5-3 Slope and Rates of Change
5-3 Slope and Rates of Change
Learn to determine the slope of a line
and to recognize constant and variable
rates of change.
5-3 Slope and Rates of Change
The slope of a line is a measure of its steepness
and is the ratio of rise to run:
y
Run
Rise
x
5-3 Slope and Rates of Change
Whenever you calculate a slope the sign will tell you
what direction the line is moving:
If a line rises from left to right, its slope is positive.
If a line falls from left to right, its slope is negative.
5-3 Slope and Rates of Change
Example A
Tell whether the slope is positive or negative.
Then find the slope.
The line falls from left to right.
The slope is negative.
The rise is 4. The run is -2.
slope = rise = 4 = -2
-2
run
5-3 Slope and Rates of Change
Example A
Tell whether the slope is positive or negative.
Then find the slope.
The line rises from left to right.
The slope is positive.
The rise is 2. The run is 3.
slope = rise = 2
3
run
5-3 Slope and Rates of Change
You Try 1
Tell whether the slope is positive or negative.
Then find the slope.
The line rises from left to right.
The slope is positive.
5-3 Slope and Rates of Change
You Try 1 continued
Tell whether the slope is positive or negative.
Then find the slope.
3
3
The rise is 3. The run is 3.
slope = rise = 3 = 1
3
run
5-3 Slope and Rates of Change
You Try 2
Tell whether the slope is positive or negative.
Then find the slope.
y
2
–2 0
–2
2
x
The line falls from right to left.
The slope is negative.
5-3 Slope and Rates of Change
You Try 2 continued
Tell whether the slope is positive or negative.
Then find the slope.
y
2
–2 0
–2
-3
2
2
x
The rise is 2. The run is -3.
slope =
rise = 2
run
-3
5-3 Slope and Rates of Change
You Try 3
Tell whether the slope is positive or negative.
Then find the slope.
The line does not point upward or downward so it
is not positive or negative.
5-3 Slope and Rates of Change
You Try 3 continued
Tell whether the slope is positive or negative.
Then find the slope.
2
M(1, –1) N(3, –1)
The rise is 0. The run is 2.
slope = rise = 0 = 0
2
run
5-3 Slope and Rates of Change
You Try 4
Tell whether the slope is positive or negative.
Then find the slope.
The line falls from left to
right.
(–2, 4)
–2
The slope is negative.
8
(0, –4)
The rise is 8. The run is –2.
slope = rise = 8 = –4
–2
run
5-3 Slope and Rates of Change
The ratio of two quantities that change, such
as slope, is a rate of change.
A constant rate of change describes changes
of the same amount during equal intervals. It
is displayed on graphs as linear (a line).
A variable rate of change describes changes
of a different amount during equal intervals.
It is displayed on graphs as nonlinear (not
a line).
5-3 Slope and Rates of Change
Example C
Tell whether each graph shows a constant or
variable rate of change.
A.
B.
The graph is nonlinear,
so the rate of change is
variable.
The graph is linear, so
the rate of change is
constant.
5-3 Slope and Rates of Change
Example C continued
Tell whether each graph shows a constant or
variable rate of change.
A.
y
B.
4
2
–4 –2 0
–2
2
4
x
–4
The graph is nonlinear,
so the rate of change is
variable.
y
4
2
–4 –2 0
–2
2
4
x
–4
The graph is linear, so
the rate of change is
constant.
5-3 Slope and Rates of Change
Example D
The graph shows the distance a monarch
butterfly travels overtime. Tell whether the
graph shows a constant or variable rate of
change. Then find how fast the butterfly is
traveling.
5-3 Slope and Rates of Change
Example D conitnued
The graph is a line, so the butterfly is traveling
at a constant rate of speed.
The amount of distance is the rise, and the
amount of time is the run. You can find the
speed by finding the slope.
(distance) = 20 miles
slope (speed) = rise
run (time)
1 hour
The butterfly travels at a rate of 20 miles per hour.
5-3 Slope and Rates of Change
6
6
5
5
Distance (mi)
Distance (mi)
You Try
The graph shows the distance a jogger travels
over time. Is he traveling at a constant or
variable rate. How fast is he traveling?
4
3
2
1
7 14 21 28 35
Time (min)
7
4
1
3
2
1
7
1
7 14 21 28 35
Time (min)
5-3 Slope and Rates of Change
You Try continued
The graph is a line, so the jogger is traveling at
a constant rate of speed.
The amount of distance is the rise, and the
amount of time is the run. You can find the
speed by finding the slope.
(distance) = 1 mi
slope (speed) = rise
run (time)
7 min
The jogger travels at a rate of 1 mile every 7
minutes.
5-3 Slope and Rates of Change
Lesson Quiz
1. Tell whether the slope is positive or negative.
Then find the slope.
Negative; -1
5-3 Slope and Rates of Change
Lesson Quiz
3. Tell whether the graph shows a constant or
variable rate of change.
variable
5-3 Slope and Rates of Change
Lesson Quiz
3. Which of the following graphs represents a
variable rate of change?
A.
B.