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Slope and Rates of Change
5-3
Vocabulary
Slope- of a line is a measure of its
steepness and is the ratio of rise
to run.
Rate of change- The ratio of two
quantities that change, such as
slope.
The slope of a line is a measure of its steepness
and is the ratio of rise to run:
y
Run
Rise
x
If a line rises from left to right, its slope is positive.
If a line falls from left to right, its slope negative.
Additional Example 1A: Identifying the Slope of the
Line
Tell whether the slope is positive or negative.
Then find the slope.
The line rises from left to right.
The slope is positive.
Additional Example 1A 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
Additional Example 1B: Identifying the Slope of the
Line
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.
Additional Example 1B 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
Check It Out: Example 1A
Tell whether the slope is positive or negative.
Then find the slope.
Check It Out: Example 1B
Tell whether the slope is positive or negative.
Then find the slope.
(–2, 4)
–2
8
(0, –4)
You can graph a line if you know its slope and
one of its points.
Additional Example 2A: Using Slope and a Point to
Graph a Line
Use the slope  2 and the point (1, –1) to graph
1
the line.
rise = -2 or 2
run
1 -1
From point (1, 1) move 2
units down and 1 unit
right, or move 2 units up
and 1 unit left. Mark the
point where you end up,
and draw a line through the
two points.
y
4
2
●
–4 –2 0
–2
–4
x
2
●
4
Remember!
You can write an integer as a fraction by putting
the integer in the numerator of the fraction and
a 1 in the denominator.
Additional Example 2B: Using Slope and a Point to
Graph a Line
Use the slope 1 and the point (–1, –1) to
graph the line.2
rise = 1
run
2
From point (–1, –1) move 1
unit up and 2 units right.
Mark the point where you
end up, and draw a line
through the two points.
y
4
2
–4 –2 0
–2
–4
●
x
2
4
Check It Out: Example 2A
Use the slope – 2 and the point (2, 0) to graph
3
the line.
rise = -2 or 2
run
3 -3
From point (2, 0) move 2
units down and 3 units
right, or move 2 units up
and 3 unit left. Mark the
point where you end up,
and draw a line through the
two points.
y
4
2
–4 –2 0
–2
–4
x
2
4
Check It Out: Example 2B
Use the slope 1 and the point (–2, 0) to graph
4
the line.
rise = 1
run
4
From point (–2, 0) move 1
unit up and 4 units right.
Mark the point where you
end up, and draw a line
through the two points.
y
4
2
–4 –2 0
–2
–4
x
2
4
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.
A variable rate of change describes changes
of a different amount during equal intervals.
The graph of a constant rate of change is a
line, and the graph of a variable rate of
change is not a line.
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.
A variable rate of change describes changes
of a different amount during equal intervals.
The graph of a constant rate of change is a
line, and the graph of a variable rate of
change is not a line.
Additional Example 3: Identifying Rates of Change in
Graphs
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.
6
6
5
5
Distance (mi)
Distance (mi)
Check It Out: Example 4
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)
Check It Out: Example 4 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)
slope (speed) = rise
run (time)