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6: Pie, Bar, and Line Graphs

# 6: Pie, Bar, and Line Graphs

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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

(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

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

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

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

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

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?

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?

484

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

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

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

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

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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

3

Since

4

4

makes the denominator zero, the domain is all real numbers except .

3

3

488

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.

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6: Pie, Bar, and Line Graphs

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