M04_SULL6651_09_AIE_C04.QXD
9/30/10
11:32 AM
Page 271
Linear and Quadratic
Functions
Outline
4.1 Linear Functions and Their Properties
4.2 Linear Models:Building Linear Functions
from Data
4.3 Quadratic Functions and Their Properties
4.4 Build Quadratic Models from Verbal
Descriptions and from Data
4.5 Inequalities Involving Quadratic
Functions
•
•
•
•
Chapter Review
Chapter Test
Cumulative Review
Chapter Projects
The Beta of a Stock
Investing in the stock market can be rewarding and fun, but how does
one go about selecting which stocks to purchase? Financial investment firms hire thousands of analysts who track individual stocks
(equities) and assess the value of the underlying company. One
measure the analysts consider is the beta of the stock. Beta
measures the relative risk of an individual company’s equity to
that of a market basket of stocks, such as the Standard &
Poor’s 500. But how is beta computed?
—See the Internet-based Chapter Project—
Up to now, our discussion has focused
on graphs of equations and functions. We learned how to graph equations using
the point-plotting method, intercepts, and the tests for symmetry. In addition, we
learned what a function is and how to identify whether a relation represents a function. We
also discussed properties of functions, such as domain/range, increasing/decreasing, even/odd,
and average rate of change.
Going forward, we will look at classes of functions. In this chapter, we focus on
linear and quadratic functions, their properties, and applications.
271
M04_SULL6651_09_AIE_C04.QXD
272
9/30/10
11:32 AM
Page 272
CHAPTER 4 Linear and Quadratic Functions
4.1 Linear Functions and Their Properties
PREPARING FOR THIS SECTION Before getting started, review the following:
• Functions (Section 3.1, pp. 200–208)
• The Graph of a Function (Section 3.2, pp. 214–217)
• Properties of Functions (Section 3.3, pp. 222–230)
• Lines (Section 2.3, pp. 167–175)
• Graphs of Equations in Two Variables; Intercepts;
Symmetry (Section 2.2, pp. 157–164)
• Linear Equations (Section 1.1, pp. 82–87)
Now Work the ‘Are You Prepared?’ problems on page 278.
OBJECTIVES 1 Graph Linear Functions (p. 272)
2 Use Average Rate of Change to Identify Linear Functions (p. 272)
3 Determine Whether a Linear Function Is Increasing, Decreasing,
or Constant (p. 275)
4 Build Linear Models from Verbal Descriptions (p. 276)
1 Graph Linear Functions
In Section 2.3 we discussed lines. In particular, for nonvertical lines we developed
the slope–intercept form of the equation of a line y = mx + b. When we write the
slope–intercept form of a line using function notation, we have a linear function.
DEFINITION
A linear function is a function of the form
f1x2 = mx + b
The graph of a linear function is a line with slope m and y-intercept b. Its
domain is the set of all real numbers.
Functions that are not linear are said to be nonlinear.
EXAMPL E 1
Graphing a Linear Function
Graph the linear function:
Solution
Figure 1
y
(0, 7)
Δx 1
This is a linear function with slope m = -3 and y-intercept b = 7. To graph this
function, we plot the point 10, 72, the y-intercept, and use the slope to find an
additional point by moving right 1 unit and down 3 units. See Figure 1.
䊉
Δy 3
5
Alternatively, we could have found an additional point by evaluating the
function at some x Z 0. For x = 1, we find f112 = -3112 + 7 = 4 and obtain the
point 11, 42 on the graph.
(1, 4)
3
1
Now Work
1
3
f1x2 = -3x + 7
PROBLEMS
13(a)
AND
(b)
5 x
2 Use Average Rate of Change to Identify Linear Functions
Look at Table 1, which shows certain values of the independent variable x and
corresponding values of the dependent variable y for the function f1x2 = - 3x + 7.
Notice that as the value of the independent variable, x, increases by 1 the value of
the dependent variable y decreases by 3. That is, the average rate of change of y with
respect to x is a constant, -3.
M04_SULL6651_09_AIE_C04.QXD
9/30/10
11:32 AM
Page 273
SECTION 4.1 Linear Functions and Their Properties
Table 1
x
y ⴝ f (x) ⴝ ⴚ3x ⴙ 7
-2
13
Average Rate of Change ⴝ
273
≤y
≤x
-3
10 - 13
=
= -3
1
- 1 - 1- 22
-1
10
7 - 10
-3
=
= -3
1
0 - 1- 12
0
7
1
4
-3
-3
2
1
3
-2
-3
It is not a coincidence that the average rate of change of the linear function
¢y
f1x2 = - 3x + 7 is the slope of the linear function. That is,
= m = -3. The
¢x
following theorem states this fact.
THEOREM
Average Rate of Change of a Linear Function
Linear functions have a constant average rate of change. That is, the average
rate of change of a linear function f1x2 = mx + b is
¢y
= m
¢x
Proof The average rate of change of f1x2 = mx + b from x1 to x2, x1 Z x2, is
1mx2 + b2 - 1mx1 + b2
f1x22 - f1x12
¢y
=
=
x2 - x1
x2 - x1
¢x
m1x
x
2
mx2 - mx1
2
1
=
=
= m
x2 - x1
x2 - x1
䊏
Based on the theorem just proved, the average rate of change of the function
2
2
g1x2 = - x + 5 is - .
5
5
Now Work
PROBLEM
13(C)
As it turns out, only linear functions have a constant average rate of change.
Because of this, we can use the average rate of change to determine whether a function
is linear or not. This is especially useful if the function is defined by a data set.
EXAMPL E 2
Using the Average Rate of Change to Identify Linear Functions
(a) A strain of E-coli Beu 397-recA441 is placed into a Petri dish at 30° Celsius
and allowed to grow. The data shown in Table 2 on page 274 are collected. The
population is measured in grams and the time in hours. Plot the ordered pairs
1x, y2 in the Cartesian plane and use the average rate of change to determine
whether the function is linear.
M04_SULL6651_09_AIE_C04.QXD
274
9/30/10
11:32 AM
Page 274
CHAPTER 4 Linear and Quadratic Functions
(b) The data in Table 3 represent the maximum number of heartbeats that a healthy
individual should have during a 15-second interval of time while exercising for
different ages. Plot the ordered pairs 1x, y2 in the Cartesian plane and use the
average rate of change to determine whether the function is linear.
Table 2
Table 3
Time
(hours), x
Population
(grams), y
(x, y)
Age, x
Maximum Number of
Heartbeats, y
(x, y)
0
0.09
(0, 0.09)
20
50
(20, 50)
1
0.12
(1, 0.12)
30
47.5
(30, 47.5)
2
0.16
(2, 0.16)
40
45
(40, 45)
3
0.22
(3, 0.22)
50
42.5
(50, 42.5)
4
0.29
(4, 0.29)
60
40
(60, 40)
5
0.39
(5, 0.39)
70
37.5
(70, 37.5)
Source: American Heart Association
Solution
Compute the average rate of change of each function. If the average rate of change
is constant, the function is linear. If the average rate of change is not constant, the
function is nonlinear.
(a) Figure 2 shows the points listed in Table 2 plotted in the Cartesian plane. Notice
that it is impossible to draw a straight line that contains all the points. Table 4
displays the average rate of change of the population.
Figure 2
Table 4
Population (grams), y
y
0.4
Time (hours), x
Population (grams), y
0
0.09
Average Rate of Change ⴝ
≤y
≤x
0.12 - 0.09
= 0.03
1 - 0
0.3
1
0.2
0.12
0.04
0.1
2
0
1
2
3
4
Time (hours), x
5
0.16
x
0.06
3
0.22
0.07
4
0.29
0.10
5
0.39
Because the average rate of change is not constant, we know that the
function is not linear. In fact, because the average rate of change is increasing as
the value of the independent variable increases, the function is increasing at an
increasing rate. So not only is the population increasing over time, but it is also
growing more rapidly as time passes.
(b) Figure 3 shows the points listed in Table 3 plotted in the Cartesian plane. We can
see that the data in Figure 3 lie on a straight line. Table 5 contains the average
rate of change of the maximum number of heartbeats. The average rate of
change of the heartbeat data is constant, - 0.25 beat per year, so the function is
linear.
M04_SULL6651_09_AIE_C04.QXD
9/30/10
11:32 AM
Page 275
SECTION 4.1 Linear Functions and Their Properties
Figure 3
Table 5
y
Heart beats
50
Age, x
Maximum Number
of Heartbeats, y
20
50
Average Rate of Change ⴝ
275
≤y
≤x
47.5 - 50
= - 0.25
30 - 20
45
30
47.5
- 0.25
40
40
45
- 0.25
20
30
40
50
60
70
x
50
42.5
Age
- 0.25
60
40
- 0.25
70
37.5
䊉
Now Work
PROBLEM
21
3 Determine Whether a Linear Function Is Increasing,
Decreasing, or Constant
Look back at the Seeing the Concept on page 169. When the slope m of a linear
function is positive 1m 7 02, the line slants upward from left to right. When the
slope m of a linear function is negative 1m 6 02, the line slants downward from left
to right. When the slope m of a linear function is zero 1m = 02, the line is horizontal.
THEOREM
Increasing, Decreasing, and Constant Linear Functions
A linear function f1x2 = mx + b is increasing over its domain if its slope, m,
is positive. It is decreasing over its domain if its slope, m, is negative. It is
constant over its domain if its slope, m, is zero.
EXAMPL E 3
Determining Whether a Linear Function Is Increasing,
Decreasing, or Constant
Determine whether the following linear functions are increasing, decreasing, or
constant.
(a) f1x2 = 5x - 2
3
(c) s1t2 = t - 4
4
Solution
(b) g1x2 = -2x + 8
(d) h1z2 = 7
(a) For the linear function f1x2 = 5x - 2, the slope is 5, which is positive. The
function f is increasing on the interval 1- q , q 2.
(b) For the linear function g1x2 = - 2x + 8, the slope is -2, which is negative. The
function g is decreasing on the interval 1- q , q 2.
3
3
(c) For the linear function s1t2 = t - 4, the slope is , which is positive. The
4
4
function s is increasing on the interval 1- q , q 2.
(d) We can write the linear function h as h1z2 = 0z + 7. Because the slope is 0,
the function h is constant on the interval 1- q , q 2.
䊉
Now Work
PROBLEM
13 (d)
M04_SULL6651_09_AIE_C04.QXD
276
9/30/10
11:32 AM
Page 276
CHAPTER 4 Linear and Quadratic Functions
4 Build Linear Models from Verbal Descriptions
When the average rate of change of a function is constant, we can use a linear
function to model the relation between the two variables. For example, if your phone
company charges you $0.07 per minute to talk regardless of the number of minutes
used, we can model the relation between the cost C and minutes used x as the linear
0.07 dollar
function C1x2 = 0.07x, with slope m =
.
1 minute
Modeling with a Linear Function
If the average rate of change of a function is a constant m, a linear function f
can be used to model the relation between the two variables as follows:
f1x2 = mx + b
where b is the value of f at 0, that is, b = f102.
EXAMPL E 4
Straight-line Depreciation
Book value is the value of an asset that a company uses to create its balance sheet.
Some companies depreciate their assets using straight-line depreciation so that the
value of the asset declines by a fixed amount each year. The amount of the decline
depends on the useful life that the company places on the asset. Suppose that a
company just purchased a fleet of new cars for its sales force at a cost of $28,000 per car.
The company chooses to depreciate each vehicle using the straight-line method over
$28,000
7 years. This means that each car will depreciate by
= $4000 per year.
7
(a) Write a linear function that expresses the book value V of each car as a function
of its age, x.
(b) Graph the linear function.
(c) What is the book value of each car after 3 years?
(d) Interpret the slope.
(e) When will the book value of each car be $8000?
[Hint: Solve the equation V1x2 = 8000.]
Solution
Figure 4
v
28,000
(a) If we let V1x2 represent the value of each car after x years, then V102 represents
the original value of each car, so V102 = $28,000. The y-intercept of the linear
function is $28,000. Because each car depreciates by $4000 per year, the slope
of the linear function is -4000. The linear function that represents the book
value V of each car after x years is
V1x2 = -4000x + 28,000
Book value ($)
24,000
(b) Figure 4 shows the graph of V.
(c) The book value of each car after 3 years is
20,000
16,000
V132 = -4000132 + 28,000
= $16,000
12,000
8000
4000
1
2 3 4 5 6 7 x
Age of vehicle (years)
(d) Since the slope of V1x2 = - 4000x + 28,000 is - 4000, the average rate of
change of book value is -$4000/year. So for each additional year that passes
the book value of the car decreases by $4000.
M04_SULL6651_09_AIE_C04.QXD
9/30/10
11:32 AM
Page 277
SECTION 4.1 Linear Functions and Their Properties
277
(e) To find when the book value will be $8000, solve the equation
V1x2 = 8000
-4000x + 28,000 = 8000
- 4000x = - 20,000
x =
Subtract 28,000 from each side.
- 20,000
= 5 Divide by -4000.
-4000
The car will have a book value of $8000 when it is 5 years old.
Now Work
EXAMPL E 5
PROBLEM
䊉
45
Supply and Demand
The quantity supplied of a good is the amount of a product that a company is willing
to make available for sale at a given price. The quantity demanded of a good is the
amount of a product that consumers are willing to purchase at a given price.
Suppose that the quantity supplied, S, and quantity demanded, D, of cellular
telephones each month are given by the following functions:
S1p2 = 60p - 900
D1p2 = - 15p + 2850
where p is the price (in dollars) of the telephone.
(a) The equilibrium price of a product is defined as the price at which quantity
supplied equals quantity demanded. That is, the equilibrium price is the price at
which S1p2 = D1p2. Find the equilibrium price of cellular telephones. What is the
equilibrium quantity, the amount demanded (or supplied) at the equilibrium price?
(b) Determine the prices for which quantity supplied is greater than quantity
demanded. That is, solve the inequality S1p2 7 D1p2.
(c) Graph S = S1p2, D = D1p2 and label the equilibrium price.
Solution
(a) To find the equilibrium price, solve the equation S1p2 = D1p2.
60p - 900 = -15p + 2850
S1p2 = 60p - 900;
D (p) = -15p + 2850
60p = - 15p + 3750 Add 900 to each side.
75p = 3750
Add 15p to each side.
p = 50
Divide each side by 75.
The equilibrium price is $50 per cellular phone. To find the equilibrium quantity,
evaluate either S1p2 or D1p2 at p = 50.
S1502 = 601502 - 900 = 2100
The equilibrium quantity is 2100 cellular phones. At a price of $50 per phone,
the company will produce and sell 2100 phones each month and have no shortages or excess inventory.
(b) The inequality S1p2 7 D1p2 is
60p - 900
60p
75p
p
7
7
7
7
-15p + 2850 S1p2 7 D1p2
- 15p + 3750 Add 900 to each side.
3750
Add 15p to each side.
50
Divide each side by 75.
If the company charges more than $50 per phone, quantity supplied will exceed
quantity demanded. In this case the company will have excess phones in
inventory.
M04_SULL6651_09_AIE_C04.QXD
278
9/30/10
11:32 AM
Page 278
CHAPTER 4 Linear and Quadratic Functions
(c) Figure 5 shows the graphs of S = S1p2 and D = D1p2 with the equilibrium
point labeled.
S, D
Quantity Supplied,
Quantity Demanded
Figure 5
S S(p)
3000 (0, 2850)
Equilibrium point
(50, 2100)
2000
D D(p)
1000
(15, 0)
50
100 p
Price ($)
䊉
Now Work
39
PROBLEM
4.1 Assess Your Understanding
‘Are You Prepared?’
Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.
1. Graph y = 2x - 3. (pp. 157–164)
2. Find the slope of the line joining the points 12, 52 and
1- 1, 32. (pp. 167–175)
3. Find the average rate of change of f1x2 = 3x2 - 2, from
2 to 4. (pp. 222–230)
4. Solve: 60x - 900 = - 15x + 2850. (pp. 82–87)
5. If f1x2 = x2 - 4, find f1- 22. (pp. 200–208)
6. True or False The graph of the function f1x2 = x2 is
increasing on the interval 10, q 2. (pp. 214–217)
Concepts and Vocabulary
7. For the graph of the linear function f1x2 = mx + b, m is the
and b is the
.
10. True or False The slope of a nonvertical line is the average
rate of change of the linear function.
8. For the graph of the linear function H1z2 = -4z + 3, the
slope is
and the y-intercept is
.
11. True or False If the average rate of change of a linear func2
tion is , then if y increases by 3, x will increase by 2.
3
12. True or False The average rate of change of f1x2 = 2x + 8
is 8.
9. If the slope m of the graph of a linear function is
, the function is increasing over its domain.
Skill Building
In Problems 13–20, a linear function is given.
(a)
(b)
(c)
(d)
Determine the slope and y-intercept of each function.
Use the slope and y-intercept to graph the linear function.
Determine the average rate of change of each function.
Determine whether the linear function is increasing, decreasing, or constant.
13. f1x2 = 2x + 3
17. f1x2 =
1
x - 3
4
14. g1x2 = 5x - 4
15. h1x2 = -3x + 4
16. p1x2 = -x + 6
2
18. h1x2 = - x + 4
3
19. F1x2 = 4
20. G1x2 = - 2
In Problems 21–28, determine whether the given function is linear or nonlinear. If it is linear, determine the slope.
21.
y ⴝ f (x)
x
22.
x
y ⴝ f (x)
23.
x
y ⴝ f (x)
24.
x
y ⴝ f (x)
-2
4
-2
1/4
-2
-8
-2
-4
-1
1
-1
1/2
-1
-3
-1
0
0
-2
0
1
0
0
0
4
1
-5
1
2
1
1
1
8
2
-8
2
4
2
0
2
12
M04_SULL6651_09_AIE_C04.QXD
9/30/10
11:32 AM
Page 279
SECTION 4.1 Linear Functions and Their Properties
25.
26.
y ⴝ f (x)
27.
x
y ⴝ f (x)
-4
-2
-1
- 3.5
0
-3
-2
1
- 10
2
x
y ⴝ f (x)
-2
- 26
-2
-1
-4
0
2
1
2
x
28.
279
x
y ⴝ f (x)
8
-2
0
-1
8
-1
1
0
8
0
4
- 2.5
1
8
1
9
-2
2
8
2
16
Applications and Extensions
29. Suppose that f1x2 = 4x - 1 and g1x2 = -2x + 5.
(a) Solve f1x2 = 0.
(b) Solve f1x2 7 0.
(c) Solve f1x2 = g1x2.
(d) Solve f1x2 … g1x2.
(e) Graph y = f1x2 and y = g1x2 and label the point that
represents the solution to the equation f1x2 = g1x2.
30. Suppose that f1x2 = 3x + 5 and g1x2 = -2x + 15.
(a) Solve f1x2 = 0.
(b) Solve f1x2 6 0.
(c) Solve f1x2 = g1x2.
(d) Solve f1x2 Ú g1x2.
(e) Graph y = f1x2 and y = g1x2 and label the point that
represents the solution to the equation f1x2 = g1x2.
31. In parts (a)–(f), use the following figure.
y
y ⫽ f (x )
34. In parts (a) and (b), use the following figure.
y
y ⫽ f(x)
(2, 5)
x
y ⫽ g(x)
(a) Solve the equation: f1x2 = g1x2.
(b) Solve the inequality: f1x2 … g1x2.
35. In parts (a) and (b), use the following figure.
y ⫽ f(x)
y
(88, 80)
(0, 12)
(40, 50)
(5, 12)
y ⫽ h (x)
x
(⫺40, 0)
x
(⫺6,⫺5)
(a) Solve f1x2 = 50.
(c) Solve f1x2 = 0.
(e) Solve f1x2 … 80.
(b) Solve f1x2 = 80.
(d) Solve f1x2 7 50.
(f) Solve 0 6 f1x2 6 80.
32. In parts (a)–(f), use the following figure.
y
y ⫽ g(x )
(0,⫺5)
y ⫽ g (x)
(a) Solve the equation: f1x2 = g1x2.
(b) Solve the inequality: g1x2 … f1x2 6 h1x2.
36. In parts (a) and (b), use the following figure.
y
(⫺15, 60)
(0, 7)
y ⫽ h (x )
(⫺4, 7)
(5, 20)
x
(15, 0)
x
(a) Solve g1x2 = 20.
(c) Solve g1x2 = 0.
(e) Solve g1x2 … 60.
(b) Solve g1x2 = 60.
(d) Solve g1x2 7 20.
(f) Solve 0 6 g1x2 6 60.
33. In parts (a) and (b) use the following figure.
y ⫽ f (x)
y
y ⫽ g(x)
(⫺4, 6)
x
(a) Solve the equation: f1x2 = g1x2.
(b) Solve the inequality: f1x2 7 g1x2.
(0, ⫺8)
(7,⫺8)
y ⫽ g (x )
y ⫽ f(x)
(a) Solve the equation: f1x2 = g1x2.
(b) Solve the inequality: g1x2 6 f1x2 … h1x2.
37. Car Rentals The cost C, in dollars, of renting a moving truck
for a day is modeled by the function C1x2 = 0.25x + 35,
where x is the number of miles driven.
(a) What is the cost if you drive x = 40 miles?
(b) If the cost of renting the moving truck is $80, how many
miles did you drive?
(c) Suppose that you want the cost to be no more than $100.
What is the maximum number of miles that you can
drive?
(d) What is the implied domain of C?
M04_SULL6651_09_AIE_C04.QXD
280
9/30/10
11:32 AM
Page 280
CHAPTER 4 Linear and Quadratic Functions
38. Phone Charges The monthly cost C, in dollars, for
international calls on a certain cellular phone plan is
modeled by the function C1x2 = 0.38x + 5, where x is the
number of minutes used.
(a) What is the cost if you talk on the phone for x = 50
minutes?
(b) Suppose that your monthly bill is $29.32. How many
minutes did you use the phone?
(c) Suppose that you budget yourself $60 per month for the
phone. What is the maximum number of minutes that
you can talk?
(d) What is the implied domain of C if there are 30 days in
the month?
39. Supply and Demand Suppose that the quantity supplied S
and quantity demanded D of T-shirts at a concert are given
by the following functions:
S1p2 = -200 + 50p
D1p2 = 1000 - 25p
where p is the price of a T-shirt.
(a) Find the equilibrium price for T-shirts at this concert.
What is the equilibrium quantity?
(b) Determine the prices for which quantity demanded is
greater than quantity supplied.
(c) What do you think will eventually happen to the price
of T-shirts if quantity demanded is greater than quantity
supplied?
40. Supply and Demand Suppose that the quantity supplied S
and quantity demanded D of hot dogs at a baseball game
are given by the following functions:
S1p2 = - 2000 + 3000p
D1p2 = 10,000 - 1000p
where p is the price of a hot dog.
(a) Find the equilibrium price for hot dogs at the baseball
game.What is the equilibrium quantity?
(b) Determine the prices for which quantity demanded is
less than quantity supplied.
(c) What do you think will eventually happen to the price
of hot dogs if quantity demanded is less than quantity
supplied?
41. Taxes The function T1x2 = 0.151x - 83502 + 835 represents the tax bill T of a single person whose adjusted gross
income is x dollars for income between $8350 and $33,950,
inclusive, in 2009.
Source: Internal Revenue Service
(a) What is the domain of this linear function?
(b) What is a single filer’s tax bill if adjusted gross income is
$20,000?
(c) Which variable is independent and which is dependent?
(d) Graph the linear function over the domain specified in
part (a).
(e) What is a single filer’s adjusted gross income if the tax
bill is $3707.50?
42. Luxury Tax In 2002, major league baseball signed a labor
agreement with the players. In this agreement, any team
whose payroll exceeded $136.5 million in 2006 had to pay a
luxury tax of 40% (for second offenses). The linear function
T1p2 = 0.401p - 136.52 describes the luxury tax T of a team
whose payroll was p (in millions of dollars).
Source: Major League Baseball
(a) What is the implied domain of this linear function?
(b) What was the luxury tax for the New York Yankees
whose 2006 payroll was $171.1 million?
(c) Graph the linear function.
(d) What is the payroll of a team that pays a luxury tax of
$11.7 million?
The point at which a company’s profits equal zero is called the
company’s break-even point. For Problems 43 and 44, let R
represent a company’s revenue, let C represent the company’s costs,
and let x represent the number of units produced and sold each day.
(a) Find the firm’s break-even point; that is, find x so that R = C.
(b) Find the values of x such that R1x2 7 C1x2.This represents the
number of units that the company must sell to earn a profit.
43. R1x2 = 8x
C1x2 = 4.5x + 17,500
44. R1x2 = 12x
C1x2 = 10x + 15,000
45. Straight-line Depreciation Suppose that a company has
just purchased a new computer for $3000. The company
chooses to depreciate the computer using the straight-line
method over 3 years.
(a) Write a linear model that expresses the book value V of
the computer as a function of its age x.
(b) What is the implied domain of the function found in
part (a)?
(c) Graph the linear function.
(d) What is the book value of the computer after 2 years?
(e) When will the computer have a book value of $2000?
46. Straight-line Depreciation Suppose that a company has
just purchased a new machine for its manufacturing facility
for $120,000. The company chooses to depreciate the
machine using the straight-line method over 10 years.
(a) Write a linear model that expresses the book value V of
the machine as a function of its age x.
(b) What is the implied domain of the function found in
part (a)?
(c) Graph the linear function.
(d) What is the book value of the machine after 4 years?
(e) When will the machine have a book value of $72,000?
47. Cost Function The simplest cost function is the linear cost
function, C1x2 = mx + b, where the y-intercept b represents
the fixed costs of operating a business and the slope m
represents the cost of each item produced. Suppose that a
small bicycle manufacturer has daily fixed costs of $1800
and each bicycle costs $90 to manufacture.
(a) Write a linear model that expresses the cost C of manufacturing x bicycles in a day.
(b) Graph the model.
(c) What is the cost of manufacturing 14 bicycles in a day?
(d) How many bicycles could be manufactured for $3780?
48. Cost Function Refer to Problem 47. Suppose that the
landlord of the building increases the bicycle manufacturer’s rent by $100 per month.
(a) Assuming that the manufacturer is open for business
20 days per month, what are the new daily fixed costs?
(b) Write a linear model that expresses the cost C of manufacturing x bicycles in a day with the higher rent.
(c) Graph the model.
(d) What is the cost of manufacturing 14 bicycles in a day?
(e) How many bicycles can be manufactured for $3780?
M04_SULL6651_09_AIE_C04.QXD
9/30/10
11:32 AM
Page 281
SECTION 4.1 Linear Functions and Their Properties
49. Truck Rentals A truck rental company rents a truck for
one day by charging $29 plus $0.07 per mile.
(a) Write a linear model that relates the cost C, in dollars,
of renting the truck to the number x of miles driven.
(b) What is the cost of renting the truck if the truck is
driven 110 miles? 230 miles?
281
50. Long Distance A phone company offers a domestic long
distance package by charging $5 plus $0.05 per minute.
(a) Write a linear model that relates the cost C, in dollars,
of talking x minutes.
(b) What is the cost of talking 105 minutes? 180 minutes?
Mixed Practice
51. Developing a Linear Model from Data The following data
represent the price p and quantity demanded per day q of
24" LCD monitor.
52. Developing a Linear Model from Data The following data
represent the various combinations of soda and hot dogs
that Yolanda can buy at a baseball game with $60.
Price, p (in dollars)
Quantity Demanded, q
Soda, s
Hot Dogs, h
150
100
20
0
200
80
15
3
250
60
10
6
300
40
5
9
(a) Plot the ordered pairs 1s, h2 in a Cartesian plane.
(b) Show that the number of hot dogs purchased h is a
linear function of the number of sodas purchased s.
(c) Determine the linear function that describes the
relation between s and h.
(d) What is the implied domain of the linear function?
(e) Graph the linear function in the Cartesian plane
drawn in part (a).
(f) Interpret the slope.
(g) Interpret the values of the intercepts.
(a) Plot the ordered pairs 1p, q2 in a Cartesian plane.
(b) Show that quantity demanded q is a linear function of
the price p.
(c) Determine the linear function that describes the relation
between p and q.
(d) What is the implied domain of the linear function?
(e) Graph the linear function in the Cartesian plane drawn
in part (a).
(f) Interpret the slope.
(g) Interpret the values of the intercepts.
Explaining Concepts: Discussion and Writing
53. Which of the following functions might have the graph
shown? (More than one answer is possible.)
(a) f1x2 = 2x - 7
y
(b) g1x2 = - 3x + 4
(c) H1x2 = 5
(d) F1x2 = 3x + 4
1
x
(e) G1x2 = x + 2
2
54. Which of the following functions might have the graph
shown? (More than one answer is possible.)
(a) f1x2 = 3x + 1
y
(b) g1x2 = -2x + 3
(c) H1x2 = 3
(d) F1x2 = -4x - 1
x
2
(e) G1x2 = - x + 3
3
55. Under what circumstances is a linear function f1x2 = mx + b odd? Can a linear function ever be even?
56. Explain how the graph of f1x2 = mx + b can be used to solve mx + b 7 0.
‘Are You Prepared?’ Answers
1.
2.
y
2
2
2
(0,3)
2 x
(1,1)
2
3
3. 18
4. 5506
5. 0
6. True