Page 1 of 7
4.5
Goal
The Slope of a Line
How steep is a roller coaster?
Find the slope of a line.
You can describe steepness by a ratio
called slope. To find the slope, divide
the rise by the run. In Exercise 39 you
will find the slope of a roller coaster.
Key Words
• rise
• run
• slope
vertical
rise 10
vertical rise
horizontal run
10
5
2
1
slope 2
EXAMPLE
1
horizontal run 5
The Slope Ratio
Find the slope of a hill that has a vertical rise of 40 feet and
a horizontal run of 200 feet. Let m represent slope.
vertical
rise 40 ft
horizontal run 200 ft
Solution
vertical rise
horizontal run
40
200
1
5
m ANSWER 1
5
The slope of the hill is .
The slope of a line is the
ratio of the vertical rise to the
horizontal run between any
two points on the line. In the
diagram, notice how you can
subtract coordinates to find the
rise and the run.
rise
2
42
slope run
5
83
y
5
(8, 4)
3
run 8 3 5
1
1
1
rise 422
(3, 2)
1
4.5
3
5
7
The Slope of a Line
9
11 x
229
Page 2 of 7
THE SLOPE OF A LINE
The slope m of a line that
passes through the
points (x1, y1) and (x2, y2) is
Student Help
READING ALGEBRA
In the slope formula, x1
is read as “x sub one”
and y1 is read as
“y sub one.”
rise
run
change in y
change in x
y
(x2, y2)
y y
x2 x1
2
1
m y2 y 1
(x1, y1)
x2 x 1
O
x
When you use the formula for slope, you can label either point as (x1, y1)
and the other as (x2, y2). After labeling the points, you must subtract the
coordinates in the same order in both the numerator and the denominator.
SLOPE
EXAMPLE
2
Positive Slope
Find the slope of the line that passes through the points (1, 0) and (3, 4).
Solution
Let (x1, y1) (1, 0) and (x2, y2) (3, 4).
y2 y1
m
x2 x1
40
31
Subtract y-values.
Use the same order
to subtract x-values.
Substitute values.
y
5
3
4
1
1
1
4
2
Simplify.
2
Slope is positive.
ANSWER The slope of the line is 2.
(3, 4)
2 3
(1, 0)
5
x
The line rises from left to right.
The slope is positive.
Find a Positive Slope
Find the slope of the line that passes through the two points. Draw a
sketch of the line to help you.
1. (x1, y1) (3, 5) and (x2, y2) (1, 4)
2. (x1, y1) (2, 0) and (x2, y2) (4, 3)
3. (x1, y1) (2, 7) and (x2, y2) (1, 3)
230
Chapter 4
Graphing Linear Equations and Functions
Page 3 of 7
Student Help
STUDY TIP
You can choose any
two points on a line
to find the slope. For
example, you can use
the points (0, 3) and
(3, 2) in Example 3 and
get the same slope.
You will see this proof
in Geometry.
3
EXAMPLE
Negative Slope
Find the slope of the line that passes through the points (0, 3) and (6, 1).
Solution
Let (x1, y1) (0, 3) and (x2, y2) (6, 1).
y y
Subtract y-values.
2
1
m
x x
2
Use the same order
to subtract x-values.
1
13
60
Substitute values.
1 (3)
60
To subtract, add
the opposite.
y
5
run 6 0
6
1
2
1
6
3
ANSWER Simplify to find the
negative slope.
1
3
The slope of the line is .
rise
13
2
(0, 3)
(6, 1)
1
1
1
3
5
x
7
9
The line falls from left to right.
The slope is negative.
Find a Negative Slope
Find the slope of the line that passes through the two points. Draw a
sketch of the line to help you.
4. (x1, y1) (2, 4) and (x2, y2) (1, 5)
5. (x1, y1) (0, 9) and (x2, y2) (4, 7)
6. (x1, y1) (2, 1) and (x2, y2) (1, 3)
EXAMPLE
4
Zero Slope
Find the slope of the line that passes through the points (1, 2) and (5, 2).
Solution
Let (x1, y1) (1, 2) and (x2, y2) (5, 2).
y2 y1
m
x2 x1
22
51
y
Substitute values.
0
0
4
Simplify to find the slope
is zero.
ANSWER 5
Subtract y-values.
Use the same order
to subtract x-values.
The slope of the line is zero.
3
(5, 2)
(1, 2)
1
1
1
1
3
5
7
x
The line is horizontal.
The slope is zero.
4.5
The Slope of a Line
231
Page 4 of 7
Student Help
Undefined Slope
Find the slope of the line that passes through the points (5, 1) and (5, 3).
MORE EXAMPLES
NE
ER T
More examples
are available at
www.mcdougallittell.com
INT
5
EXAMPLE
Solution
Let (x1, y1) (5, 1) and (x2, y2) (5, 3).
y y
2
y
Subtract y-values.
2
1
m
x x
Use the same order
to subtract x-values.
1
3 (1)
55
31
55
Subtracting a negative
is the same as adding
a positive.
4
0
Division by zero is
undefined.
(5, 3)
1
Substitute values.
ANSWER 3
1
1
1
3
x
(5, 1)
3
The line is vertical.
The slope is undefined.
Because division by zero is undefined, the expression
4
has no meaning. The slope of the line is undefined.
0
Find the Slope of a Line
For each line, determine whether the slope is positive, negative, zero, or
undefined. If the slope is defined, find the slope.
7.
8.
y
5
(2, 4)
(4, 4)
3
(1, 4)
3
1
3
y
5
(2, 4)
3
(1, 1)
1
1
9.
y
5
5 x
(4, 2)
1
3
5 x
1
3
SUMMARY
Slopes of Lines
A line with positive
slope rises from left
to right.
y
A line with zero
slope is horizontal.
y
x
232
A line with negative
slope falls from left
to right.
Chapter 4
y
x
Graphing Linear Equations and Functions
A line with
undefined slope
is vertical.
y
x
x
5 x
Page 5 of 7
4.5 Exercises
Guided Practice
Vocabulary Check
Use the photo of a ramp.
1. What is the rise of the ramp?
2. What is the run of the ramp?
15 ft
3. What is the slope of the
25 ft
ramp?
Skill Check
Plot the points and draw the line that passes through them. Without
finding the slope, determine whether the slope is positive, negative, zero,
or undefined.
4. (1, 5) and (5, 5)
5. (2, 2) and (0, 1)
6. (4, 2) and (4, 1)
7. (3, 1) and (1, 3)
8. (2, 1) and (5, 3)
9. (4, 3) and (0, 3)
Find the slope of the line.
10.
11.
y
5
12.
y
y
3
(2, 5)
1
1
1
1
(3, 2)
1
(3, 0)
1
5
(2, 4)
1
3
x
(1, 1)
1
(3, 1)
3
x
x
1
1
Practice and Applications
THE SLOPE RATIO Plot the points and draw a line that passes through
them. Use the rise and run to find the slope.
13. (2, 3) and (0, 6)
14. (1, 4) and (3, 2)
15. (3, 1) and (3, 2)
16. (2, 2) and (6, 1)
17. (2, 1) and (2, 4)
18. (1, 3) and (4, 0)
GRAPHICAL REASONING Find the slope of the line.
Student Help
19.
Example 1: Exs. 13–18,
29–34
Example 2: Exs. 21–28
Example 3: Exs. 19,
23–28
Example 4: Exs. 20,
29–34
Example 5: Exs. 29–34
20.
y
21.
y
5
HOMEWORK HELP
5
y
3
(2, 2)
(2, 4)
3
(3, 4)
1
(2, 3)
1
3
1
1
1
1
(1, 0)
x
1
3 x
(2, 2)
1
3 x
22. CRITICAL THINKING Is the slope always positive if the coordinates of two
points on the line are positive? Justify your answer.
4.5
The Slope of a Line
233
Page 6 of 7
FINDING SLOPE Find the slope of the line that passes through
the points.
23. (4, 3) and (8, 5)
24. (2, 4) and (1, 6)
25. (3, 8) and (7, 7)
26. (3, 4) and (9, 4)
27. (3, 5) and (5, 8)
28. (6, 7) and (4, 4)
ZERO OR UNDEFINED SLOPE Determine whether the slope is zero,
undefined, or neither.
29. (0, 4) and (5, 7)
30. (1, 2) and (1, 6)
31. (6, 2) and (9, 2)
32. (5, 8) and (3, 8)
33. (8, 7) and (14, 1)
34. (3, 10) and (3, 5)
35.
History
jib sail
The photo shows the
U.S.S. Constitution. Built in the late
1700s, it is the oldest warship afloat.
Find the slope of the edge of the
Constitution’s jib sail.
72 ft
48 ft
36. LADDER The top of a ladder is 12 feet from the ground. The base of the
ladder is 5 feet to the left of the wall. What is the slope of the ladder? Make
a sketch to help you.
37. INDUCTIVE REASONING Choose
U.S.S. CONSTITUTION was
nicknamed “Old Ironsides”
by the crew in 1812 after the
defeat of the 38-gun British
frigate Guerrière.
three different pairs of points on the
line. Find the slope of the line
using each pair. What do you notice?
What conclusion can you draw?
y
3
E
D
C
B
5
A
1
1
1
3
5 x
38. INDUCTIVE REASONING Based on your conclusion from Exercise 37,
complete the following sentence: No matter what pair of points you choose
on a line, the ____
? is constant.
ROLLER COASTER In Exercises 39 and 40, use the following information.
You are supervising the construction of a roller coaster for young children. For
the first 20 feet of horizontal distance, the track must rise off the ground at a
constant rate. After your crew has constructed 5 feet of horizontal distance, the
track is 1 foot off the ground.
1 ft
5 ft
20 ft
39. Plot points for the heights of the track in 5-foot intervals. Draw a line
through the points. Find the slope of the line. What does it represent?
40. After 20 feet of horizontal distance is constructed, you are at the highest
point of your roller coaster. How high off the ground is the track?
234
Chapter 4
Graphing Linear Equations and Functions
Page 7 of 7
Road Grade
EXAMPLE
Road signs sometimes describe the
slope of a road in terms of its grade.
The grade of a road is given as a
positive percent. Find the grade of
the road shown in the sketch.
4 ft
50 ft
Solution
vertical rise
horizontal run
4
50
Find the slope: .
4
50
4
50
8
100
Write as a fraction whose denominator is 100: .
8
100
Write as a percent: 8%.
ANSWER The grade of the road is 8%.
1
41. Find the grade of a road that rises 1 feet for every horizontal distance of
2
25 feet.
42. Find the grade of a road that rises 70 feet for every horizontal distance of
1000 feet.
Standardized Test
Practice
43. MULTIPLE CHOICE What is the slope of the line through the points
(4, 3) and (11, 5)?
A
7
2
B
2
7
C
2
7
D
7
2
44. MULTIPLE CHOICE Which word describes the slope of a vertical line?
F
Mixed Review
zero
G
positive
H
undefined
J
negative
SOLVING EQUATIONS Solve the equation. (Lesson 3.1)
45. x 7 12
46. x 3 11
47. x (2) 6
REWRITING EQUATIONS Rewrite the equation so that y is a function
of x. (Lesson 3.7)
Maintaining Skills
48. 5y 10x 5
1
2
49. y x 3
3
3
50. 4x y 11
51. 8x 2y 10
52. 3x 6y 12
2
53. x y 1
5
OPERATIONS WITH DECIMALS Determine whether the equation is true
or false. (Skills Review p. 759)
54. 1.3 2.7 1.4
1.8
55. 1 0
1.8
2.7
56. 1 10 0
0.3
57. 14.4 0.14 2.88
58. (7.8)(1.5) 4.6 16.3 59. 12 0 p 7.18 12
4.5
The Slope of a Line
235