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2.3 The Slope of a Line

Learn about how slope relates to the graph of a linear function and how to interpret slope. Adapted from Intermediate Algebra with Applications and Visualization 3e by Rockswold and Krieger.
by

Steve Grosteffon

on 20 February 2016

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Transcript of 2.3 The Slope of a Line

The Slope of a Line
Let’s go for a ride on the Gainesville-Hawthorne Trail. We’ll start at Boulware Springs (Mile 0) and ride out to the Lochloosa Trailhead (Mile 15) and back.
https://www.floridastateparks.org/trail/Gainesville-Hawthorne
Suppose it takes us 1 hour to get to Hawthorne but only 45 minutes to return after taking a 15-minute break.
Let's make a numerical representation to get started.
Suppose it takes us 1 hour to get to Hawthorne but only 45 minutes to return after taking a 15-minute break.
Initially we are at Mile 0.
Suppose it takes us 1 hour to get to Hawthorne but only 45 minutes to return after taking a 15-minute break.
After 1 hour we reach Mile 15.
Fifteen minutes (0.25 hours) later we are still at Mile 15.
Suppose it takes us 1 hour to get to Hawthorne but only 45 minutes to return after taking a 15-minute break.
Finally, after a total of 2 hours, we arrive back at Mile 0.
Suppose it takes us 1 hour to get to Hawthorne but only 45 minutes to return after taking a 15-minute break.
From the table we may create a line graph.
The first line segment rises from left to right.
The second is horizontal.
The third falls
from left to right.
Each line segment has a different "tilt." To quantify the tilt we use the concept of
slope
, which is the
ratio of rise over run
.
The increasing line segment rises 15 miles for 1 hour of run so its
slope
is
15 miles per hour
.
15 miles
1 hour
The horizontal line segment does not rise or fall so its
slope
is
0 miles per hour
.
The decreasing line segment falls 15 miles for 3/4 hour of run so its
slope
is (-15 miles)/(3/4 hour) =
-20 miles per hour
.
-15 miles
3/4 hour
15 mph
We may interpret the
slope
as a
rate of change
, the velocity at which we cycled.
0 mph
-20 mph
In this lesson we will learn more about slope and how it relates to the graph of a linear function.
rise
run
The graph of a linear function is a line.
The "tilt" of the line is called the slope and equals
rise
over
run
.
rise
=
y
₂ -
y

run
=
x
₂ -
x

Slope
(
x
₁,
y
₁)
y
₂ -
y

x
₂ -
x

slope =
(
x₂
,
y₂
)
Slope Man
A
positive slope
indicates that the line rises from left to right.
A
negative slope
indicates that the line falls from
left to right.
This formula is equivalent to
y
=
mx
+
b
where
m
is the slope
and
b
is the
y
-intercept
.
In a previous lesson we learned that the formula for a linear function is
f
(
x
) =
ax
+
b
where
a
is the average rate of change
and
b
is the beginning value
.
f
(
x
) =
a
x
+
b
y
=
m
x
+
b
A horizontal line has
0 slope
.
0
0
A vertical line has
undefined slope
.
The graph of a linear function is a
line
.
y
=
mx
+
b
(0,
b
)
increasing
m
>0
1
m
Plot the
y
-intercept and then use the slope to find other points on the line.
y
=
mx
+
b
y
=
m
x
+
b
y
=
mx
+
b
decreasing
m
<0
(0,
b
)
y
=
mx
+
b
y
=
m
x
+
b
1
m
Slope-Intercept Form for a Line
y
=
mx
+
b
m
is the slope
b
is the
y
-intercept
y
=
m
x
+
b
y
=
m
x
+
b
When a linear function is used to model physical quantities, the slope of its graph provides certain information.
The slope of the graph of a linear function indicates the rate at which a quantity is either increasing or decreasing.
Slope as a Rate of Change
From 1981 to 2000, average public college tuition and fees can be modeled by

f
(
x
) =
136
x
+ 772,

where
x
= 1 corresponds to 1981. The slope of the graph of
f
is
m
= 136
and indicates that, on average,
tuition and fees increased by $136 per year between 1981 and 2000
.
From 1987 to 2004 the number of federally insured banks could be approximated by
N
(
t
) =
-358.4
t
+ 13,723,

where
t
= 0 corresponds to 1987. The slope of the graph of
N
is
m
= -358.4
and indicates that, on average,
the number of federally insured banks decreased by approximately 358 per year from 1987 to 2004
.
Now it's your turn!
Try the following exercises and then watch the video solutions to check.
Determine if
f
(
x
) = 6 - 8
x
is a linear function. If it is, write it in the form
f
(
x
) =
ax
+
b
.
Find the slope of the line passing through (-3,7) and (6,-2).
Determine the slope of the line shown in the graph.
Let
f
be a linear function. Find the slope,
x
-intercept, and
y
-intercept of the graph of
f
.
The line graph shows the number of welfare beneficiaries in millions for selected years.
(a) Find the slope of each line segment.
(b) Interpret each slope as a rate of change.
Year
Welfare Beneficiaries (millions)
Starting at home, a driver travels away from home on a straight highway for 2 hours at 60 miles per hour, stops for 1 hour, and then drives home at 40 miles per hour. Sketch a graph that shows the distance between the driver and home.
Slope
can be interpreted as a
rate of change
of a quantity.
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