3 Equations: Describing Relationships in Data
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26
CHAPTER 1
■
Data, Functions, and Models
Dennis Sabo/Shutterstock.com 2009
For example, an ocean diver observes that the deeper she dives, the higher the
water pressure—she can feel the water pressing on her ears. How deep can she dive
before the pressure becomes dangerously high? To answer this question, she must be
able to predict what the pressure is at different depths without having to endanger
her life by diving to these depths. So she begins by diving to safe depths and measuring the water pressure. The data she obtains help her find a model (or equation)
that she can use to predict the pressure at depths to which she cannot possibly dive.
This situation is summarized as follows.
■
■
■
The data give the water pressure at different depths.
The model is an equation that represents the data.
Our goal is to use the model to predict the pressure at depths that are not in
the data.
The pressure-depth data and model are given below. Note that the single equation
P = 14.7 + 0.45d contains all the data and more. For instance, we can use this equation to predict the pressure at a depth of 200 ft, a value that is not available from the data.
Data
Depth
(ft)
Pressure
(lb/in2)
0
10
20
30
200
14.7
19.2
23.7
28.2
?
Model
Making a model
P = 14.7 + 0.45d
Using the model
In this section we learn how to make such models for data. The depth-pressure model
is obtained in Example 3.
2
■ Making a Linear Model from Data
A model is a mathematical representation (such as an equation) of a real-world situation. Modeling is the process of finding mathematical models. Once a model is
found, it can be used to answer questions about the thing being modeled.
Many real-world data start with an initial value for the output variable, and then
a fixed amount is added to the output variable for each unit increase in the input variable. For example, the production cost for manufacturing a certain number of cars consists of an initial fixed cost for setting up the equipment plus an additional unit cost for
manufacturing each car. In these cases we use a linear model to describe the data.
Linear Models
A linear model is an equation of the form y = A + Bx. In this model, A is the
initial value of y, that is, the value of y when x is zero, and B is the constant
amount by which y changes (increases or decreases) for each unit increase in x.
Add B for each
unit change in x
y =
r
Initial value
of y
r
Linear equations are studied in more
detail in Section 2.2.
A
+
Bx
SECTION 1.3
■
Equations: Describing Relationships in Data
27
In the next three examples we find some linear models from data.
e x a m p l e 1 Data, Equation, Graph
table 1
x
(chairs)
C
(dollars)
0
1
2
3
4
80
92
104
116
128
A furniture maker collects the data in Table 1, giving his cost C of producing x chairs.
(a) Find a linear model for the cost C of making x chairs.
(b) Draw a graph of the equation you found in part (a).
(c) What does the shape of the graph tell us about his cost of making chairs?
Solution
(a) The initial cost (the cost of producing zero chairs) is $80. We can see from the
table that each chair produced costs an additional $12. That is, the unit cost is
$12. So an equation that models the relationship between C and x is
r
r
Initial cost
Add $12 for each
(or fixed cost) chair produced
C = 80 + 12x
✓ C H E C K To check that this equation correctly models the data, let’s try
some values for x. If four chairs are produced, then we can use the equation to
calculate the cost.
C = 80 + 12x
Model
C = 80 + 12142
Replace x by 4
= 128
Graphing equations is reviewed
in Algebra Toolkit D.2, page T71.
Calculate
This matches the cost given in the table for making four chairs. You can check
that the other values in the table also satisfy this equation.
(b) The ordered pairs in the table are solutions of the equation, so we plot them in
Figure 1(a). We can see that the points lie on a straight line, so we complete
the graph of the equation by drawing the line containing the plotted points as
in Figure 1(b).
C
150
140
130
120
110
100
90
80
0
C
150
140
130
120
110
100
90
80
1
2
3
4
5
(a) Graph from table
6 x
0
1
2
3
4
5
6 x
(b) Graph of equation
figure 1
(c) From the graph, it appears that cost increases steadily as the number of chairs
produced increases.
■
NOW TRY EXERCISES 7 AND 19
■
28
CHAPTER 1
■
Data, Functions, and Models
How can we tell whether a given set of data has a linear model? Let’s consider
data sets whose inputs are evenly spaced. For instance, the inputs 0, 1, 2, 3, . . . in
Example 1 are evenly spaced—successive inputs are one unit apart. For data with
evenly spaced inputs, there is a linear model for the data if the outputs increase (or
decrease) by a constant amount between successive inputs. We can test whether data
satisfy this condition by making a table of first differences. Each entry in the first
difference column of the table is the difference between an output and the immediately preceding output.
First Differences
For data with evenly spaced inputs:
■
■
The first differences are the differences in successive outputs.
If the first differences are constant, then there is a linear model for
the data.
The next example illustrates how we make and use a first difference table.
e x a m p l e 2 A Model for Temperature and Elevation
table 2
Elevation
(km)
Temperature
(°C)
0
1
2
3
4
5
20
10
0
- 10
- 20
- 30
10 - 20 = - 10
0 - 10 = - 10
- 10 - 0 = - 10
- 20 - 1- 102 = - 10
- 30 - 1- 202 = - 10
A mountain climber knows that the higher the elevation, the colder is the temperature. Table 2 gives data on the temperature at different elevations above ground level
on a certain day, gathered by using weather balloons.
(a) Show that a linear model is appropriate for these data.
(b) Find a linear model for the relationship between temperature and elevation.
(c) Draw a graph of the equation you found in part (b).
Solution
(a) We first observe that the inputs for these data are evenly spaced. To see
whether a linear model is appropriate, let’s make a table of first differences.
The entries in the first difference column are obtained by subtracting from
each output the preceding output.
Elevation
(km)
Temperature
(°C)
First
difference
0
1
2
3
4
5
20
10
0
- 10
- 20
- 30
—
- 10
- 10
- 10
- 10
- 10
We see that the first differences are constant (each is - 10), so there is a linear
model for these data.
(b) The linear model we seek is an equation of the form
T = A + Bh
where T represents temperature and h elevation.
SECTION 1.3
■
Equations: Describing Relationships in Data
29
When h is zero (ground level), the temperature is 20°C, so the initial value
A is 20. The first differences are the constant - 10, so the number B in the model
is - 10. We can now express the model as
r
Subtract 10 for each
kilometer of elevation
r
Temperature
at elevation 0
T = 20 -
10h
Notice how this equation fits the data. From the data we see that for each 1-km
increase in elevation, the temperature decreases by 10°C. So at an elevation of
h km we must subtract 10h degrees from the ground temperature.
✓ C H E C K To check that this equation correctly models the data, we try
some values for h from the table. For example, if the elevation is 5 km, we replace h by 5 in the model:
T
20
10
0
_10
1
2
3
4
5
6 h
_20
_30
_40
figure 2
Graph of T = 20 - 10h
T = 20 - 10h
Model
T = 20 - 10152
Replace h by 5
T = - 30
Calculate
This matches the temperature given in the table for an elevation of 5 km. You
can check that the other values in the table also satisfy this equation.
(c) We plot the points in the table and then complete the graph by drawing the
line that contains the plotted points. (See Figure 2.)
■
NOW TRY EXERCISES 11 AND 21
■
e x a m p l e 3 A Model for Depth-Pressure Data
table 3
Depth
(ft)
Pressure
(lb/in2)
0
10
20
30
40
50
14.7
19.2
23.7
28.2
32.7
37.2
A scuba diver obtains the depth-pressure data shown in Table 3.
(a) Show that a linear model is appropriate for these data.
(b) Find a linear model that describes the relationship between depth and pressure.
Solution
(a) The inputs for these data are evenly spaced. We make a table of first differences.
Depth
(ft)
Pressure
(lb/in2)
First
differences
0
10
20
30
40
50
14.7
19.2
23.7
28.2
32.7
37.2
—
4.5
4.5
4.5
4.5
4.5
We see that the first differences are constant (each is 4.5), so there is a linear
model for these data.
(b) The linear model we seek is an equation of the form
P = A + Bd
where P represents pressure and d represents depth.
30
CHAPTER 1
■
Data, Functions, and Models
When the depth d is 0, the pressure is 14.7 lb/in2, so the initial value A is
14.7. From the first difference column in the table we see that pressure increases
by 4.5 lb/in2 for each 10-ft increase in depth. So for each 1-ft increase in depth
the pressure increases by
4.5
= 0.45 lb/in 2
10
Pressure at
depth 0
Add 0.45 lb/in2 for
each foot of depth
r
r
So the number B in the model is 0.45. We can now express the model as
P = 14.7 + 0.45d
If d is 0, then P = 14.7 + 0.4510 2 = 14.7.
If d is 10, then P = 14.7 + 0.45110 2 = 19.2.
If d is 30, then P = 14.7 + 0.45130 2 = 28.2.
✓ C H E C K Let’s check whether this model fits the data. For instance, when
d is 20, we get P = 14.7 + 0.451202 = 23.7, which agrees with the table. In the
margin we check the model against other entries in the table.
■
2
■
■ Getting Information from a Linear Model
IN CONTEXT ➤
© Ralph White/CORBIS
NOW TRY EXERCISE 23
The point of making a model is to use it to predict conditions that are not directly observed in our data.
In the next example we use the depth-pressure model of Example 3 to find the
pressure at depths to which no human can dive unaided. This illustrates the power of
the modeling process: It allows us to explore properties of the real world that are beyond our physical experience. In the first-ever attempt to explore the ocean depths,
Otis Barton and William Beebe built a steel sphere (see the photo) with a diameter
of 4 ft 9 in., which they called the bathysphere (bathys is the Greek word for deep).
They needed to build their craft to be strong enough to withstand the crushing water
pressure at the great depths to which they planned to descend. They used the depthpressure model to estimate the pressure at those depths and then built the bathysphere accordingly. On August 15, 1934, they successfully descended to a depth of
3028 ft below the surface of the Atlantic. From the bathysphere’s portholes they observed exciting new marine species that had never before been seen by humans.
e x a m p l e 4 Using the Depth-Pressure Model
The bathysphere described above is lowered to the bottom of a deep ocean trench.
Use the depth-pressure model P = 14.7 + 0.45d to predict the pressure at a depth
of 3000 ft.
Solution
Since the depth is 3000 ft, we replace d by 3000 in the model and solve for P:
P = 14.7 + 0.45d
Model
P = 14.7 + 0.45130002
Replace d by 3000
P = 1364.7
Calculate
2
So the pressure is 1364.7 lb/in .
■
NOW TRY EXERCISE 25
■
SECTION 1.3
■
Equations: Describing Relationships in Data
31
Test your skill in graphing equations in two variables. You can review this topic
in Algebra Toolkit D.2 on page T71.
1. An equation is given. Determine whether the given point (x, y) is a solution
of the equation.
(a) y = 5x - 3; (3, 12)
(c) y = 25 - 3x 3; 1- 2, 12
(b) 5x - 2y = 4; (2, 3)
2. An equation is given. Determine whether the given point (x, y) is on the
graph of the equation.
(a) y = 3x - 7; 12, - 1 2
(c) y = 2x 2 + 1; 1- 1, 32
(b) 3y - x = - 6; (2, 0)
3. An equation is given. Complete the table and graph the equation.
(a) y = 2 + x
x
0
y
2
(b) y = 1 - 3x
1
2
3
4
5
(c) y = - 7 + 4x
x
0
y
-7
1
x
-2
y
7
-1
0
1
2
3
(d) y = - 3 - 2x
2
3
4
5
x
-3
y
3
-2
-1
0
1
2
4. Find the x- and y-intercepts of the given equation.
(a) y = x - 2
(b) 2y = 3x + 2
(c) 2x - 6y = 12
(d) 4x - 5y = 8
5. Graph the given equation, and find the x- and y-intercepts.
(a) y = 2x
(b) y = 6 + 2x
(c) y = - 6 + 3x
(d) 3y = 6 - 2x
1.3 Exercises
CONCEPTS
Fundamentals
1. (a) What is a model? Give some examples of models you use every day.
(b) If you work for $15 an hour, describe the relation between your pay and the number
of hours you work, using (i) a table, (ii) a graph, (iii) an equation.
x
y
First difference
0
45
—
1
39
2
33
3
27
4
21
5
15
2. For data with evenly spaced inputs, if the first differences are _______, then a linear
model is appropriate for the data. In the data shown in the margin, x represents the input
and y represents the output. What are the first differences? Is a linear model appropriate?
3. The equation L = 4S is a linear model for the total number of legs L that S sheep have.
Using the model, we find that 12 sheep have L = 4 *
legs.
=
ٗ ٗ
4. Suppose digital cable service costs $49 a month with an initial installation fee of $110.
We make a linear model for the total cost C of digital cable service for x months by
writing the equation ______________.
Think About It
5. What is the purpose of making a model? Support your answer by examples.
6. Explain how data, equations, and graphs work together to describe a real-world situation.
Give an example of a real-world situation that can be described in these three ways.
32
CHAPTER 1
SKILLS
■
Data, Functions, and Models
7–10 ■
(a)
(b)
(c)
7.
A set of data is given.
Find a linear model for the data.
Use the model to complete the table.
Draw a graph of the model.
x
y
0
8.
9.
10.
a
b
55
0
- 10
1
52
1
-4
86
2
49
2
2
74
3
46
3
8
u
v
A
B
5
0
110
0
1
12
1
98
2
19
2
3
26
3
4
4
4
4
5
5
5
5
6
6
6
6
11–14 ■ A set of data is given.
(a) Find the first differences.
(b) Is a linear model appropriate? If so, find a linear model for the data.
(c) If there is a linear model, use it to complete the table.
11.
First
difference
12.
x
y
205
0
60
1
218
1
54
2
231
2
48
3
244
3
42
x
y
0
4
4
5
5
6
6
13.
First
difference
14.
x
y
23
0
17
1
19
1
38
2
16
2
59
3
11
3
80
x
y
0
4
4
5
5
6
6
First
difference
First
difference
SECTION 1.3
■
15–18
15.
Equations: Describing Relationships in Data
■
Find a linear model for the data graphed in the scatter plot.
y
30
16.
20
10
0
17.
1
Tony Ayling
CONTEXTS
Male and female angler fish
2
18.
0.1
0.2
0.3
0.4 x
y
100
80
60
40
20
0
4 x
3
y
1000
800
600
400
200
0
33
1
2
3
4 x
y
2.50
2.00
1.50
1.00
0.50
0
5 10 15 20 25 30 35 40 x
19. Truck Rental A home improvement store provides short-term truck rentals for their
customers to take large items home. The store charges a base rate of $19 plus a time
charge for every half hour that the truck is used. The table gives rental rate data for
different rental periods.
(a) Find a linear model for the relation between the rental cost and rental period (in hours).
(b) Draw a graph of the equation you found.
(c) Use the model to predict the rental cost for 5 hours.
Period (h)
Rental cost ($)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
19.00
24.00
29.00
34.00
39.00
44.00
49.00
20. Aquatic Life in the Midnight Zone In Example 6 we used the model
P = 14.7 + 0.45d to find the pressure (lb/in2) at various ocean depths. The deepest
ocean trenches plunge to an astounding 7 miles below sea level, far too deep for
sunlight to penetrate. Yet our planet is so teeming with life that even at these depths
there are living creatures. Anglerfish can live at depths up to 11,000 ft and are
characterized by luminescent appendages, which they use to lure their prey. The
scientific name of the bizarre-looking anglerfish shown here, linophryne arborifera,
means, roughly, “toad that fishes with a tree-like net.”
(a) Use the model to predict the pressure at the 11,000-ft depth where anglerfish can live.
(b) One atmosphere (atm) is defined as a pressure of 14.7 lb/in2, which is the normal
air pressure we experience at sea level. Convert your answer in part (a) to
atmospheres. How many times greater is the pressure under which anglerfish live
than the pressure under which we live?
21. Boiling Point Most high-altitude hikers know that cooking takes longer at higher
elevations. This is because the atmospheric pressure decreases as the elevation increases,
causing water to boil at a lower temperature, and food cooks more slowly at that lower
34
CHAPTER 1
■
Data, Functions, and Models
Peter Zaharov/Shutterstock.com 2009
temperature. The table below gives data for the boiling point of water at different elevations.
(a) Use first differences to show that a linear model is appropriate for the data.
(b) Find a linear model for the relation between boiling point and elevation.
(c) Use the model to predict the boiling point of water at the peak of Mount
Kilimanjaro, 19,340 ft above sea level.
Mount Kilimanjaro
Elevation
(؋ 1000 ft)
Boiling point
(°F)
First
difference
0
212.0
—
1
210.2
2
208.4
3
206.6
4
204.8
5
203.0
22. Temperatures on Mount Kilimanjaro Mount Kilimanjaro is the highest mountain
in Africa. Its snow-covered peak rises 4800 m above the surrounding plain. It is located
in northern Tanzania near the equator, and conditions on the mountain vary from
equatorial, to tropical, to arctic, because of the steadily decreasing temperature as the
altitude increases. The table below gives data for the temperature on a typical day on
Kilimanjaro at various elevations above the base of the mountain.
(a) Use first differences to show that a linear model is appropriate for the data.
(b) Find a linear model for the relation between the temperature on Kilimanjaro and
the elevation above the base of the mountain.
(c) Use the model to predict what the temperature will be on a typical day at the peak
of Kilimanjaro.
Elevation
above base (m)
Miles
driven
Pounds of
chocolate used
0
20
40
60
80
100
0
17
34
51
68
85
Temperature
(°C)
First
difference
0
30
—
400
28
800
26
1200
24
1600
22
2000
20
23. Chocolate-Powered Car Two British entrepreneurs, Andy Pag and John Grimshaw,
drove 4500 miles from England to Timbuktu, Mali, in a truck powered by chocolate. They
used an ethanol that is made from old, unusable chocolate, and it took about 17 pounds of
chocolate to make 1 gallon of ethanol. The table in the margin gives data for the
relationship between the amount of chocolate used and the number of miles driven.
(a) Use first differences to show that a linear model is appropriate for the data.
(b) Find a linear model for the relation between the amount of chocolate used and the
number of miles driven.
(c) Use the model found in part (b) to predict how many pounds of chocolate it took to
drive from England to Timbuktu.
SECTION 1.4
Profit ($)
0
10
20
30
40
50
- 80.00
- 40.00
0.00
40.00
80.00
120.00
Salary (ϫ $1000)
Number
Books (thousands)
2 3
Year
4
5 x
y
40
30
20
10
0
2
1
1
Functions: Describing Change
35
24. Profit An Internet company sells cell phone accessories. The table in the margin gives
the profit they make on selling battery chargers. (Note that negative numbers in the
table represent a loss.) Because of storage costs, the company needs to sell at least 20
chargers before they begin to make a profit.
(a) Find a linear model for the relation between profit and the number of battery
chargers sold.
(b) Draw a graph of the equation you found.
(c) Use the model to predict the profit from selling 150 battery chargers.
y
200
190
180
170
160
150
0
■
2 3 4 5 6 x
Years since 2001
25. Salary A woman is hired as CEO of a small company and is offered a salary of
$150,000 for the first year. In addition, she is promised regular salary increases. The
graph in the margin shows her potential salary (in thousands of dollars) for the first few
years that she works for the company.
(a) Find a linear model for the relation between her salary and the number of years she
works for the company.
(b) Use the model to predict what her salary will be after she has worked 10 years for
the company.
26. Library Book Collection A city library remodeled and expanded in the year 2001
and increased its maximum capacity to about 100,000 books. In 2001 the library held
about 20,000 books, and each subsequent year the library adds a fixed number of books
to its collection. The graph in the margin plots the number of books (in thousands) the
library held each year from 2001 to 2007.
(a) Find a linear model for the relation between the number of books in the library and
the number of years since 2001.
(b) Use the model to predict the number of books in the library after 25 years (in 2026).
1.4 Functions: Describing Change
■
Definition of Function
■
Which Two-Variable Data Represent Functions?
■
Which Equations Represent Functions?
■
Which Graphs Represent Functions?
■
Four Ways to Represent a Function
IN THIS SECTION… we begin our study of functions. There are four basic ways to
represent functions: words, data, equations, and graphs. The concept of function is a
versatile tool for modeling the real world. In succeeding chapters we study more properties
of functions; each new property provides a new modeling tool.
GET READY… by reviewing how to solve equations in Algebra Toolkit C.1. Test your
skill by doing the Algebra Checkpoint at the end of this section.
In Sections 1.1–1.3 we saw how analyzing two-variable data can reveal relationships
between the variables. Such relationships can be seen from the data themselves, from
a graph, or from an equation. But this is not the whole story; in many real-world situations we are interested in how a change in one variable results in a change in the
other variable. We’ll study these types of relations using the concept of function.
Equipped with this concept, we will be able make a great leap in our understanding
of our ever-changing world.
36
2
CHAPTER 1
■
Data, Functions, and Models
■ Definition of Function
We use the term function to describe the dependence of one changing quantity on another. For example, we say that the height of a child is a function of the child’s age,
the weather is a function of the date, the cost of mailing a package is a function of its
weight, and so on. Situations like these, which involve change, have the property that
each input (or each value for the first variable) results in exactly one output (exactly
one value for the other variable). For instance, mailing a package does not result in
two different costs. This leads us to the following definition of function.
Definition of Function
A function is a relation in which each input gives exactly one output.
We can easily tell whether a relation is a function by making a diagram, as we did in
Section 1.2. The diagram in Figure 1(a) represents a function because for each input
there is exactly one output. But the relation described by the diagram in Figure 1(b)
is not a function—the input 2 corresponds to two different outputs, 20 and 30.
10
20
30
40
1
2
3
4
Inputs
Outputs
10
20
30
40
50
1
2
3
4
Inputs
(a) A function
Outputs
(b) Not a function
f i g u r e 1 When is a relation a function?
e x a m p l e 1 Which Relations Are Functions?
A relation is given by a table. The input is in the first column, and the output in the
second column. Is the relation a function?
(a) Table 1 gives the number of women in the U.S. Senate between 1997 and 2007.
(b) Table 2 gives the ages of women in a certain neighborhood and the number of
children each woman has.
table 1
table 2
Women in U.S. Senate
Number of children
Year
Number
Age
Number
1997
1999
2001
2003
2005
2007
9
9
14
14
14
16
31
32
24
35
31
22
3
0
1
1
2
0