6: Pie, Bar, and Line Graphs
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478
Chapter 11 • Additional Topics
are usually expressed in percents. For example, a student activities group is planning a ski trip
and knows that the expenses, per person, are as follows:
Expenses for a Ski Trip, per person
Category
Food
Transportation
Equipment rental
Lift tickets
Lodging
Total
Expense
Percent
$155
360
200
85
200
$1000
15.5%
36%
20%
8.5%
20%
The pie chart in Figure 11.13 is a useful graph to show the ski club how the expenses of
the trip are distributed.
Ski trip expenses
Transportation
36%
Lift tickets
8.5%
Voter preference on a proposition
No
Food
15.5%
Lodging
20%
Equipment rental
20%
Figure 11.13
Yes
Figure 11.14
A glance at a pie chart can often tell the story and eliminate the need to compare numbers. Consider the pie chart in Figure 11.14. Without any numerical values presented, the
graph gives us the information that a majority of voters are in favor of the proposition,
because more than half the circle is shaded for yes.
Classroom Example
Use the pie chart shown in Figure
11.15 to answer these questions.
(a) What percent of the bagels sold
were garlic, onion, or blueberry?
(b) What percent of the bagels sold
were not poppy seed or sesame
seed?
(c) Were less than one-fourth of the
bagels sold either blueberry or
cinnamon?
EXAMPLE 1
Use the pie chart in Figure 11.15 to answer the questions.
Bagels sold
Garlic
7%
Blueberry
10%
Whole
wheat
12%
Onion
5%
Plain
25%
Cinnamon
14%
Poppy seed
13%
Sesame seed
14%
Figure 11.15
Copyright 2009 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
11.6 • Pie, Bar, and Line Graphs
479
(a) What percent of the bagels sold were plain, sesame seed, or poppy seed?
(b) What percent of the bagels sold were not onion or garlic?
(c) Were more than half the bagels sold either plain or whole wheat?
Solution
(a) Of the bagels sold, 25% were plain, 14% were sesame seed, and 13% were poppy
seed. Together these add up to 52% of the bagels sold.
(b) Five percent were onion bagels, and 7% were garlic bagels. Therefore, onion bagels
together with garlic bagels accounted for 12% of the sales. Since the pie chart represents the whole, or 100%, of the bagels sold, the percent of bagels sold that were
not onion or garlic is 100% – 12%, or 88%.
(c) By inspection the sectors for plain and whole wheat do not make up more than half
of the circle. Mathematically, plain bagels were 25% of the sales, and whole wheat
bagels were 12% of the sales. Together, plain and whole wheat bagels made up 37%
of the sales, which is less than half.
Bar Graph
Another type of graph is the bar graph. Bars are drawn either vertically or horizontally to
show amounts. Bar graphs are very useful for comparisons.
Consider this information on the number of students in certain college majors at a
university:
College majors
Number of students
Business
Computer science
Natural science
English
Fine arts
Education
2400
850
700
1800
400
1000
The information is displayed in the bar graph in Figure 11.16. The graph has a title, a vertical
axis that gives the numbers of students, and a horizontal axis that shows the majors. Vertical
bars are drawn for each major; the height of the bar is determined by the number of students
(per the scale on the vertical axis). The bars also could be displayed horizontally.
College majors
Number of students
3000
2500
2000
1500
1000
500
Figure 11.16
N
Ed
n
uc
at
io
ts
ar
ish
gl
ne
Fi
nc
e
En
sc
ie
at
ur
al
ci
e
rs
Co
m
pu
te
Bu
sin
es
s
nc
e
0
Chapter 11 • Additional Topics
Bar graphs can display multiple bars and be used to compare information about two or
more groups. Suppose you are trying to decide whether there is a difference in the number
of music CDs purchased by men and those purchased by women in the various age groups.
The following table contains the information, and the bar graph in Figure 11.17 displays this
information.
Ages
Men
Women
18–21
22–35
36–50
35
28
8
31
12
14
Number of CDs
Music CDs purchased in a year
40
30
20
10
0
Ages
18–21
Ages
36–50
Ages
22–35
Men
Women
Figure 11.17
Line Graph
Line graphs are used to show the relationship between two variables. Each graph has two
perpendicular axes with a variable assigned to each axis. Line graphs are useful for indicating trends. Consider the following information regarding a corporation’s profit from 2005 to
2010. The profit is shown in millions of dollars.
Year
Proﬁt
2005
258
2006
110
2007
165
2008
205
2009
224
2010
185
A line graph for this information is shown in Figure 11.18. From the graph, you can see
the trends in profits. There was a large decrease in profits between 2005 and 2006. After that,
profits rose from 2006 to 2009. Then there was a decrease in profits from 2009 to 2010.
Profit
300
Millions of dollars
480
250
200
150
100
50
0
2005 2006 2007 2008 2009 2010
Year
Figure 11.18
Copyright 2009 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
11.6 • Pie, Bar, and Line Graphs
The graph in Figure 11.19 displays information about the interest rates charged by a bank for an
automobile loan over the last 7 months. Use the graph to answer these questions.
Interest rates for an automobile loan
9.0
Interest rate, %
8.5
8.5
8.0
8.0 8.2
7.8
7.5
7.5
7.0
7.0
7.1
6.5
er
N
ov
em
D ber
ec
em
be
r
ob
m
O
ct
be
r
us
t
te
Se
p
A
ug
y
e
6.0
Ju
l
(a) The largest increase in interest
rates was between which two
months?
(b) What was the change in interest
rates between October and
November?
(c) In which month was the second
highest interest rate?
EXAMPLE 2
Ju
n
Classroom Example
Use Figure 11.19 to answer these
questions.
481
Figure 11.19
(a) The largest decrease in interest rates was between which two months?
(b) What was the change in interest rates between July and August?
(c) If you wanted the lowest possible interest rate, in what month should you have gotten a loan?
Solution
(a) The largest decrease was between October and November.
(b) The change between July and August was 7.8 Ϫ 8.5, or Ϫ0.7. So the interest rate
dropped by 0.7 point.
(c) The interest rate was the lowest in December.
One of the challenging aspects of creating graphs is deciding which type of graph to
use. The information in the following table about a budget is displayed in both a pie chart
(Figure 11.20) and a bar graph (Figure 11.21). Each graph displays the information in a
different format; the choice of which format to use is up to you.
Budget Expenses
Item
Automobile
Rent
Groceries
Utilities
Phones
Entertainment
Clothes
Percent of income
26
33
12
10
5
6
8
482
Chapter 11 • Additional Topics
Budget expenses
Phones
5%
Entertainment
6%
Clothes
8%
Budget expenses
Percent
40
Rent
33%
Utilities
10%
30
20
10
Groceries
12%
R
Gr ent
oc
er
ie
Ut s
il
Ce ities
ll
En ph
ter on
tai e
nm
en
Cl t
ot
he
s
Au
to
0
Auto
26%
Figure 11.21
Figure 11.20
Concept Quiz 11.6
For Problems 1 – 7, answer true or false.
1.
2.
3.
4.
5.
6.
7.
Pie charts are also called circle graphs.
Circle graphs and line graphs are basically the same.
A line graph can effectively show a comparison between two variables.
Constructing a pie chart requires finding fractional parts of 360°.
Constructing a bar graph requires finding fractional parts of 180°.
The bars on a bar graph can be displayed either vertically or horizontally.
Sometimes the choice of which kind of graph to use to display certain information is
simply a matter of personal preference.
Problem Set 11.6
For Problems 1– 5, use the pie chart in Figure 11.22. (Objective 2)
Boat rentals
For Problems 6 –11, use the pie chart in Figure 11.23.
(Objective 2)
Science electives
Sailboats
6%
Physics
8%
Kayaks
8%
Pontoon
boats
38%
Jon boats
20%
Chemistry
13%
Geology
15%
Astronomy
16%
Ski boats
28%
Figure 11.22
1. What percent of the boat rentals were kayaks or sailboats?
2. Were more than half the rentals pontoon boats or ski boats?
3. If there were 2400 rentals, how many times were Jon
boats rented?
Biology
26%
Oceanography
22%
Figure 11.23
6. Which science elective is the most popular?
7. Which science elective is the least popular?
8. What percent of the students chose biology or geology?
9. What percent of the students chose chemistry or
physics?
4. Is the ratio of sailboat rentals to kayak rentals the same
as the ratio of ski boat rentals to pontoon boat rentals?
(Justify your answer.)
10. What percent of the students did not choose biology or
oceanography?
5. What percent of the rentals were not sailboats or ski
boats?
11. What percent of the students did not choose oceanography or astronomy?
11.6 • Pie, Bar, and Line Graphs
For Problems 12–15, use the bar graph in Figure 11.24.
(Objective 2)
Favorite Florida vacation activity
Daytona speedway
483
21. In June what was the difference in the interest rates
between the bank and credit union?
22. What was the change in interest rates for the bank
between May and June?
For Problems 23 – 28, use the graph in Figure 11.26.
Golf course
(Objective 2)
Tips for waiter
Beach
200
Space center
180
Water park
160
Theme park
140
4000
Number of people
Figure 11.24
120
100
80
12. How many more people preferred the theme park to the
beach?
60
13. How many more people preferred the space center to the
water park?
20
40
0
15. What is the difference in the number of people who
chose the beach over the golf course?
For Problems 16 – 22, use the bar graph in Figure 11.25.
(Objective 2)
Automobile loan interest rates
Figure 11.26
23. What is the total amount the waiter earned in tips for
Friday, Saturday, and Sunday?
24. What is the total amount the waiter earned in tips for
Monday through Thursday?
8.4
8.2
Interest rate, %
Su
nd
ay
14. What is the order of activities from the most popular to
the least popular?
Sa
tu
rd
ay
3000
Fr
id
ay
2000
M
on
da
y
Tu
es
da
y
W
ed
ne
sd
ay
Th
ur
sd
ay
1000
Dollars
0
25. To avoid loss of income, what would be the best day for
the waiter to take off work (according to the information
in the graph)?
8.0
7.8
26. What was the difference in tips between Saturday and
Wednesday?
7.6
7.4
7.2
27. How much did the waiter earn in tips for the week?
7.0
28. What was the average daily amount of tips?
6.8
Jan.
Feb.
Mar.
Apr.
May.
Jun.
For Problems 29 – 35, use the graph in Figure 11.27.
(Objective 2)
Bank
Credit Union
Annual total return of fund
Figure 11.25
17. Between which two months did the interest rate for banks
go up while the interest rate for credit unions went down?
18. Between which two months did the banks raise the interest rate by 0.2%?
Percent
20
16. The greatest difference in the interest rates between
banks and credit unions occurs in which month?
15
10
5
0
2005 2006 2007 2008 2009 2010
High Tech
Utility
19. Between which two months did the credit union keep
the interest rate constant?
Figure 11.27
20. For which month and for which type of institution was
the interest rate the lowest?
29. What is the difference in annual total returns between
the Utility Fund and the High Tech Fund for 2005?
Copyright 2009 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
484
Chapter 11 • Additional Topics
30. What is the difference in annual total returns between
the Utility Fund and the High Tech Fund for 2007?
31. What was the change in annual total returns for the
Utility Fund between 2006 and 2007?
32. What was the change in annual total returns for the High
Tech Fund between 2009 and 2010?
33. The greatest annual change in annual total returns for the
High Tech Fund occurred between which two years?
34. The greatest annual change in annual total returns
for the Utility Fund occurred between which two
years?
35. (a) What is the average annual total return for the High
Tech Fund for the 6 years shown?
(b) What is the average annual total return for the
Utility Fund for the 6 years shown?
(c) Which fund has the highest average annual total
return?
Thoughts Into Words
Passing rate on final exam
% passing
98.5
Boss’s graph of sales
80
Number of sales
36. Mrs. Guenther says that each of her classes performed
quite differently on the final exam according to the
graph in Figure 11.28. Do you agree or disagree with her
statement?
97.5
75
70
65
60
55
50
1
96.5
95.5
Class #1
Class #2
Class #3
2
3
Year
4
5
Figure 11.29
Figure 11.28
Lauren’s graph of sales
Number of sales
200
37. Lauren’s boss says her sales over the past 5 years have
been dropping significantly and steadily and shows her
the graph in Figure 11.29. Lauren shows her boss the
graph in Figure 11.30 and claims that there has been
only a slight decrease. If you were the boss, would you
accept the portrayal of Lauren’s sales as shown in her
graph?
160
140
120
80
40
0
1
2
3
Year
4
Figure 11.30
Further Investigations
The first step in constructing a pie chart using a compass and
protractor is to find the degrees for each sector. Remember,
we are dividing up the 360° of the entire circle. Consider the
information in the table below.
Pizza Sales for Saturday
Type
Pepperoni
Sausage
Supreme
Veggie
Cheese
Total
Number sold
60
45
40
15
20
180
5
11.7 • Relations and Functions
Number sold of that type
* 360°
Total number sold
60
Degrees for pepperoni ϭ
* 360° ϭ 120°
180
Degrees for any type ϭ
Degrees for sausage
38. Construct a pie chart for the following data:
Marching Rams Membership
Category
45
ϭ
* 360° ϭ 90°
180
Degrees for supreme ϭ
40
* 360° ϭ 80°
180
Degrees for veggie
ϭ
15
* 360° ϭ 30°
180
Degrees for cheese
20
ϭ
* 360° ϭ 40°
180
Number of students
Band
Colorguard
Majorettes
105
42
33
39. Construct a pie chart for these data:
Investment Portfolio for Mr. Jordan
Category
Dollars invested
Stocks
Mutual funds
Bonds
Annuities
Gold
Using a compass, draw a circle with a diameter of your choosing.
From the center of the circle draw a radius. With the protractor located at the center of the circle and the radius as a side of the angle, draw
the angle for the desired number of degrees in the first sector.
Continue until all the sectors are shown. Be sure to label the pie chart
and the sectors as in Figure 11.31.
$ 8,000
14,000
6,000
5,000
3,000
40. If you have access to a computer with a spreadsheet
application, try producing the pie charts in Problems 38
and 39 using the software.
Pizza sales for Saturday
Cheese
Veggie
485
Pepperoni
Supreme
Sausage
Figure 11.31
Answers to the Concept Quiz
1. True
2. False
3. True
4. True
11.7
5. False
6. True
7. True
Relations and Functions
OBJECTIVES
1
Understand the definitions of a function and a relation
2
Use function notation
3
Specify the domain and range
In the next two sections of this chapter, we will work with a concept that has an important
role throughout mathematics—namely, the concept of a function. Functions are used to unify
mathematics and also to serve as a meaningful way of applying mathematics to many realworld problems. They provide us with a means of studying quantities that vary with one
486
Chapter 11 • Additional Topics
another—that is, when a change in one quantity causes a corresponding change in another.
We will consider the general concept of a function in this section and then deal with the application of functions in the next section.
Mathematically, a function is a special kind of relation, so we begin our discussion with
a simple definition of a relation.
Definition 11.7
A relation is a set of ordered pairs.
Thus a set of ordered pairs such as {(1, 2), (3, 7), (8, 14)} is a relation. The domain of a relation is the set of all first components of the ordered pair. The range of a relation is the set of
all second components of the ordered pair. The relation {(1, 2), (3, 7), (8, 14)} has a domain
of {1, 3, 8} and a range of {2, 7, 14}.
Classroom Example
State the domain and range for each
relation:
(a) The month and corresponding
birthstone:{(January, garnet),
(February, amethyst), (March,
aquamarine), (April, diamond),
(May, emerald)}
(b) January high temperatures in
cities of the world: {Berlin, 30°F),
(Dublin, 46°F), (London, 43°F),
(Mexico City, 72°F)}
(c) An integer between Ϫ4 and 3, and
its absolute value {(Ϫ3, 3),
(Ϫ2, 2), (Ϫ1, 1) (0, 0) (1, 1) (2, 2)}
(d) A number and its cube:
{(Ϫ3, Ϫ27), (Ϫ2, Ϫ8), (Ϫ1, Ϫ1),
(2, 8), (4, 64)}
EXAMPLE 1
Here are four examples of relations. State the domain and range for each relation:
(a) The presidential election year and who won the election: {(1928, Hoover), (1960,
Kennedy), (1964, Johnson), (1976, Carter), (1992, Clinton), (1996, Clinton)}
(b) A radioactive element and its half-life in hours: {(Iodine-133, 20.9), (Barium-135,
28.7), (Technetium-99m, 6)}
(c) A natural number less than 5 and its opposite: {(1, Ϫ1), (2, Ϫ2), (3, Ϫ3), (4, Ϫ4)}
(d) A number and its square: {(Ϫ3, 9), (Ϫ2, 4), (Ϫ1, 1), (0, 0), (1, 1), (2, 4), (3, 9)}
Solution
(a) Domain ϭ {1928, 1960, 1964, 1976, 1992, 1996}; Range ϭ {Hoover, Kennedy,
Johnson, Carter, Clinton}
(b) Domain ϭ {Iodine-133, Barium-135, Technetium-99m}; Range ϭ {6, 20.9, 28.7}
(c) Domain ϭ {1, 2, 3, 4}; Range ϭ {Ϫ1, Ϫ2, Ϫ3, Ϫ4}
(d) Domain ϭ {Ϫ3, Ϫ2, Ϫ1, 0, 1, 2, 3}; Range ϭ {0, 1, 4, 9}
The ordered pairs we refer to in Definition 11.7 may be generated in various ways, such
as from a graph, a chart, or a description of the relation. However, one of the most common
ways of generating ordered pairs is from equations. Since the solution set of an equation in
two variables is a set of ordered pairs, such an equation describes a relation. Each of the following equations describes a relation between the variables x and y. We list some of the
infinitely many ordered pairs (x, y) of each relation.
1. x2 ϩ y2 ϭ 4: 10, 22, 10, -22, 12, 02, 1- 2, 02
2. x ϭ y2: 116, 42, 116, -42, 125, 52, 125, -52
3. 2x Ϫ y ϭ -3: 1- 2, -12, 1-1, 12, 10, 32, 11, 52, 12, 72
1
1
1
1
4. y ϭ
: a- 2, - b, a- 1, - b, a0, - b, 11, -12, 13, 12
xϪ2
4
3
2
5. y ϭ x2: 1- 2, 42, 1- 1, 12, 10, 02, 11, 12, 12, 42
Now direct your attention to the ordered pairs of the last three relations. These relations are
a special type called functions.
11.7 • Relations and Functions
487
Definition 11.8
A function is a relation in which each member of the domain is assigned one and only
one member of the range. A function is a relation in which no two different ordered
pairs have the same first component.
Notice that the relation described by equation 1 is not a function because two different ordered
pairs, (0, 2) and (0, Ϫ2), have the same first component. Likewise, the relation described by
equation 2 is not a function because (16, 4) and (16, Ϫ4) have the same first component.
Classroom Example
Specify the domain and range for
each relation, and state whether or
not the relation is a function:
(a) {(3, 7), (2, 8), (1, 7),(0, 8), (Ϫ1, 9)}
(b) {(2, 3), (1, 4), (Ϫ2, 5), (2, 6)}
(c) {(94, A), (96, A), (99, A)}
(d) {(B, 82), (B, 85), (B, 87), (B, 89)}
EXAMPLE 2
Specify the domain and range for each relation, and state whether or not the relation is a
function:
(a) {(Ϫ3, 3), (Ϫ2, 2), (Ϫ1, 1), (0, 0), (1, 1), (2, 2), (3, 3)}
(b) {(1, 73), (2, 73), (3, 73)}
(c) {(5, 10), (6, 20), (5, Ϫ10), (6, Ϫ20)}
(d) {(Shaquille O’Neal, Magic), (Shaquille O’Neal, Heat), (Shaquille O’Neal, Lakers)}
Solution
(a) Domain ϭ {Ϫ3, Ϫ2, Ϫ1, 0, 1, 2, 3}; Range ϭ {0, 1, 2, 3}. It is a function.
(b) Domain ϭ {1, 2, 3}; Range ϭ {73}. It is a function.
(c) Domain ϭ {5, 6}; Range ϭ {Ϫ20, Ϫ10, 10, 20}. It is not a function because
(5, 10) and (5, Ϫ10) have the same first component.
(d) Domain ϭ {Shaquille O’Neal}; Range ϭ {Magic, Heat, Lakers}. It is not a function
because (Shaquille O’Neal, Magic) (Shaquille O’Neal, Heat) and (Shaquille O’Neal,
Lakes) have the same first component.
The domain of a function is frequently of more concern than the range. You should be
aware of any necessary restrictions on x. Consider the next examples.
Classroom Example
Specify the domain for each relation:
1
xϪ4
2
1
(b) y ϭ
xϩ4
3x
(c) y ϭ
4x Ϫ 7
(a) y ϭ
EXAMPLE 3
(a) y ϭ 2x ϩ 3
Specify the domain for each relation:
(b) y ϭ
1
xϪ3
(c) y ϭ
5x
3x Ϫ 4
Solution
(a) The domain of the relation described by y ϭ 2x ϩ 3 is the set of all real numbers,
because we can substitute any real number for x.
(b) We can replace x with any real number except 3, because 3 makes the denominator
zero. Thus the domain is all real numbers except 3.
(c) We need to find the value of x that makes the denominator equal to zero. To do that
we set the denominator equal to zero and solve for x.
3x Ϫ 4 ϭ 0
3x ϭ 4
4
xϭ
3
Since
4
4
makes the denominator zero, the domain is all real numbers except .
3
3
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488
Chapter 11 • Additional Topics
Functional Notation
Thus far we have been using the regular notation for writing equations to describe functions;
1
that is, we have used equations such as y ϭ x2 and y ϭ
, where y is expressed in terms
xϪ2
of x, to specify certain functions. There is a special functional notation that is very
convenient to use when working with the function concept.
The notation f1x2 is read “f of x” and is defined to be the value of the function f at x.
[Do not interpret f1x2 to mean f times x!] Instead of writing y ϭ x2, we can write f1x2 ϭ x2.
Therefore, f122 means the value of the function f at 2, which is 22 ϭ 4. So we write f122 ϭ 4.
This is a convenient way of expressing various values of the function. We illustrate that idea
with another example.
Classroom Example
If f(x) ϭ x3 ϩ 2, find f(Ϫ1), f (2),
f(Ϫ3), f(0), and f(h).
EXAMPLE 4
If f1x2 ϭ x2 Ϫ 6, find f102, f112, f122, f132, f1-12, and f1h2.
Solution
f(x) ؍x2 ؊ 6
f102 ϭ 02 Ϫ 6 ϭ -6
f112 ϭ 12 Ϫ 6 ϭ -5
f122 ϭ 22 Ϫ 6 ϭ -2
f132 ϭ 32 Ϫ 6 ϭ 3
f1 -12 ϭ 1- 122 Ϫ 6 ϭ -5
f1h2 ϭ h2 Ϫ 6
When we are working with more than one function in the same problem, we use different letters to designate the different functions, as the next example demonstrates.
Classroom Example
If f(x) ϭ 4x Ϫ 3 and
g(x) ϭ 2x2 ϩ x Ϫ 3, find f(Ϫ1),
f(3), g(1), and g(Ϫ3).
EXAMPLE 5
If f1x2 ϭ 2x ϩ 5 and g1x2 ϭ x2 Ϫ 2x ϩ 1 , find f122, f1-32, g1-12, and g142.
Solution
f(x) ؍2x ؉ 5
g(x) ؍x2 ؊ 2x ؉ 1
g1- 12 ϭ 1-12 2 Ϫ 21 -12 ϩ 1 ϭ 4
f122 ϭ 2122 ϩ 5 ϭ 9
f1 -32 ϭ 21- 32 ϩ 5 ϭ -1 g142 ϭ 42 Ϫ 2142 ϩ 1 ϭ 9
Concept Quiz 11.7
For Problems 1– 8, answer true or false.
1.
2.
3.
4.
5.
A function is a special type of relation.
The relation {(John, Mary), (Mike, Ada), (Kyle, Jenn), (Mike, Sydney)} is a function.
Given f(x) ϭ 3x ϩ 4, the notation f(7) means to find the value of f when x ϭ 7.
The set of all first components of the ordered pairs of a relation is called the range.
The domain of a function can never be the set of all real numbers.
x
6. The domain of the function f (x) ϭ
is the set of all real numbers.
xϪ3
7. The range of the function f (x) ϭ x ϩ 1 is the set of all real numbers.
8. If f(x) ϭ - x2 Ϫ 1, then f(2) ϭ -5.