LESSON
Page 1 of 8
8.4
The Slope of a Line
Now
BEFORE
You graphed lines in a
coordinate plane.
Vocabulary
slope, p. 404
rise, p. 404
run, p. 404
You’ll find and interpret slopes
of lines.
WHY?
So you can compare animal
speeds, as in Ex. 17.
Wakeboarding How steep
is a wakeboard ramp like
the one shown? To find
out, you can calculate the
ramp’s slope. The slope of
a line is the ratio of the
line’s vertical change,
called the rise , to its
horizontal change, called
the run .
Example 1
Finding Slope
A wakeboard ramp has a rise of 6 feet
and a run of 10 feet. Find its slope.
rise
ru n
6
10
3
5
slope rise 6 ft
run 10 ft
3
Answer The wakeboard ramp has a slope of .
5
To determine the slope of a line in a coordinate plane, you can find the
ratio of the vertical change between two points on the line and the
horizontal change between the points.
Slope of a Line
Note Worthy
You may find it helpful to use
colors when you include
examples in your notebook. In
the notebook shown, notice
how colors are used to
associate the rise and run in
the slope formula with the rise
and run in the graph.
Given two points on a nonvertical line,
you can find the slope m of the line
using this formula:
rise
ru n
m difference of y-coordinates
difference of x-coordinates
Example
404
Chapter 8
Linear Functions
41
53
3
2
m y
5
4
3
2
1
O
(5, 4)
rise
(3, 1)
run
1
3
4
5
6 x
Page 2 of 8
Comparing Slopes You can use the diagrams below to compare the
slopes of different lines. Imagine that you are walking to the right.
y
y
4
3
2
1
4
3
2
1
O
2
3
4
5
6 x
Positive slope
If the line rises,
the slope is positive.
1
2
3
4
5
Negative slope
If the line falls,
the slope is negative.
y
y
4
4
3
3
2
1
2
O
x
O
1
2
3
4
5
6 x
O
Zero slope
If the line is horizontal,
the slope is zero.
Example 2
1
2
3
4
6 x
Undefined slope
If the line is vertical,
the slope is undefined.
Finding Positive and Negative Slope
Find the slope of the line shown.
Watch Out
When you calculate a slope,
be sure to use the x- and
y-coordinates of the two points
in the same order. In part (a)
of Example 2, for instance, the
following expression for the
slope would be incorrect:
52
14
m rise
difference of y-coordinates
y
a. m ru n
difference of x-coordinates
(4, 5)
5
4
3
52
41
2
3
1
3
(1, 2)
O
1
2
3
4
5 x
Answer The slope is 1.
rise
difference of y-coordinates
y
b. m ru n
difference of x-coordinates
(0, 1)
2
3 1
30
O
2
3
4
4
4
3
3
2
4 x
(3, 3)
4
3
Answer The slope is .
Checkpoint
Find the slope of the line through the given points.
1. (1, 2), (4, 7)
2. (2, 5), (6, 1)
3. (0, 0), (3, 9)
Lesson 8.4
4. (5, 0), (7, 8)
The Slope of a Line
405
Page 3 of 8
Example 3
Zero and Undefined Slope
Find the slope of the line shown.
difference of y-coordinates
rise
y
a. m ru n
difference of x-coordinates
4
33
41
2
1
0
3
O
0
(1, 3)
1
(4, 3)
2 3
4
5
6 x
4
5
6 x
Answer The slope is 0.
difference of y-coordinates
rise
y
b. m ru n
difference of x-coordinates
3
2
1
3 (1)
22
4
0
(2, 3)
O
Division by zero
is undefined.
1
3
(2, 1)
2
Answer The slope is undefined.
Checkpoint
Find the slope of the line through the given points. Tell whether the
slope is positive, negative, zero, or undefined.
Example 4
6. (6, 3), (6, 1)
7. (7, 4), (5, 4)
The graph shows the distance
traveled by a wakeboarder as a
function of time. The slope of the line
gives the wakeboarder’s speed, which
is the rate of change in distance
traveled with respect to time. Find
the wakeboarder’s speed.
In the
Real World
Wakeboarding Experts
recommend that wakeboarders
travel at speeds from 16 to
19 miles per hour. Is the
speed of the wakeboarder in
Example 4 within this interval?
Explain.
8. (1, 5), (4, 1)
Interpreting Slope as a Rate of Change
Solution
Use the points (2, 52) and (7, 182) to
find the slope of the line.
difference of y-coordinates
difference of x-coordinates
m 182 ft 52 ft
7 sec 2 sec
Wakeboard Distance
200
Distance traveled (feet)
5. (2, 3), (4, 5)
(7, 182)
160
120
80
(2, 52)
40
0
0
2
4
6
Time (seconds)
130 ft
5 sec
26 ft/sec
Answer The wakeboarder’s speed is 26 feet per second.
406
Chapter 8
Linear Functions
8
Page 4 of 8
8.4
Exercises
INTERNET
More Practice, p. 810
CLASSZONE.COM
eWorkbook Plus
Guided Practice
Vocabulary Check
1. Copy and complete: The vertical change between two points on a line
is called the _?_, and the horizontal change is called the _?_.
2. Why is the slope of a vertical line undefined?
Skill Check
3. Error Analysis Describe and
correct the error in calculating
the slope of the line through
the points (5, 4) and (0, 2).
24
50
2
5
m Tell whether the slope of the line is positive, negative, zero, or undefined.
Then find the slope.
4.
5.
y
3
2
1
O
O
(4, 2)
1 2
3
4
1
2
4
3
y
(3, 2)
x
(3, 1)
2
5 x
(1, 1)
7.
6.
y
1
4
3
(2, 2)
1
(0, 3)
3 2
O
1
2 x
Writing
A wakeboard ramp has a rise of 5 feet and a run of 12 feet.
Find the slope of the ramp. Compare this slope with the slope of the
ramp in Example 1.
Practice and Problem Solving
Homework Help
Example
1
2
3
4
Exercises
35–38
8–16, 18–33
8–16, 18–33
17
Online Resources
Tell whether the slope of the line is positive, negative, zero, or undefined.
Then find the slope.
8.
9.
y
5
4
3
(2, 4)
2
1
O
y
(1, 3)
(1, 2)
(2, 1)
3 2
10.
y
3
2
1
2 x
O
2
1
3 x
(3, 1)
4 3 2
3
2
1
O
1 x
(1, 2)
CLASSZONE.COM
• More Examples
• eTutorial Plus
Find the coordinates of two points on the line with the given equation.
Then use the points to find the slope of the line.
11. y 2x 4
12. y 1
3
13. y x 5
2
14. x 2y 6
15. 4x 3y 12
16. x 3
Lesson 8.4
The Slope of a Line
407
Page 5 of 8
17. Extended Problem Solving The graph shows the distance run by
a cheetah as a function of time.
a. Find the slope of the line.
Distance Run by a Cheetah
Distance (meters)
b. Interpret What information about
the cheetah can you obtain from
the slope?
c. Compare and Contrast A gazelle’s top
speed is about 22 meters per second.
Suppose you made a graph showing
the distance run by a gazelle as a
function of time. How would the
graph for the gazelle compare with
the graph for the cheetah? Explain
your thinking.
80
70
60
50
40
30
20
10
0
(2, 54)
(1, 27)
0
1 2 3 4 5
Time (seconds)
Sketch an example of the type of line described.
18. A line with zero slope
19. A line with undefined slope
20. A line with positive slope
21. A line with negative slope
Find the slope of the line through the given points.
22. (3, 3), (5, 7)
23. (6, 1), (4, 3)
24. (7, 3), (7, 2)
25. (3, 5), (6, 11)
26. (4, 1), (12, 8)
27. (5, 7), (0, 7)
28. (1, 0), (0, 5)
29. (3, 2), (8, 2)
30. (2, 6), (2, 6)
31. (8, 8), (2, 6) 32. (65, 87), (82, 16)
34.
33. (10, 10), (10, 0)
Writing
Describe the difference between a line with zero slope and
a line with undefined slope.
35. Wheelchair Ramp You are building a wheelchair ramp that leads to a
doorway 22 inches above the ground. The slope of the ramp must be
1
. Find the length of ground (in feet) that the ramp covers.
12
36. Cinder Cones A cinder cone is a type of volcano. To describe the
steepness of a cinder cone from one point on the cone to another,
you can find the gradient between the two points.
Change in elevation (in feet)
Horizontal change (in miles)
Gradient The graph shows a cross
section of a cinder cone. Use
the information in the graph to
find the gradient between the
given points on the cinder
cone. Include units in your
answers.
a. A and B
b. B and C
c. A and C
408
Chapter 8
Linear Functions
Cinder Cone Cross Section
600
Elevation (feet)
The island shown above is
a cinder cone in Crater Lake
National Park, Oregon.
C
B
400
200
0
A
0
0.1 0.2 0.3 0.4
Horizontal distance (miles)
Page 6 of 8
37. Roads The grade of a road is its slope written as a percent. A warning
sign must be posted if a section of road has a grade of at least 8% and is
more than 750 feet long.
a. Interpret and Apply A road rises 63 feet over a horizontal distance
of 840 feet. Should a warning sign be posted? Explain your thinking.
b. Critical Thinking The grade of a section of road that stretches over a
horizontal distance of 1000 feet is 9%. How many feet does the road
rise over that distance?
38. Horseback Riding A riding instructor takes students on mountain
trails. The instructor wants to avoid steep trails. On the steepest part of
trail A, the path rises 15 feet over a horizontal distance of 50 feet. On
the steepest part of trail B, the path rises 30 feet over a horizontal
distance of 75 feet. Which trail should the instructor take? Explain.
39. Logical Reasoning Choose three different pairs of points on the given
line, and find the slope of the line using each pair. What conclusion can
you draw from your results?
a.
b.
y
3
2
y
E
Q
D
R
C
4
B
A
O
1
2
3
4
4 3 2
5 x
2
3
3
2
1
O
2
3
4
5 x
S
2
3
T
40. Challenge Without graphing, choose a point P so that the slope of the
1
line through (1, 1) and P is .
9
Mixed Review
Solve the equation. Check your solution. (Lessons 2.5, 2.6)
41. x 7 5
42. x 3 21
43. 3y 33
m
10
44. 5
Find the greatest common factor of the numbers. (Lesson 4.2)
45. 15, 48
46. 64, 56
47. 105, 125
48. 121, 132
Find the intercepts of the equation’s graph. Then graph the equation.
(Lesson 8.3)
49. 2x y 2
Standardized Test
Practice
50. 9x 2y 18
51. 3x 4y 24
52. Multiple Choice What is the slope of the line that passes through the
points (1, 14) and (5, 4)?
A. 3
1
3
B. 1
3
C. D. 3
53. Multiple Choice The slope of a line through the point (0, 0) is 2.
Which point is also on the line?
F. (4, 2)
G. (2, 4)
H. (2, 4)
Lesson 8.4
I. (2, 4)
The Slope of a Line
409
Page 7 of 8
First Pages
Parallel, Perpendicular,
and Skew Lines
Review these
topics in preparation
for solving problems
that involve parallel
and perpendicular
lines in Lesson 8.5.
Parallel Lines
▼
Two lines are parallel lines if they lie in the same plane and do not intersect.
The symbol is used to state that two lines are parallel. Triangles ( ) are used
in a diagram to indicate that lines are parallel. In the diagram below, lines t
and v are parallel.
v
t
H
Example Name one pair of parallel lines that lie in plane P.
Because lines a and c are marked as being
parallel, you know that a c.
a
b
P
c
Perpendicular Lines
Two lines are perpendicular lines if they intersect to form a right angle. The
symbol ∏ is used to state that two lines are perpendicular. In the diagram,
lines m and n are perpendicular.
m
B
n
Example Name two lines that are perpendicular to line f.
Because lines g and j intersect line f at right
angles, you know that g ∏ f and j ∏ f.
f
j
R
410
Chapter 8
1
ChapterFunctions
Linear
Title
g
h
Page 8 of 8
First Pages
Skew Lines
r
Two lines are skew lines if they do not lie in
the same plane and do not intersect. In the
diagram, lines r and s are skew lines.
Q
s
u
Example Name two lines that are skew.
Lines u and w are skew. Note that lines u
and v are not skew because they intersect.
v
w
Y
Checkpoint
Test your
knowledge of
parallel,
perpendicular, and
skew lines by solving
these problems.
Tell whether the lines are parallel or perpendicular.
1. Lines a and b
a
2. Lines a and c
b
F
3. Lines d and b
c
d
4. Lines c and d
Tell whether the lines are skew. Explain.
j
5. Lines k and m
k
6. Lines k and j
7. Lines j and m
m
H
In Exercises 8–10, use the radio shown. The
radio has the shape of a box with rectangular
sides. Consider the antenna and each edge of
the radio as part of a line.
8. Name three lines perpendicular to ^&
GE*(.
J
B
A
D
H
C
9. Name two lines parallel to ^&
AC*(.
F
^&*(.
10. Name two lines that are skew to CD
Student Reference
G
E
Parallel, Perpendicular, and Skew Lines
411