4.6
Applications of Two-Variable
Graphs
OBJECTIVES
4.6
1. Solve an application of a linear function
2. Identify the dependent and independent
variables for an application
3. Create a scatter plot
It is unusual to find a newspaper that does not have several two-variable graphs. Each of the
following graphs was found in a daily newspaper.
8
64
66
68
70
Employment from business starts
1,000
900
800
700
1995
96
97
98
99
Year
y
Cost ($1000) of 30-second
commercial during super bowl
1600
1400
Cost ($1000s)
© 2001 McGraw-Hill Companies
1200
1000
800
600
400
200
x
0
81 85 89 93 97 00
Year
All three of the graphs above use the principles described in this chapter. There are, however, a few noticeable differences. Notice the units on the x and y axes for each graph. This
leads to the following rule of graphing applications.
271
APPLICATIONS OF TWO-VARIABLE GRAPHS
SECTION 4.6
273
Note that the graph of equation (1) does not extend beyond the first quadrant because of
the nature of our problem, in which solutions are only realistic when s 0.
CHECK YOURSELF 1
A salesperson’s monthly salary S is based on a fixed salary of $1200 plus 8% of all
monthly sales x. The linear equation relating S and x is
S 0.08x 1200
Graph the relationship between S and x. Hint: Find the monthly salary for sales of
$0, $10,000, and $20,000.
In our second example, we will find and graph a linear equation from just two points.
Example 2
An Application of a Linear Function
In producing a new product, a manufacturer predicts that the number of items produced x
and the cost in dollars C of producing those items will be related by a linear equation.
Suppose that the cost of producing 100 items will be $5000 and the cost of producing
500 items will be $15,000. Find the linear equation relating x and C.
To solve this problem, we must find the equation of the line passing through points
(100, 5000) and (500, 15,000).
Although the numbers are considerably larger than we have encountered thus far in this
section, the process is exactly the same.
First, we find the slope:
m
15,000 5000
10,000
25
500 100
400
We can now use the point-slope form as before to find the desired equation.
C 5000 25(x 100)
C 5000 25x 2500
C 25x 2500
To graph the equation we have just derived, we must choose the scaling on the x and
C axes carefully to get a “reasonable” picture. Here we choose increments of 100 on the
x axis and 2500 on the C axis because those seem appropriate for the given information.
© 2001 McGraw-Hill Companies
C
15,000
(500, 15,000)
12,500
NOTE Notice how the change
in scaling “distorts” the slope of
the line.
10,000
7500
5000
(100, 5000)
2500
x
100 200 300 400 500
APPLICATIONS OF TWO-VARIABLE GRAPHS
SECTION 4.6
273
Note that the graph of equation (1) does not extend beyond the first quadrant because of
the nature of our problem, in which solutions are only realistic when s 0.
CHECK YOURSELF 1
A salesperson’s monthly salary S is based on a fixed salary of $1200 plus 8% of all
monthly sales x. The linear equation relating S and x is
S 0.08x 1200
Graph the relationship between S and x. Hint: Find the monthly salary for sales of
$0, $10,000, and $20,000.
In our second example, we will find and graph a linear equation from just two points.
Example 2
An Application of a Linear Function
In producing a new product, a manufacturer predicts that the number of items produced x
and the cost in dollars C of producing those items will be related by a linear equation.
Suppose that the cost of producing 100 items will be $5000 and the cost of producing
500 items will be $15,000. Find the linear equation relating x and C.
To solve this problem, we must find the equation of the line passing through points
(100, 5000) and (500, 15,000).
Although the numbers are considerably larger than we have encountered thus far in this
section, the process is exactly the same.
First, we find the slope:
m
15,000 5000
10,000
25
500 100
400
We can now use the point-slope form as before to find the desired equation.
C 5000 25(x 100)
C 5000 25x 2500
C 25x 2500
To graph the equation we have just derived, we must choose the scaling on the x and
C axes carefully to get a “reasonable” picture. Here we choose increments of 100 on the
x axis and 2500 on the C axis because those seem appropriate for the given information.
© 2001 McGraw-Hill Companies
C
15,000
(500, 15,000)
12,500
NOTE Notice how the change
in scaling “distorts” the slope of
the line.
10,000
7500
5000
(100, 5000)
2500
x
100 200 300 400 500
274
CHAPTER 4
GRAPHS OF LINEAR EQUATIONS AND FUNCTIONS
CHECK YOURSELF 2
A company predicts that the value in dollars V and the time that a piece of
equipment has been in use t are related by a linear equation. If the equipment is
valued at $1500 after 2 years and at $300 after 10 years, find the linear equation
relating t and V.
When an equation is written such that the left side is y and the right side is an expression
involving the variable x, such as
y 3x 2
we can rewrite the equation as a function. In this case, we have
f(x) 3x 2
This implies that
y f(x)
We can say that y is a function of x, or y is dependent on x. That leads to the following
definitions.
Definitions: Independent Variable and Dependent Variable
Given that y f(x),
x is called the independent variable and y is called the dependent variable.
Identifying which variable is independent and which is dependent is important in many
applications.
Example 3
Identifying the Dependent Variable
From each pair, identify which variable is dependent on the other.
(a) The age of a car and its resale value.
(b) The amount of interest earned in a bank account and the amount of time the money
has been in the bank.
NOTE If you think about it,
you will see that time will be
the independent variable in
most ordered pairs. Most
everything depends on time
rather than the reverse.
The interest depends on the time, so interest is the dependent variable (y) and time is the
independent variable (x).
(c) The number of cigarettes you have smoked and the probability of dying from a
smoking-related disease.
The number of cigarettes is the independent variable (x), and the probability of dying
from a smoking-related disease is the dependent variable (y).
© 2001 McGraw-Hill Companies
The resale value depends on the age, so we would assign the age of the car the independent variable (x) and the resale value the dependent variable (y).
APPLICATIONS OF TWO-VARIABLE GRAPHS
SECTION 4.6
275
CHECK YOURSELF 3
From each pair, identify which variable is dependent on the other.
(a) The number of credits taken and the amount of tuition paid.
(b) The temperature of a cup of coffee and the length of time since it was poured.
In the next example, you will combine the skills you have learned to this point of the
section.
Example 4
Modeling with a Function
Shaquille and Kobe are interested in renting a gym for summer basketball. They are told
that they must pay a flat rate of $200 plus $75 per hour.
(a) Identify the dependent and independent variables.
(b) Find the equation of the relationship.
(c) Scale the axes and graph the relationship.
(d) Find f(2) and f(5).
(a) The independent variable is the number of hours of use. The dependent variable is the
total cost.
(b) The equation is y 200 75x, or we could write f(x) 200 75x.
© 2001 McGraw-Hill Companies
(c) Using only the first quadrant (why?), we get the graph
600
(5, 575)
500
400
(2, 350)
300
200
100
2
NOTE f(2) is the cost of a 2-h
rental.
5
(d) f(2) $350, f(5) $575
GRAPHS OF LINEAR EQUATIONS AND FUNCTIONS
CHECK YOURSELF 4
Tiger and Sergio are selling instructional videos. Their contract gives them $10,000
plus $1.25 per video (think of this as $1250 for each 1000 videos sold)
(a)
(b)
(c)
(b)
Identify the dependent and independent variables.
Find the equation of the relationship.
Scale the axes and graph the relationship.
Find f(3000) and f(50,000).
In Example 2, we found a graph given two points. Although this is occasionally useful, it is
far more common that we have many points. To graph several points we use a scatter plot.
A scatter plot is a graph of a set of ordered pairs. Scatter plots help us see the relationship between two sets of data. For example, the following graph represents the relationship
between the number of wins and the number of losses that a professional football team
might have in a full season. We can use this graph to determine the number of losses that a
team with 10 wins would have.
15
10
Losses
5
5
10
15
Wins
The ordered pair (10, 6) indicates that a team with 10 wins would have 6 losses. Notice that
the ordered pairs form a perfect line with slope 1.
The set of ordered pairs graphed below shows the relationship between the number of
miles driven and the amount of gas purchased the last 12 times that Allie filled her gas tank.
Notice that the points almost form a straight line.
10
8
6
4
2
100
200
Miles
300
© 2001 McGraw-Hill Companies
CHAPTER 4
Gallons
276
APPLICATIONS OF TWO-VARIABLE GRAPHS
NOTE A prediction line is a line
that gives us a “reasonable”
estimation for y when we have
a given x. We will reserve
definition of the word
“reasonable” for future
mathematics classes.
SECTION 4.6
277
Suppose that you were asked to estimate the amount of gas Allie will need to buy if she
drives 250 miles. Even though there is no ordered pair associated with 250 miles, you can
comfortably guess that Allie will need about 9 gallons of gas. You arrived at this answer by
noting that the points fell in a fairly straight line, and you estimated where that line would
be when the x was 250. Essentially, you created a prediction line, which is a line that is
used to estimate the y value when you are given a value for x.
In a subsequent mathematics or statistics class, you will learn how to find a prediction
line. The first step in finding a prediction line is to create and sketch a scatter plot, as
Example 5 illustrates.
Example 5
Creating a Scatter Plot
© 2001 McGraw-Hill Companies
Carlotta kept the following chart next to her treadmill. Create a scatter plot for the ordered
pairs.
Minutes
Miles
53
48
55
30
40
62
35
50
65
6.4
5.7
6.8
4.5
5.2
7.0
4.9
6.0
7.2
Each combination of minutes and miles makes an ordered pair. The first ordered pair is
(53, 6.4). The scatter plot is the graph of all nine ordered pairs.
6
(30, 4.5)
4
2
25
50
GRAPHS OF LINEAR EQUATIONS AND FUNCTIONS
CHECK YOURSELF 5
Whitney keeps track of use of the copy machine in the library. She created the
following chart:
Month
School Days
Duplication Count
September
October
December
January
February
March
April
May
June
9
21
8
15
19
17
21
22
10
1230
3268
1124
2253
2872
2597
3410
3502
1470
Create a scatter plot for the ordered pairs.
CHECK YOURSELF ANSWERS
1.
Sy
3000
2000
(20,000, 2800)
(10,000, 2000)
2. V 150t 1800
3. (a) Tuition is dependent, number of credits
taken is independent; (b) The temperature is
dependent, the time since the coffee was
poured is independent.
1000
10,000
30,000
x
20,000
Sales
4. (a) The number of videos is independent. The total amount is dependent;
(b) y 10,000 1.25x;
(c)
5.
C
80,000
3000
(50,000, 72,500)
70,000
2000
60,000
1000
50,000
40,000
10
20
30,000
20,000
10,000
(3000, 13,750)
x
10,000 20,000 30,000 40,000 50,000
(d) f (3000) 13,750, f(50,000) 72,500
© 2001 McGraw-Hill Companies
CHAPTER 4
Salary
278
Name
4.6 Exercises
Section
Date
1. Consumer Affairs. A car rental agency charges $20 per day and 16¢ per mile for the
use of a compact automobile. The cost of the rental C and the number of miles driven
per day s are related by the equation
ANSWERS
C 0.16s 20
1.
Graph the relationship between C and s. Be sure to select appropriate scaling for the
C and s axes.
2.
Cost
C
s
Miles
2. Checking Account Charges. A bank has the following structure for charges on
checking accounts. The monthly charge consists of a fixed amount of $8 and an
additional charge of 12¢ per check. The monthly cost of an account C and the number
of checks written per month n are related by the equation
C 0.12n 8
Graph the relationship between C and n.
Cost
© 2001 McGraw-Hill Companies
C
n
Checks
279
ANSWERS
3. Tuition Charges. A college has tuition charges based on the following pattern.
3.
Tuition is $35 per credit-hour plus a fixed student fee of $75.
(a) Write a linear equation that shows the relationship between the total tuition
charge T and the number of credit-hours taken h.
(b) Graph the relationship between T and h.
4.
5.
T
h
4. Weekly Salary. A salesperson’s weekly salary is based on a fixed amount of $200
plus 10% of the total amount of weekly sales.
(a) Write an equation that shows the relationship between the weekly salary S and
the amount of weekly sales x (in dollars).
(b) Graph the relationship between S and x.
Salary
S
x
Weekly sales
5. Science. A temperature of 10°C corresponds to a temperature of 50°F. Also 40°C
corresponds to 104°F. Find the linear equation relating F and C.
280
°C
40
30
20
10
0
–10
–20
°F
110
100
90
80
70
60
50
40
30
20
10
0
–10
–20
°C
40
30
20
10
0
–10
–20
© 2001 McGraw-Hill Companies
°F
110
100
90
80
70
60
50
40
30
20
10
0
–10
–20
ANSWERS
6. A realtor receives $500 a month plus 4% commission on sales. The equation that
describes the total monthly income, I (in dollars), of the realtor is I .04s 500, in
which s is the amount of sales. (a) Graph this equation for 0 s 160, where s is
the sales in thousands. (b) Plot the point whose coordinates are (90, 4100) on the
graph. Write a sentence to describe the meaning of this ordered pair.
6.
7.
8.
7. An electrician charges $55 plus $1 per minute to wire an addition to a house. The
equation that describes the total cost, C, of the job is C t 55 in which t is the
number of minutes the electrician works. (a) Graph the equation for 0 t 80.
(b) Plot the point whose coordinates are (30, 85) on the graph. Write a sentence to
describe the meaning of this ordered pair.
8. A business purchases a new duplicating machine for $5,000. The depreciated value,
© 2001 McGraw-Hill Companies
v, after t years is given by v 5000 250t. Sketch a graph of this equation.
281
ANSWERS
9. Trac Hunyh’s weekly cost of operating a taxi is $100 plus 15 cents a mile. The
9.
equation that describes Trac’s cost is C 100 0.15m, in which m is the number of
miles driven in a week. (a) Graph the equation for 0 m 200. (b) How many
miles would Trac have to drive for the weekly cost to be $127?
10.
11.
10. Business. In planning for a new item, a manufacturer assumes that the number of
12.
items produced, x, and the cost in dollars, C, of producing these items are related by a
linear equation. Projections are that 100 items will cost $10,000 to produce and that
300 items will cost $22,000 to produce. Find the equation that relates C and x.
13.
11. Business. Mike bills a customer at the rate of $35 per hour plus a fixed service call
14.
charge of $50.
15.
16.
17.
(a) Write an equation that will allow you to compute the total bill for any number of
hours, x, that it takes to complete a job.
(b) What will the total cost of a job be if it takes 3.5 hours to complete?
(c) How many hours would a job have to take if the total bill were $160.25?
18.
12. Business. Two years after an expansion, a company had sales of $42,000. Four years
later the sales were $102,000. Assuming that the sales in dollars, S, and the time, t, in
years are related by a linear equation, find the equation relating S and t.
In exercises 13 to 18, identify which variable is dependent and which is independent.
13. The amount of a phone bill and the length of the call.
14. The cost of filling a car’s gas tank and the size of the tank.
16. The amount of penalty on an unpaid tax bill and the length of the time unpaid.
17. The length of time needed to graduate from college and the number of credits taken
per semester.
18. The amount of snowfall in Boston and the length of the winter.
282
© 2001 McGraw-Hill Companies
15. The height of a ball thrown in the air and the time in the air.
ANSWERS
In exercises 19 to 24, create a scatter plot from the given information.
80
70
60
19. In a local industrial plant, the number of work-hours in safety training and the
number of work-hours lost as a result of accidents have been recorded for
10 divisions.
Division
No. of Work-Hours
in Safety Training
No. of Work-Hours
Lost from Accidents
1
2
3
4
5
6
7
8
9
10
10
15
20
25
30
40
45
50
60
65
80
75
72
70
60
53
50
48
42
35
50
40
10 20 30 40 50 60 70
19.
100
90
80
70
80
90
100
20.
500
400
300
200
100
4
8 12 16 20 24 28
21.
20. In a statistics class, the mid-term and final exam scores were collected for
10 students. Each exam was worth a total of 100 points.
Mid-Term Exam Scores
Final Exam Scores
71
79
84
76
62
93
88
91
68
77
80
85
88
81
75
90
87
96
82
83
21. A rental car agency has collected data relating the number of miles traveled and the
© 2001 McGraw-Hill Companies
total cost in dollars.
Miles Traveled (in thousands)
Cost (in $)
2
6
10
14
8
5
12
16
3
18
21
60
100
200
275
175
90
290
400
75
450
475
283
ANSWERS
22. A math placement test was given to all entering freshmen at Bucks County
100
Community College. The placement test scores and the score on the first test were
recorded for students in a college algebra class.
80
60
6 12 18 24 30 36 42
22.
60
50
40
0.5 1.0 1.5 2.0 2.5 3.0 3.5
23.
4
Placement Test Scores (max. of 40)
First Test Score
25
18
30
14
10
32
12
16
22
27
38
78
75
88
65
62
85
68
73
78
82
93
3
2
80
24.
100
120
140
23. Students claim they can tell the cost of a textbook by the thickness of the book. They
picked nine books of roughly the same height and weight. The following data were
collected.
Thickness (in cm)
Cost (in $)
1.0
0.8
3.0
2.4
1.6
1.9
0.5
1.2
3.2
44
43
53
50
46
48
42
45
54
24. The following table shows the IQ of 12 students along with their cumulative grade
284
IQ
GPA
117
93
102
110
88
75
107
111
120
95
115
99
3.2
2.6
2.9
3.1
2.4
1.9
3.1
3.2
3.5
2.7
3.4
2.9
© 2001 McGraw-Hill Companies
point average (GPA) after 4 years of college.
ANSWERS
25. Exercise and Age. Aerobic exercise requires that your heartbeat be at a certain rate
for 12 minutes or more for full physical benefit. To determine the proper heart rate
for a healthy person, start with the number 220 and subtract the person’s age. Then
multiply by 0.70. The result is the target aerobic heart rate, the rate to maintain during
exercise.
25.
1. Write a formula for the relation between a person’s age (A) and the person’s target
2.
3.
4.
5.
aerobic heart rate (R).
Using at least 10 different ages, construct a table of target heart rates by age.
Draw a graph of this table of values.
What are reasonable limits for the person’s age that you would use with your
formula? Would it make sense to use A 2? Or A 150? In other words, what is
a reasonable domain for A?
What are the benefits of aerobic exercise over other types of exercise?
6. List some different types of exercise that are nonaerobic. Describe the differences
© 2001 McGraw-Hill Companies
between the two different types of exercise.
285
Answers
1. C 0.16s 20
3. (a) T 35h 75 and (b) see graph
C
Cost
$80
T
$60
$600
$40
$400
$20
$200
s
h
100 200 300
Miles
9
C 32
5
10
15
20
(b) The electrician
7. (a)
Cost ($)
5. F 5
160
140
120
100
80
60
40
20
charges $85 for
30 minutes of work.
0
10 20 30 40 50 60 70 80
Time (in min.)
9. (a)
11. (a) C 35x 50; (b) $172.50;
(c) 3.15 h
200
Cost ($)
13. Independent: length of call; dependent:
amount of bill
100
15. Independent: time in air; dependent:
height of ball
17. Independent: number of credits;
0
dependent: time to graduate
100
200
Miles
(b) 180 mi
21.
80
500
70
400
60
300
50
200
40
100
10 20 30 40 50 60 70
23.
25.
60
50
40
0.5 1.0 1.5 2.0 2.5 3.0 3.5
286
4
8 12 16 20 24 28
© 2001 McGraw-Hill Companies
19.