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3 Relations, Functions, and Graphs

3 Relations, Functions, and Graphs

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Section 4.3



2



᭤ Determine if relations are functions

by inspection or by using the Vertical Line

Test.



239



Relations, Functions, and Graphs



Functions

In the study of mathematics and its applications, the focus is mainly on a special

type of relation called a function.



Definition of Function

A function is a relation in which no two ordered pairs have the same first

component and different second components.



This definition means that a given first component cannot be paired with two

different second components. For instance, the pairs (1, 3) and ͑1, Ϫ1͒ could not

be ordered pairs of a function.

Consider the relations described at the beginning of this section.

Relation



Ordered Pairs



Sample Relation



1

2

3

4



(person, month)

(hours, pay)

(instructor, course)

(time, temperature)



ͭ(A, May), (B, Dec), (C, Oct), . . .ͮ

ͭ(12, 84), (4, 28), (6, 42), (15, 105), . . .ͮ

ͭ(A, MATH001), (A, MATH002), . . .ͮ

ͭ͑8, 70Њ͒, ͑10, 78Њ͒, ͑12, 78Њ͒, . . .ͮ



The first relation is a function because each person has only one birth month. The

second relation is a function because the number of hours worked at a particular

job can yield only one paycheck amount. The third relation is not a function

because an instructor can teach more than one course. The fourth relation is a

function. Note that the ordered pairs ͑10, 78Њ͒ and ͑12, 78Њ͒ do not violate the

definition of a function.

Study Tip

The ordered pairs of a relation can

be thought of in the form (input,

output). For a function, a given

input cannot yield two different

outputs. For instance, if the input

is a person’s name and the output

is that person’s month of birth,

then your name as the input can

yield only your month of birth as

the output.



EXAMPLE 2



Testing Whether a Relation Is a Function



Decide whether each relation represents a function.

a. Input: a, b, c

b.

c.

a

1

Input Output

ͧx, yͨ

Output: 2, 3, 4

x

y

2

ͭ͑a, 2͒, ͑b, 3͒, ͑c, 4͒ͮ

b

͑3, 1͒

3

1

3

c



4



4



3



͑4, 3͒



Input



Output



5



4



͑5, 4͒



͑3, 2͒

3

2

Solution

a. This set of ordered pairs does represent a function. No first component has two

different second components.

b. This diagram does represent a function. No first component has two different

second components.

c. This table does not represent a function. The first component 3 is paired with

two different second components, 1 and 2.

CHECKPOINT Now try Exercise 7.



240



Chapter 4



Graphs and Functions



y



( x, y 1)



x



( x, y 2)



In algebra, it is common to represent functions by equations in two variables

rather than by ordered pairs. For instance, the equation y ϭ x2 represents the

variable y as a function of x. The variable x is the independent variable (the

input) and y is the dependent variable (the output). In this context, the domain

of the function is the set of all allowable values of x, and the range is the

resulting set of all values taken on by the dependent variable y.

From the graph of an equation, it is easy to determine whether the equation

represents y as a function of x. The graph in Figure 4.22 does not represent a function of x because the indicated value of x is paired with two y-values. Graphically,

this means that a vertical line intersects the graph more than once.



Vertical Line Test

A set of points on a rectangular coordinate system is the graph of y as a

function of x if and only if no vertical line intersects the graph at more than

one point.



Figure 4.22



Using the Vertical Line Test for Functions



EXAMPLE 3



Use the Vertical Line Test to determine whether y is a function of x.

y

y

a.

b.



−1



c.



3



3



2



2



1



1

x

1



−1



2



x

−1



3



y



d.



x



1



3



−1

y



x



Solution

a. From the graph, you can see that no vertical line intersects more than one point

on the graph. So, the relation does represent y as a function of x.

b. From the graph, you can see that a vertical line intersects more than one point

on the graph. So, the relation does not represent y as a function of x.

c. From the graph, you can see that a vertical line intersects more than one point

on the graph. So, the relation does not represent y as a function of x.

d. From the graph, you can see that no vertical line intersects more than one point

on the graph. So, the relation does represent y as a function of x.

CHECKPOINT Now try Exercise 27.



Section 4.3



3 ᭤ Use function notation and evaluate

functions.



Relations, Functions, and Graphs



241



Function Notation

To discuss functions represented by equations, it is common practice to give them

names using function notation. For instance, the function

y ϭ 2x Ϫ 6

can be given the name “f ” and written in function notation as

f ͑x͒ ϭ 2x Ϫ 6.



Function Notation

In the notation f (x):

f is the name of the function.

x is a domain (or input) value.

f(x) is a range (or output) value y for a given x.

The symbol f (x) is read as the value of f at x or simply f of x.



The process of finding the value of f (x) for a given value of x is called

evaluating a function. This is accomplished by substituting a given x-value

(input) into the equation to obtain the value of f (x) (output). Here is an example.

Function



x-Values (input)



Function Values (output)



f ͑x͒ ϭ 4 Ϫ 3x



x ϭ Ϫ2



f ͑Ϫ2͒ ϭ 4 Ϫ 3͑Ϫ2͒ ϭ 4 ϩ 6 ϭ 10



x ϭ Ϫ1



f ͑Ϫ1͒ ϭ 4 Ϫ 3͑Ϫ1͒ ϭ 4 ϩ 3 ϭ 7



xϭ0



f ͑0͒ ϭ 4 Ϫ 3͑0͒ ϭ 4 Ϫ 0 ϭ 4



xϭ2



f ͑2͒ ϭ 4 Ϫ 3͑2͒ ϭ 4 Ϫ 6 ϭ Ϫ2



xϭ3



f ͑3͒ ϭ 4 Ϫ 3͑3͒ ϭ 4 Ϫ 9 ϭ Ϫ5



Although f and x are often used as a convenient function name and

independent (input) variable, you can use other letters. For instance, the equations

f ͑x͒ ϭ x2 Ϫ 3x ϩ 5, f ͑t͒ ϭ t2 Ϫ 3t ϩ 5, and



g͑s͒ ϭ s2 Ϫ 3s ϩ 5



all define the same function. In fact, the letters used are just “placeholders” and

this same function is well described by the form

f ͑᭿͒ ϭ ͑᭿͒2 Ϫ 3͑᭿͒ ϩ 5

where the parentheses are used in place of a letter. To evaluate f ͑Ϫ2͒, simply

place Ϫ2 in each set of parentheses, as follows.

f ͑Ϫ2͒ ϭ ͑Ϫ2͒2 Ϫ 3͑Ϫ2͒ ϩ 5

ϭ4ϩ6ϩ5

ϭ 15

It is important to put parentheses around the x-value (input) and then simplify the

result.



242



Chapter 4



Graphs and Functions



Evaluating a Function



EXAMPLE 4



Let f ͑x͒ ϭ x2 ϩ 1. Find each value of the function.

a. f ͑Ϫ2͒



b. f ͑0͒



Solution

a.



f ͑x͒ ϭ x2 ϩ 1



Write original function.



f ͑Ϫ2͒ ϭ ͑Ϫ2͒ ϩ 1

2



ϭ4ϩ1ϭ5

b. f ͑x͒ ϭ



x2



ϩ1



Substitute Ϫ2 for x.

Simplify.

Write original function.



f ͑0͒ ϭ ͑0͒2 ϩ 1



Substitute 0 for x.



ϭ0ϩ1ϭ1



Simplify.



CHECKPOINT Now try Exercise 45.



EXAMPLE 5



Evaluating a Function



Let g͑x͒ ϭ 3x Ϫ x 2. Find each value of the function.

a. g͑2͒



b. g͑0͒



Solution

a. Substituting 2 for x produces g͑2͒ ϭ 3͑2͒ Ϫ ͑2͒2 ϭ 6 Ϫ 4 ϭ 2.

b. Substituting 0 for x produces g͑0͒ ϭ 3͑0͒ Ϫ ͑0͒2 ϭ 0 Ϫ 0 ϭ 0.

CHECKPOINT Now try Exercise 47.



4



᭤ Identify the domain of a function.



Finding the Domain of a Function

The domain of a function may be explicitly described along with the function, or

it may be implied by the context in which the function is used. For instance, if

weekly pay is a function of hours worked (for a 40-hour work week), the implied

domain is 0 Յ x Յ 40. Certainly x cannot be negative in this context.

EXAMPLE 6



Finding the Domain of a Function



Find the domain of each function.

a. f:ͭ͑Ϫ3, 0͒, ͑Ϫ1, 2͒, ͑0, 4͒, ͑2, 4͒, ͑4, Ϫ1͒ͮ

b. Area of a square: A ϭ s 2

Solution

a. The domain of f consists of all first components in the set of ordered pairs. So,

the domain is ͭϪ3, Ϫ1, 0, 2, 4ͮ.

b. For the area of a square, you must choose positive values for the side s. So, the

domain is the set of all real numbers s such that s > 0.

CHECKPOINT Now try Exercise 53.



Section 4.3



Relations, Functions, and Graphs



243



Concept Check

1. Explain the difference between a relation and a

function.



3. In your own words, explain how to use the Vertical

Line Test.



2. Explain the meanings of the terms domain and

range in the context of a function.



4. What is the meaning of the notation f ͑3͒?



Go to pages 284–285 to

record your assignments.



4.3 EXERCISES



Developing Skills

In Exercises 1–6, find the domain and range of the

relation. See Example 1.

1. ͭ͑Ϫ4, 3͒, ͑2, 5͒, ͑1, 2͒, ͑4, Ϫ3͒ͮ

2. ͭ͑Ϫ1, 5͒, ͑8, 3͒, ͑4, 6͒, ͑Ϫ5, Ϫ2͒ͮ

3. ͭ ͑2, 16͒, ͑Ϫ9, Ϫ10͒, ͑12, 0͒ͮ



4.



ͭ͑23, Ϫ4͒, ͑Ϫ6, 14 ͒, ͑0, 0͒ͮ



5. ͭ͑Ϫ1, 3͒, ͑5, Ϫ7͒, ͑Ϫ1, 4͒, ͑8, Ϫ2͒, ͑1, Ϫ7͒ͮ

6. ͭ͑1, 1͒, ͑2, 4͒, ͑3, 9͒, ͑Ϫ2, 4͒, ͑Ϫ1, 1͒ͮ



In Exercises 7–26, determine whether the relation

represents a function. See Example 2.

7. Domain

−2

−1

0

1

2



Range

5

6

7

8



8. Domain

−2

−1

0

1

2



Range

3

4

5



9. Domain

−2

−1

0

1

2



Range 10. Domain

7

−2

9

−1

0

1

2



Range

3

4

5

6

7



11. Domain

0

2

4

6

8



Range 12. Domain

25

10

30

20

30

40

50



Range

5

10

15

20

25



13. Domain

0

1

2

3

4



Range 14. Domain

1

−4

2

−3

5

−2

9

−1



Range

3

4



15. Domain



Range

60 Minutes

CSI

Survivor

Dateline

Law & Order

Conan O’Brien



CBS



NBC



244



Chapter 4



Domain

60 Minutes

CSI

Survivor

Dateline

Law & Order

Conan O’Brien



16.



17. Domain



Year

2002

2003

2004

2005



Graphs and Functions



Range



21.



22.



Input Output

ͧx, yͨ

x

y



CBS



NBC



Range

Single women

in the labor force

(in percent)

67.4

66.2

65.9

66.0



Input Output

ͧx, yͨ

x

y



1



1



͑1, 1͒



2



1



͑2, 1͒



3



2



͑3, 2͒



4



1



͑4, 1͒



5



3



͑5, 3͒



6



1



͑6, 1͒



3



4



͑3, 4͒



8



1



͑8, 1͒



1



5



͑1, 5͒



10



1



͑10, 1͒



23. ͭ͑0, 25͒, ͑2, 25͒, ͑4, 30͒, ͑6, 30͒, ͑8, 30͒ͮ

24. ͭ͑10, 5͒, ͑20, 10͒, ͑30, 15͒, ͑40, 20͒, ͑50, 25͒ͮ



(Source: U.S. Bureau of Labor Statistics)

Domain

Percent daily value

of vitamin C

per serving



18.



25. Input: a, b, c

Output: 0, 4, 9



Range



ͭ͑a, 0͒, ͑b, 4͒, ͑c, 9͒ͮ



Cereal



26. Input: 3, 5, 7

Output: d, e, f



Corn Flakes

Wheaties

Cheerios

Total



10%

100%



19.



ͭ͑3, d͒ ͑5, e͒, ͑7, f ͒, ͑7, d͒ͮ

In Exercises 27–36, use the Vertical Line Test to determine whether y is a function of x. See Example 3.



20.



Input Output

ͧx, yͨ

x

y



Input Output

ͧx, yͨ

x

y



0



2



͑0, 2͒



0



2



͑0, 2͒



1



4



͑1, 4͒



1



4



͑1, 4͒



2



6



͑2, 6͒



2



6



͑2, 6͒



3



8



͑3, 8͒



1



8



͑1, 8͒



4



10



͑4, 10͒



0



10



͑0, 10͒



y



27.



y



28.



4



4



2



2

x



−4 −2



2



−4



4



3



4



y



30.

4



1

x

1

−1



2



−4



y



29.



x



−4 −2



4



2



3



3

2

1



−2



x

1



2



Section 4.3

y



31.



y



32.



4



2



3



1



42. f ͑s͒ ϭ 4 Ϫ 23s



x



2

−1



1

−2 −1



1



2



y



33.



2



3



4



(a) f ͑60͒



4



1

2



3



2



4



−2



(d) f ͑12 ͒



43. f ͑v͒ ϭ 12 v2



(a) f ͑Ϫ4͒

(c) f ͑0͒



(b) f ͑4͒

(d) f ͑2͒



44. g ͑u͒ ϭ Ϫ2u2



(a) g͑0͒

(c) g͑3͒



(b) g͑2͒

(d) g͑Ϫ4͒



45. f ͑x͒ ϭ 4x2 ϩ 2



(a) f ͑1͒

(c) f ͑Ϫ4͒



(b) f ͑Ϫ1͒

(d) f ͑Ϫ 32 ͒



46. g͑t͒ ϭ 5 Ϫ 2t2



(a) g͑52 ͒

(c) g͑0͒



(b) g͑Ϫ10͒

(d) g͑34 ͒



47. g͑x͒ ϭ 2x2 Ϫ 3x ϩ 1



(a) g͑0͒



(b) g͑Ϫ2͒



x



−2 −1

y



35.



1



2



2

1

x

1 2



−2



−2



38. g͑x͒ ϭ Ϫ 45x



39. f ͑x͒ ϭ 2x Ϫ 1

40. f ͑t͒ ϭ 3 Ϫ 4t

41. h ͑t͒ ϭ 14t Ϫ 1



(d) g͑12 ͒



48. h͑x͒ ϭ 1 Ϫ 4x Ϫ x2



(a) h͑0͒

(c) h͑10͒



(b) h͑Ϫ4͒

(d) h͑32 ͒



Խ



(a) g͑2͒



(b) g͑Ϫ2͒



x

−1



1



2



(a) f ͑2͒



Խ



49. g ͑u͒ ϭ u ϩ 2



−2



In Exercises 37–52, evaluate the function as indicated,

and simplify. See Examples 4 and 5.

37. f ͑x͒ ϭ 12x



(c) g͑1͒



y



36.



2

1

−2 −1



(b) f ͑Ϫ15͒



(c) f ͑Ϫ18͒



3

x

1



245



y



34.



2



−1



1



−2



x



Relations, Functions, and Graphs



(b) f ͑5͒



(c) f ͑Ϫ4͒



(d) f ͑Ϫ 23 ͒



(a) g͑5͒



(b) g͑0͒



(d) g͑Ϫ 54 ͒



(a) f ͑0͒



(b) f ͑3͒



(c) f ͑Ϫ3͒



(d) f ͑



(a) f ͑0͒



(b) f ͑1͒



51. h͑x͒ ϭ x3 Ϫ 1

52. f ͑x͒ ϭ 16 Ϫ x4



(c) g͑Ϫ3͒



Ϫ 12



ԽԽ



50. h͑s͒ ϭ s ϩ 2



͒



(c) g͑10͒



(d) g͑Ϫ 52 ͒



(a) h͑4͒



(b) h͑Ϫ10͒



(c) h͑Ϫ2͒



(d) h͑32 ͒



(a) h͑0͒



(b) h͑1͒



(c) h͑3͒



(d) h͑12 ͒



(a) f ͑Ϫ2͒



(b) f ͑2͒



(c) f ͑1͒



(d) f ͑3͒



In Exercises 53–60, find the domain of the function. See

Example 6.

53. f :ͭ͑0, 4͒, ͑1, 3͒, ͑2, 2͒, ͑3, 1͒, ͑4, 0͒ͮ



(c) f ͑Ϫ2͒



(d) f ͑34 ͒



54. f:ͭ͑Ϫ2, Ϫ1͒, ͑Ϫ1, 0͒, ͑0, 1͒, ͑1, 2͒, ͑2, 3͒ͮ



(a) h͑200͒



(b) h͑Ϫ12͒



55. g:ͭ͑Ϫ8, Ϫ1͒, ͑Ϫ6, 0͒, ͑2, 7͒, ͑5, 0͒, ͑12, 10͒ͮ



(c) h͑8͒



(d) h͑Ϫ 52 ͒



56. g:ͭ͑Ϫ4, 4͒, ͑3, 8͒, ͑4, 5͒, ͑9, Ϫ2͒, ͑10, Ϫ7͒ͮ



246



Chapter 4



Graphs and Functions



57. h:ͭ͑Ϫ5, 2͒, ͑Ϫ4, 2͒, ͑Ϫ3, 2͒, ͑Ϫ2, 2͒, ͑Ϫ1, 2͒ͮ



59. Area of a circle: A ϭ ␲ r 2



58. h:ͭ͑10, 100͒, ͑20, 200͒, ͑30, 300͒, ͑40, 400͒ͮ



60. Perimeter of a square: P ϭ 4s



Solving Problems

61. Demand The demand for a product is a function

of its price. Consider the demand function

f ͑ p͒ ϭ 20 Ϫ 0.5p



Interpreting a Graph In Exercises 65–68, use the

information in the graph. (Source: U.S. National

Center for Education Statistics)

y



Enrollment (in millions)



where p is the price in dollars.

(a) Find f ͑10͒ and f ͑15͒.

(b) Describe the effect a price increase has on

demand.

62. Maximum Load The maximum safe load L (in

pounds) for a wooden beam 2 inches wide and d

inches high is L͑d͒ ϭ 100d 2.

(a) Complete the table.

d



2



4



6



8



Lͧd ͨ



18.0

17.5

17.0

16.5

16.0

15.5

15.0

14.5



High school

College



t



2000 2001 2002 2003 2004 2005



Year



65. Is the high school enrollment a function of the year?

66. Is the college enrollment a function of the year?



(b) Describe the effect of an increase in height on

the maximum safe load.

63. Distance The function d͑t͒ ϭ 50t gives the distance

(in miles) that a car will travel in t hours at an average

speed of 50 miles per hour. Find the distance traveled

for (a) t ϭ 2, (b) t ϭ 4, and (c) t ϭ 10.



64. Speed of Sound The function S(h) ϭ 1116 Ϫ 4.04h

approximates the speed of sound (in feet per second)

at altitude h (in thousands of feet). Use the function

to approximate the speed of sound for (a) h ϭ 0,

(b) h ϭ 10, and (c) h ϭ 30.



67. Let f ͑t͒ represent the number of high school students

in year t. Find f (2001).

68. Let g͑t͒ represent the number of college students in

year t. Find g(2005).

69.



Geometry Write the formula for the perimeter

P of a square with sides of length s. Is P a function

of s? Explain.



70.



Geometry Write the formula for the volume V

of a cube with sides of length t. Is V a function of t?

Explain.



Section 4.3



72. SAT Scores and Grade-Point Average The graph

shows the SAT scores x and the grade-point averages

(GPA) y for 12 students.

y



Grade-point average



Length of time (in hours)



71. Sunrise and Sunset The graph approximates the

length of time L (in hours) between sunrise and

sunset in Erie, Pennsylvania for the year 2007. The

variable t represents the day of the year.

L



18

16

14

12

10

8



4

3

2

1

x

800



t



50



247



Relations, Functions, and Graphs



1200



1600



2000



2400



SAT score



100 150 200 250 300 350 400



Day of the year



(a) Is the GPA y a function of the SAT score x?



(a) Is the length of time L a function of the day of the

year t ?

(b) Estimate the range of this relation.



(b) Estimate the range of this relation.



Explaining Concepts

73. Is it possible to find a relation that is not a function?

If it is, find one.

74. Is it possible to find a function that is not a relation?

If it is, find one.

75.



76.



Is it possible for the number of elements in the

domain of a relation to be greater than the number of

elements in the range of the relation? Explain.



Determine whether the statement uses the word

function in a way that is mathematically correct.

Explain your reasoning.

(a) The amount of money in your savings account is

a function of your salary.



(b) The speed at which a free-falling baseball strikes

the ground is a function of the height from which

it is dropped.



Cumulative Review

In Exercises 77–80, rewrite the statement using

inequality notation.

77.

78.

79.

80.



x is negative.

m is at least Ϫ3.

z is at least 85, but no more than 100.

n is less than 20, but no less than 16.



In Exercises 81–88, solve the equation.



ԽԽ

Խ4hԽ ϭ 24

Խx ϩ 4Խ ϭ 5

Խ6b ϩ 8Խ ϭ 2b



ԽԽ



81. x ϭ 8



82. g ϭ Ϫ4



83.



84.



85.

87.



Խ mԽ ϭ 2

1

5



Խ

Խ

Խn Ϫ 2Խ ϭ Խ2n ϩ 9Խ



86. 2t Ϫ 3 ϭ 11

88.



248



Chapter 4



Graphs and Functions



Mid-Chapter Quiz

Take this quiz as you would take a quiz in class. After you are done, check

your work against the answers in the back of the book.



2. Determine the quadrant(s) in which the point ͑3, y͒ is located, or the axis on

which the point is located, without plotting it. (y is a real number.)

3. Determine whether each ordered pair is a solution of the equation y ϭ 9 Ϫ x .



ԽԽ



240

220



(a) (2, 7)

(b) ͑Ϫ3, 12͒

(c) ͑Ϫ9, 0͒

(d) ͑0, Ϫ9͒

4. The scatter plot at the left shows the amounts (in billions of dollars) spent on

prescription drugs in the United States for the years 2000 through 2005.

Estimate the amount spent on prescription drugs for each year from 2000 to

2005. (Source: National Association of Chain Drug Stores)



200

180

160



05



04



20



03



20



02



20



20



20



20



01



140

00



Amount spent on

prescription drugs

(in billions of dollars)



1. Plot the points ͑4, Ϫ2͒ and ͑Ϫ1, Ϫ 52 ͒ on a rectangular coordinate system.



Year



In Exercises 5 and 6, find the x- and y-intercepts of the graph of the

equation.

5. x Ϫ 3y ϭ 12



Figure for 4



6. y ϭ Ϫ7x ϩ 2



In Exercises 7–9, sketch the graph of the equation.

7. y ϭ 5 Ϫ 2x



8. y ϭ ͑x ϩ 2͒2



Խ



Խ



9. y ϭ x ϩ 3



In Exercises 10 and 11, find the domain and range of the relation.

10. ͭ͑1, 4͒, ͑2, 6͒, ͑3, 10͒, ͑2, 14͒, ͑1, 0͒ͮ

11. ͭ͑Ϫ3, 6͒, ͑Ϫ2, 6͒, ͑Ϫ1, 6͒, ͑0, 6͒ͮ

12. Determine whether the relation in the figure is a function of x using the

Vertical Line Test.



y

4

3

2



In Exercises 13 and 14, evaluate the function as indicated, and simplify.



1

−3 −2 −1

−2

−3

−4



Figure for 12



x

1



2



3



4



13. f ͑x͒ ϭ 3͑x ϩ 2͒ Ϫ 4

14. g͑x͒ ϭ 4 Ϫ x 2

(a) f ͑0͒

(b) f ͑Ϫ3͒

(a) g͑Ϫ1͒

(b) g͑8͒

15. Find the domain of the function f: {(10, 1), (15, 3), (20, 9), (25, 27)}.

16.

Use a graphing calculator to graph y ϭ 3.6x Ϫ 2.4. Graphically

estimate the intercepts of the graph. Explain how to verify your estimates

algebraically.

17. A new computer system sells for approximately $2000 and depreciates at

the rate of $500 per year.

(a) Find an equation that relates the value of the computer system to the

number of years.

(b) Sketch the graph of the equation.

(c) What is the y-intercept of the graph, and what does it represent in the

context of the problem?



Section 4.4



249



Slope and Graphs of Linear Equations



4.4 Slope and Graphs of Linear Equations

What You Should Learn



Stockbyte/Getty Images



1 ᭤ Determine the slope of a line through two points.

2 ᭤ Write linear equations in slope-intercept form and graph the equations.

3 ᭤ Use slopes to determine whether lines are parallel, perpendicular, or neither.



The Slope of a Line

Why You Should Learn It

Slopes of lines can be used in many

business applications. For instance, in

Exercise 94 on page 261, you will

interpret the meaning of the slope of

a line segment that represents the

average price of a troy ounce of gold.



1



The slope of a nonvertical line is the number of units the line rises or falls

vertically for each unit of horizontal change from left to right. For example, the

line in Figure 4.23 rises two units for each unit of horizontal change from left to

right, and so this line has a slope of m ϭ 2.

y



y



m=2



᭤ Determine the slope of a line through



y2



(x 2, y 2)



two points.

y2 − y1



2 units



(x 1, y 1)

y1

1 unit



x2 − x1



x



x1

Figure 4.23



y2 − y1

x2 − x1

x



x2



Figure 4.24



Study Tip

In the definition at the right, the rise

is the vertical change between the

points and the run is the horizontal

change between the points.



m=



Definition of the Slope of a Line

The slope m of a nonvertical line passing through the points ͑x1, y1͒ and

͑x2, y2͒ is





y2 Ϫ y1 Change in y Rise

ϭ

ϭ

x2 Ϫ x1 Change in x

Run



where x1



x2. (See Figure 4.24.)



When the formula for slope is used, the order of subtraction is important.

Given two points on a line, you are free to label either of them ͑x1, y1͒ and the

other ͑x2, y2͒. However, once this has been done, you must form the numerator

and denominator using the same order of subtraction.





y2 Ϫ y1

x2 Ϫ x1



Correct







y1 Ϫ y2

x1 Ϫ x2



Correct







y2 Ϫ y1

x1 Ϫ x2



Incorrect







y1 Ϫ y2

x2 Ϫ x1



Incorrect



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