept) is to be determinedfrom the line’s algebraic representation. Here is a typical example.

Find the slope and y intercept of the line 3y � 2x � 6 and draw the graph.

The Slope-Intercept Form of the Equation of a Line �

The equation

y = mx + b

is the equation of the line whose slope is m and whose y intercept is (0, b).

�mx2 � mx1

x2 � x1�

m(x2 � x1)

x2 � x1� m

Slope �y2 � y1

x2 � x1�

(mx2 � b) � (mx1 � b)

x2 � x1

THE SLOPE-INTERCEPT FORMOF THE EQUATION OF A LINE

change in y

change in x


m(0, b) 1

x

y

FIGURE 1.20 The slope and yintercept of the line y � mx � b.

E x p l o r e !E x p l o r e !

Store Y1 � L1*X � 1 in the

equation editor, and determine

the slopes needed to create the

screen shown.EXAMPLE 3 .3EXAMPLE 3 .3

SolutionFirst put the equation 3y � 2x � 6 in slope-intercept form y � mx � b. To do this,solve for y to get

or

It follows that the slope is and the y intercept is (0, 2).

To graph a linear function, plot two of its points and draw a straight line throughthem. In this case, you already know one point, the y intercept (0, 2). A convenientchoice for the x coordinate of the second point is x � 3, since the corresponding y

coordinate is . Draw a line through the points (0, 2) and (3, 0)

to obtain the graph shown in Figure 1.21.

Geometric information about a line can be obtained readily from the slope-interceptformula y � mx � b. There is another form of the equation of a line, however, thatis usually more efficient for problems in which the geometric properties of a line areknown and the goal is to find the equation of the line.

The point-slope form of the equation of a line is simply the formula for slope indisguise. To see this, suppose the point (x, y) lies on the line that passes through agiven point (x0, y0) and that has slope m. Using the points (x, y) and (x0, y0) to com-pute the slope, you get

which you can put in point-slope form

y � y0 � m(x � x0)

by simply multiplying both sides by x � x0.

The use of the point-slope form of the equation of a line is illustrated in the nexttwo examples.

y � y0

x � x0� m

The Point-Slope Form of the Equation of a Line � Theequation

y � y0 � m(x � x0)

is an equation of the line that passes through the point (x0, y0) and that has slopeequal to m.

THE POINT-SLOPE FORMOF THE EQUATION OF A LINE

y � �2

3(3) � 2 � 0

�2

3

y � �2

3x � 23y � �2x � 6


E x p l o r e !E x p l o r e !

Write Y1 � .5X � L1 in the

equation editor and determine

the values of the list L1 needed

to create the screen shown.

y

x

(0, 2)

(3, 0)

FIGURE 1.21 The line 3y � 2x � 6.

Find the equation of the line that passes through the point (5, 1) and whose slope is

equal to .

Solution

Use the formula y � y0 � m(x � x0) with (x0, y0) � (5, 1) and m � to get

which you can rewrite as

The graph is shown in Figure 1.22.

Instead of the point-slope form, the slope-intercept form could have been used tosolve the problem in Example 3.4. For practice, solve the problem this way. Noticethat the solution based on the point-slope formula is more efficient.

The next example illustrates how you can use the point-slope form to find anequation of a line that passes through two given points.

Find the equation of the line that passes through the points (3, �2) and (1, 6).

SolutionFirst compute the slope

Then use the point-slope formula with (1, 6) as the given point (x0, y0) to get

or

Convince yourself that the resulting equation would have been the same if youhad chosen (3, �2) to be the given point (x0, y0). The graph is shown in Figure 1.23.

y � �4x � 10y � 6 � �4(x � 1)

m �6 � (�2)

1 � 3�

8

�2� �4

y �1

2x �

3

2

y � 1 �1

2(x � 5)

1

2

1

2




y

x

(0, 23/2)

(5, 1)

FIGURE 1.22 The line

y �12

x �32

.

y

x

(3, –2)

(1, 6)

10

FIGURE 1.23 The line y � �4x � 10.

If the rate of change of one quantity with respect to a second quantity is constant, thefunction relating the quantities must be linear. The constant rate of change is the slopeof the corresponding line. The next two examples illustrate techniques you can use tofind the appropriate linear functions in such situations.

Since the beginning of the year, the price of whole wheat bread at a local discountsupermarket has been rising at a constant rate of 2 cents per month. By Novemberfirst, the price had reached $1.56 per loaf. Express the price of the bread as a func-tion of time and determine the price at the beginning of the year.

SolutionLet x denote the number of months that have elapsed since the first of the year andy the price of a loaf of bread (in cents). Since y changes at a constant rate with respectto x, the function relating y to x must be linear, and its graph is a straight line. Sincethe price y increases by 2 each time x increases by 1, the slope of the line must be2. The fact that the price was 156 cents ($1.56) on November first, 10 months afterthe first of the year, implies that the line passes through the point (10, 156). To writean equation defining y as a function of x, use the point-slope formula

with

to get or

The corresponding line is shown in Figure 1.24. Notice that the y intercept is (0, 136), which implies that the price of bread at the beginning of the year was $1.36per loaf.

FIGURE 1.24 The rising price of bread: y � 2x � 136.

x

y

(0, 136)

(Jan. 1) (Nov. 1)

(10, 156)

10

y � 2x � 136y � 156 � 2(x � 10)

m � 2, x0 � 10, y0 � 156

y � y0 � m(x � x0)

PRACTICAL APPLICATIONS



Sometimes it is hard to tell how two quantities, x and y, in a data set are relatedby simply examining the data. In such cases, it may be useful to graph the data tosee if the points (x, y) follow a clear pattern (say, lie along a line). Here is an exam-ple of this procedure.

Table 1.2 lists federal employment information for the United States in the period1950–1989. Plot a graph with federal employment on the y axis and time (measuredfrom 1950) on the x axis. Do the points follow a clear pattern? Based on this data,what do you think the federal employment will be in the year 2000?

TABLE 1.2 Employment Information for 1950-1989

Federal CivilianYear Employees (Millions)

1950 1.9

1955 2.2

1960 2.3

1965 2.4

1970 2.7

1975 2.7

1980 2.9

1985 2.9

1989 3.0

Source: U.S. Bureau of the Census, Statistical Abstract of the United States, 1990, p. 339.

SolutionThe graph is shown in Figure 1.25. Note that it is roughly linear. Does this mean fed-eral employment is linearly related to time? Not really, but it does suggest that wemay be able to get useful information by using a line to approximate the data. Sucha line is shown in Figure 1.25. Note that by extending the line to the right, we might“guess” that federal employment will reach approximately 3.4 million in the year2000.


E x p l o r e !E x p l o r e !Place the data in Table 1.2 into

L1 and L2, where L1 is the year

beyond 1900 and L2 is the

number of federal civilian em-

ployees (in millions). Set the

graph type to Scatterplot by

accessing the STAT PLOT menu,

and verify that the line

y � 0.027x � 2.017 fits the data

well, as claimed in the note on

page 39.


It is often useful to find a line that “best approximates” data in some mean-ingful way. A procedure for doing this, called least-squares approximation,is developed in Chapter 7. When least-squares approximation is applied to thedata in this example, it yields the line y � 0.027x � 2.017, so in the year2000 (when x � 50), we may expect the federal civilian employment to be

0.027(50) � 2.017 � 3.37 million.

In applications, it is sometimes necessary or useful to know whether two given linesare parallel or perpendicular. A vertical line is parallel only to other vertical lines andis perpendicular to any horizontal line. Cases involving nonvertical lines can be han-dled by the following criteria.

These criteria are illustrated in Figure 1.26. Geometric proofs are outlined inExercises 54 and 55. We close this section with an example illustrating one way thecriteria can be used.

Parallel and Perpendicular Lines � Let m1 and m2 be the slopesof the nonvertical lines L1 and L2. Then

L1 and L2 are parallel if and only if m1 � m2

L1 and L2 are perpendicular if and only if m2 �

PARALLEL AND PERPENDICULARLINES

FIGURE 1.25 Growth of federal civilian employment in the United States (1950–1989).

5 10 15 20 25 30 35 40 45 500

0

1.0

1.5

2.0

2.5

3.0

3.5

Mill

ions

of

empl

oyee

s

20001950 Years past 1950

y

x


Note

�1

m1

Let L be the line 4x � 3y � 6.(a) Find the equation of a line L1 parallel to L through P(�1, 4).(b) Find the equation of a line L2 perpendicular to L through Q(2, �3).

Solution

By rewriting the equation 4x � 3y � 6 in the slope-intercept form , we

see that L has slope .

(a) Any line parallel to L must also have slope . The required line L1 contains

P(�1, 4), so

(b) A line perpendicular to L must have slope m ��1/mL � . Since the required

line L2 contains Q(2, �3), we have

y �3

4x �

9

2

y � 3 �3

4(x � 2)

3

4

y � �4

3x �

8

3

y � 4 � �4

3(x � 1)

m � �4

3

mL � �4

3

y � �4

3x � 2

FIGURE 1.26

y

x

y

x

L2

L1 L1L2

(a) Parallel lines have m1 � m2 (b) Perpendicular lines have m2 � �1/m1


EXAMPLE 3 .8EXAMPLE 3 .8E x p l o r e !E x p l o r e !Store f(x) � A*x � 2 and g(x)

� (�1/A)*x � 5 into the equa-

tion editor of your graphing cal-

culator. On the home screen,

store different values into A,

and then graph the functions us-

ing a square viewing window.

What do you notice for differ-

ent A values? Can you predict

where the point of intersection

will be?

The given line L and the required lines L1 and L2 are shown in Figure 1.27.

In Problems 1 through 6, find the slope (if possible) of the line that passes throughthe given pair of points.

1. (2, �3) and (0, 4) 2. (�1, 2) and (2, 5)

3. (2, 0) and (0, 2) 4. (5, �1) and (–2, –1)

5. (2, 6) and (2, �4) 6. and

In Problems 7 through 18, find the slope and intercepts of the given line and drawa graph.

7. y � 3x 8. y � 5x � 2

9. y � 3x � 6 10. x � y � 2

11. 3x � 2y � 6 12. 2x � 4y � 12

13. 5y � 3x � 4 14. 4x � 2y � 6

15. 16. y � 2

17. x � �3 18.x � 3

�5�

y � 1

2� 1

x

2�

y

5� 1

��1

7,

1

8��2

3, �

1

5�

FIGURE 1.27 Lines parallel and perpendicular to a given line L.

y

x

L L2

L1

P(�1, 4)

Q(2, �3)


P . R . O . B . L . E . M . S 1.3P . R . O . B . L . E . M . S 1.3

In Problems 19 through 34, write an equation for the line with the given properties.

19. Through (2, 0) with slope 1 20. Through (–1, 2) with slope

21. Through (5, –2) with slope 22. Through (0, 0) with slope 5

23. Through (2, 5) and parallel to the 24. Through (2, 5) and parallel to the x axis y axis

25. Through (1, 0) and (0, 1) 26. Through (2, 5) and (1, �2)

27. Through and 28. Through (�2, 3) and (0, 5)

29. Through (1, 5) and (3, 5) 30. Through (1, 5) and (1, �4)

31. Through (4, 1) and parallel to the 32. Through (�2, 3) and parallel to the line 2x � y � 3 line x � 3y � 5

33. Through (3, 5) and perpendicular 34. Through and perpendicular to the line x � y � 4

to the line 2x + 5y = 3

MANUFACTURING COST 35. A manufacturer’s total cost consists of a fixed overhead of $5,000 plus produc-tion costs of $60 per unit. Express the total cost as a function of the number ofunits produced and draw the graph.

CAR RENTAL 36. A certain car rental agency charges $35 per day plus 55 cents per mile.(a) Express the cost of renting a car from this agency for 1 day as a function of

the number of miles driven and draw the graph.(b) How much does it cost to rent a car for a 1-day trip of 50 miles?(c) How many miles were driven if the daily rental cost was $72?

COURSE REGISTRATION 37. Students at a state college may preregister for their fall classes by mail during thesummer. Those who do not preregister must register in person in September. Theregistrar can process 35 students per hour during the September registrationperiod. Suppose that after 4 hours in September, a total of 360 students (includ-ing those who preregistered) have been registered.(a) Express the number of students registered as a function of time and draw the

graph.(b) How many students were registered after 3 hours?(c) How many students preregistered during the summer?

MEMBERSHIP FEES 38. Membership in a swimming club costs $250 for the 12-week summer season. Ifa member joins after the start of the season, the fee is prorated; that is, it is reducedlinearly.(a) Express the membership fee as a function of the number of weeks that have

elapsed by the time the membership is purchased and draw the graph.

��1

2, 1�

�2

3,

1

4��1

5, 1�

�1

2

2

3


(b) Compute the cost of a membership that is purchased 5 weeks after the startof the season.

LINEAR DEPRECIATION 39. A doctor owns $1,500 worth of medical books which, for tax purposes, areassumed to depreciate linearly to zero over a 10-year period. That is, the value ofthe books decreases at a constant rate so that it is equal to zero at the end of 10years. Express the value of the books as a function of time and draw the graph.

LINEAR DEPRECIATION 40. A manufacturer buys $20,000 worth of machinery that depreciates linearly so thatits trade-in value after 10 years will be $1,000.(a) Express the value of the machinery as a function of its age and draw the graph.(b) Compute the value of the machinery after 4 years.(c) When does the machinery become worthless? The manufacturer might not wait

this long to dispose of the machinery. Discuss the issues the manufacturer mayconsider in deciding when to sell.

WATER CONSUMPTION 41. Since the beginning of the month, a local reservoir has been losing water at a con-stant rate. On the 12th of the month the reservoir held 200 million gallons ofwater, and on the 21st it held only 164 million gallons.(a) Express the amount of water in the reservoir as a function of time and draw

the graph.(b) How much water was in the reservoir on the 8th of the month?

CAR POOLING 42. To encourage motorists to form car pools, the transit authority in a certain met-ropolitan area has been offering a special reduced rate at toll bridges for vehiclescontaining four or more persons. When the program began 30 days ago, 157 vehi-cles qualified for the reduced rate during the morning rush hour. Since then, thenumber of vehicles qualifying has been increasing at a constant rate, and today247 vehicles qualified.(a) Express the number of vehicles qualifying each morning for the reduced rate

as a function of time and draw the graph.(b) If the trend continues, how many vehicles will qualify during the morning rush

hour 14 days from now?

TEMPERATURE CONVERSION 43. (a) Temperature measured in degrees Fahrenheit is a linear function of tempera-ture measured in degrees Celsius. Use the fact that 0° Celsius is equal to 32°Fahrenheit and 100° Celsius is equal to 212° Fahrenheit to write an equationfor this linear function.

(b) Use the function you obtained in part (a) to convert 15° Celsius to Fahrenheit.(c) Convert 68° Fahrenheit to Celsius.

COLLEGE ADMISSIONS 44. The average scores of incoming students at an eastern liberal arts college in theSAT mathematics examination have been declining at a constant rate in recentyears. In 1990, the average SAT score was 575, while in 1995 it was 545.(a) Express the average SAT score as a function of time.(b) If the trend continues, what will the average SAT score of incoming students

be in 2000?(c) If the trend continues, when will the average SAT score be 527?


APPRECIATION OF ASSETS 45. The value of a certain rare book doubles every 10 years. In 1900, the book wasworth $100.(a) How much was it worth in 1930? In 1990? What about the year 2000?(b) Is the value of the book a linear function of its age? Answer this question by

interpreting an appropriate graph.

AIR POLLUTION 46. In certain parts of the world, the number of deaths N per week have been observedto be linearly related to the average concentration x of sulfur dioxide in the air.Suppose there are 97 deaths when x � 100 mg/m3 and 110 deaths when x � 500mg/m3.(a) What is the functional relationship between N and x?(b) Use the function in part (a) to find the number of deaths per week when x �

300 mg/m3. What concentration of sulfur dioxide corresponds to 100 deathsper week?

(c) Research data on how air pollution affects the death rate in a population.*Summarize your results in a one-paragraph essay.

NUTRITION 47. Each ounce of Food I contains 3 gm of carbohydrate and 2 gm of protein, andeach ounce of Food II contains 5 gm of carbohydrate and 3 gm of protein. Sup-pose x ounces of Food I are mixed with y ounces of Food II. The foods are com-bined to produce a blend that contains exactly 73 gm of carbohydrate and 46 gmof protein.(a) Explain why there are 3x � 5y gm of carbohydrate in the blend and why we

must have 3x � 5y � 73. Find a similar equation for protein. Sketch the graphsof both equations.

(b) Where do the two graphs in part (a) intersect? Interpret the significance of thispoint of intersection.

ALCOHOL ABUSE CONTROL 48. Ethyl alcohol is metabolized by the human body at a constant rate (independentof concentration). Suppose the rate is 10 ml per hour.(a) How much time is required to eliminate the effects of a liter of beer contain-

ing 3% ethyl alcohol?(b) Express the time T required to metabolize the effects of drinking ethyl alco-

hol as a function of the amount A of ethyl alcohol consumed.(c) Discuss how the function in part (b) can be used to determine a reasonable

“cutoff” value for the amount of ethyl alcohol A that each individual may beserved at a party.

49. Graph and on the same set of coordinate axes

using [�10, 10]1 by [�10, 10]1. Are the two lines parallel?

y �144

45x �

630

229y �

25

7x �

13

2


* You may find the following articles helpful: D. W. Dockery, J. Schwartz, and J. D. Spengler, “AirPollution and Daily Mortality: Associations with Particulates and Acid Aerosols,” Environ. Res., Vol.59, 1992, pp. 362–373; Y. S. Kim, “Air Pollution, Climate, Socioeconomics Status and Total Mortalityin the United States,” Sci. Total Environ., Vol. 42, 1985, pp. 245–256.

50. Graph and on the same set of coordinate axes

using [�10, 10]1 by [�10, 10]1 for a starting range. Adjust the range settingsuntil both lines are displayed. Are the two lines parallel?

51. A rental company rents a piece of equipment for a $60.00 flat fee plus an hourlyfee of $5.00 per hour.(a) Make a chart showing the number of hours the equipment is rented and the cost

for renting the equipment for 2 hours, 5 hours, 10 hours, and t hours of time.(b) Write an algebraic expression representing the cost y as a function of the num-

ber of hours t. Assume t can be measured to any decimal portion of an hour.(In other words, assume t is any nonnegative real number.)

(c) Graph the expression from part (b).(d) Use the graph to approximate, to two decimal places, the number of hours the

equipment was rented if the bill is $216.25 (before taxes).

ASTRONOMY 52. The following table gives the length of year L (in earth years) of each planet inthe solar system along with the mean (average) distance D of the planet from thesun, in astronomical units (1 astronomical unit is the mean distance of the earthfrom the sun).(a) Plot the point (D, L) for each planet on a coordinate plane. Do these quanti-

ties appear to be linearly related?

(b) For each planet, compute the ratio . Interpret what you find by expressing

L as a function of D.(c) What you have discovered in part (b) is one of Kepler’s laws, named for the

German astronomer Johannes Kepler (1571–1630). Read an article aboutKepler and describe his place in the history of science.

Mean Distance Length ofPlanet from Sun, D Year, L

Mercury 0.388 0.241

Venus 0.722 0.615

Earth 1.000 1.000

Mars 1.523 1.881

Jupiter 5.203 11.862

Saturn 9.545 29.457

Uranus 19.189 84.013

Neptune 30.079 164.783

Pluto 39.463 248.420

Source: Kendrick Frazier, The Solar System, Alexandria, VA:Time/Life Books, p. 37.

L2

D3

y �139

695x �

346

14y �

54

270x �

63

19


EMPLOYMENT RATES 53. In the note after Example 3.7, we observed that the line that “best approximates”the data in the example has the equation y � 0.027x � 2.017. Interpret the slopeof this line in terms of the rate of growth of federal employment. Would youexpect the rate of growth of private employment to be larger or smaller than thefederal growth rate? Write a paragraph on the relationship between federal andprivate employment.

PARALLEL LINES 54. Show that two nonvertical lines are parallel if and only if they have the same slope.

PERPENDICULAR LINES 55. Show that if a nonvertical line L1 with slope m1 is perpendicular to a line L2 withslope m2, then m2 � �1/m1. [Hint: Find expressions for the slopes of the linesL1 and L2 in the accompanying figure. Then apply the Pythagorean theorem alongwith the distance formula from Problem 39, Section 2 of this chapter, to the righttriangle OAB to obtain the required relationship between m1 and m2.]

A mathematical representation of a practical situation is called a mathematicalmodel. In preceding sections, you saw models representing such quantities as manu-facturing cost, air pollution levels, population size, supply, and demand. In this sec-tion, you will see examples illustrating some of the techniques you can use to buildmathematical models of your own.

In the first two examples, the quantity you are seeking is expressed most naturally interms of two variables. You will have to eliminate one of these variables before youcan write the quantity as a function of a single variable.

The highway department is planning to build a picnic area for motorists along a majorhighway. It is to be rectangular with an area of 5,000 square yards and is to be fencedoff on the three sides not adjacent to the highway. Express the number of yards offencing required as a function of the length of the unfenced side.

ELIMINATION OF VARIABLES

L1

L2

A

B

(a, b)

(a, c)

0 x

y


FunctionalModels

4


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Chapter 1 Section 3 Linear Functions 29 - Lamar.edu calculus/hoffman/ch0… · B2 4A2 B 2A Chapter 1 Section 3 Linear Functions 29 EXAMPLE 3.1EXAMPLE 3.1 Linear Functions 3. Solution