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 3͑͒ 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
mϭ
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.
mϭ
y2 Ϫ y1
x2 Ϫ x1
Correct
mϭ
y1 Ϫ y2
x1 Ϫ x2
Correct
mϭ
y2 Ϫ y1
x1 Ϫ x2
Incorrect
mϭ
y1 Ϫ y2
x2 Ϫ x1
Incorrect