Page 1 of 5
Slope
BEFORE
Now
WHY?
You used a table to graph a
linear function.
You’ll find the slope of a line.
So you can compare pay rates for
baby-sitting, as in Ex. 16.
You can use ratios to describe the slope of a line.
Word Watch
1 Draw an x-axis and a y-axis on graph paper.
y
4
3
2
1
slope, p. 378
2 Start at the origin. Move 2 units to the right
and 3 units up. Plot this point and label it A.
A
rise 3
O run 2 3
3 Start at A. Move 4 units to the right and
6 units up. Plot this point and label it B.
4
5
6
7 x
4 Draw a line through A, B, and the origin.
with
Solving
Run is positive when
moving to the right and
negative when moving to
the left.
Rise is positive when
moving up and negative
when moving down.
The slope of a nonvertical line is the ratio of the rise (vertical change)
to the run (horizontal change) between any two points on the line, as
shown below. A line has a constant slope.
NO PRINT ZONE
5 Find the ratio of rise to run for each of the movements described in Steps 2
and 3. What do you notice?
y
4
3
2
1
rise
2
2
slope ru n
5
5
run 5
rise 2
x
O
1
2
3
4
5
6
7
8
Examples of lines with positive, negative, and zero slopes are shown
below. The slope of a vertical line is undefined.
A line that rises from
left to right has a
positive slope.
A line that falls from
left to right has a
negative slope.
y
y
378
Chapter 8
Ratios and Proportions
x
O
y
x
O
x
PAGE 378
O
A horizontal line
has a slope of 0.
Page 2 of 5
1
EXAMPLE
Finding the Slope of a Line
To find the slope of a line, find the ratio of the rise to the run between
two points on the line.
a.
b.
y
O
y
4
3
6
5
4
3
2
3
2
3 2
4
1
2
3
rise
ru n
4
5
6
3
O
3 4
2
7 x
3
3
slope 2
4
slope rise
ru n
1
2
1
NO PRINT ZONE
Slope as a Rate When the graph of a line represents a real-world
situation, the slope of the line can often be interpreted as a rate.
Science
EXAMPLE
2
Interpreting Slope as a Rate
Volcanoes The graph represents the
distance traveled by a lava flow over time.
To find the speed of the lava flow, find the
slope of the line.
6 mi
4h
Write rise over run.
1.5 mi
1h
Find unit rate.
Lava Flow
Distance (mi)
rise
ru n
slope ANSWER The speed of the lava flow is
1.5 miles per hour.
■
8
7
6
5
4
3
2
1
0
(4, 6)
0 1 2 3 4 5 6
Time (h)
Volcanoes
The molten lava that erupts
from volcanoes can be as
hot as 1200C. Find this
temperature in degrees
Fahrenheit using the
PAGE 379
5 x
9
5
equation F C 32.
Your turn now
Complete the following exercises.
1. Plot the points (3, 4) and (6, 3). Then find the slope of the line that
passes through the points.
2. In Example 2, suppose the line starts at the origin and passes
through the point (3, 6). Find the speed of the lava flow.
Lesson 8.3
Slope
379
Page 3 of 5
EXAMPLE
3
Using Slope to Draw a Line
Draw the line that has a slope of 3 and passes through (2, 5).
1 Plot (2, 5).
y
6
5
4
3
2
1
2 Write the slope as a fraction.
rise
ru n
3
1
slope 3 Move 1 unit to the right and 3 units
down to plot the second point.
1
3
O
1
2
3
5 x
4 Draw a line through the two points.
Your turn now
Refer to Example 3.
NO PRINT ZONE
1
3. Draw the line that has a slope of and passes through (2, 5).
3
INTERNET
Exercises
eWorkbook Plus
CLASSZONE.COM
More Practice, p. 712
Getting Ready to Practice
1. Vocabulary Copy and complete: The slope of a nonvertical line is the
ratio of the _?_ to the _?_ between any two points on the line.
Find the slope of the line.
O
y
2
5
3
1
2
3
3
4 5 x
3
5. How far did you travel in 2 hours?
380
Chapter 8
Ratios and Proportions
y
O
2
3
2
1
x
2
x
O
4
Canoe Trip The graph represents the distance
that you traveled in a canoe over time.
6. What was your speed?
4.
2
1
5
4
3
2
1
0
1
0
2
3
4 5
1
2
Time (h)
3
PAGE 380
3.
y
3
2
1
Distance (mi)
2.
Page 4 of 5
Practice and Problem Solving
with
Example
1
2
3
Homework
Exercises
7–12, 18–23
14–16, 27–28
24–26
Without finding the slope of the line, tell whether the slope is
positive, negative, or zero.
7.
8.
y
2
2
Online Resources
O
2
3 x
2
CLASSZONE.COM
9.
y
y
4
4
2
1
2
1
O
1
2
3
4
3 2
5 x
• More Examples
• eTutorial Plus
O
1
2 x
1
2 x
Find the slope of the line.
10.
2
NO PRINT ZONE
11.
y
12.
y
y
4
3
4
3
4
1
2
1
2
1
O
1
2 x
2
O
1
3 2
2 3 x
O
13. Describe You are given a graph that shows the distance you travel over
Baby-Sitting In Exercises 14–16, use
the graph, which shows two pay rates
for baby-sitting.
14. Find Joe’s pay rate for baby-sitting.
15. Find Melanie’s pay rate for baby-sitting.
Dollars
time while walking at a steady rate. Describe how to use the graph to
find the rate at which you are walking.
24
20
16
12
8
4
0
Joe
Melanie
0
1
16. Who has a greater pay rate?
2
3
Time (h)
4
17. Critical Thinking Give an example of a rate that can be represented
as a line in a coordinate plane. Explain your choice.
Draw the graph of the line that passes through the points. Then
find the slope of the line.
PAGE 381
with
18. (3, 4), (5, 6)
19. (0, 0), (8, 7)
20. (2, 5), (5, 2)
21. (7, 2), (1, 2)
22. (1, 3), (3, 4)
23. (1, 5), (1, 1)
Review
Need help with plotting
points in a coordinate
plane? See p. 293.
Draw the line that has the given slope and passes through the
given point.
24. slope 3; (3, 1)
3
5
25. slope ; (2, 1) 26. slope ; (5, 5)
4
6
Lesson 8.3
Slope
381
Page 5 of 5
Extended Problem Solving In Exercises 27
and 28, use the graph showing the
distance covered by two sea turtles.
Distance (ft)
27. Identify Just by looking at the graphs,
tell which turtle swam at a greater
speed. Explain your choice.
28. Compare A third turtle swims 18 feet
Green sea turtle
Sea Turtles
in 8 seconds. How does the swimming
rate of the third turtle compare with
the rates of the other two turtles?
Explain your reasoning.
35
30
25
20
15
10
5
0
Leatherback
sea turtle
Green
sea turtle
0 2 4 6 8 10 12 14 16
Time (sec)
29. Copy and complete the table.
Side length
1
2
3
4
Perimeter
? ? ?
x-coordinates are the side lengths and
y-coordinates are the perimeters. Draw a line through the points.
?
30. Graph your results as points whose
31. Challenge Explain what the slope of the line in Exercise 30 tells about
the relationship between the side length of a square and its perimeter.
Mixed Review
NO PRINT ZONE
Geometry In Exercises 29–31, use the table which lists the side
lengths of four squares.
Solve the equation. Check your solution. (Lesson 7.4)
32. 4x 20
33. 36x 6
x
34. 3.2
12
x
35. 7
11.1
$5.60
36. Write as a unit rate. (Lesson 8.2)
7 lb
Basic Skills Write the decimal as a fraction in simplest form.
37. 0.4
38. 0.75
39. 0.05
40. 0.32
Test-Taking Practice
41. Multiple Choice What is the slope of the blue line?
State Test Practice
CLASSZONE.COM
3
2
A. 2
3
B. 2
3
C. y
4
3
2
D. 42. Multiple Choice What is the slope of the red line?
1
O
3
F. 2
382
Chapter 8
Ratios and Proportions
2
G. 3
2
H. 3
3
I. 2
1
3 x
PAGE 382
INTERNET