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Do NowGraph the linear function.
y = 2x + 4
Course 2
5-3 Slope and Rates of Change
Hwk: p 44
EQ: How do I determine the slope of a line and to recognize constant and variable rates of change?
Course 2
5-3 Slope and Rates of Change
M7A3.a Plot points on a coordinate plane; M7A3.b Represent, describe, and analyze relations from tables, graphs, and formulas
Vocabulary
slope
Insert Lesson Title Here
Course 2
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
If a line points downward as x increases, its slope is negative. If a line points upward as x increases, its slope is positive.
x
Run
Rise
Course 2
5-3 Slope and Rates of Change
Tell whether the slope is positive or negative. Then find the slope.
Additional Example 1A: Identifying the Slope of the Line
Course 2
5-3 Slope and Rates of Change
The line rises from left to right.
The slope is positive.
Tell whether the slope is positive or negative. Then find the slope.
Additional Example 1A Continued
The rise is 3. The run is 3.
slope = riserun
= 33
= 1
3
3
Course 2
5-3 Slope and Rates of Change
Tell whether the slope is positive or negative. Then find the slope.
Additional Example 1B: Identifying the Slope of the Line
Course 2
5-3 Slope and Rates of Change
The line falls from right to left.
The slope is negative.
0 2
2
–2–2
y
x
Tell whether the slope is positive or negative. Then find the slope.
Additional Example 1B Continued
The rise is 2. The run is -3.
slope = riserun
= 2 -3
2
-3
Course 2
5-3 Slope and Rates of Change
0 2
2
–2–2
y
x
Check It Out: Example 1A
The line does not point upward or downward so it is not positive or negative.
Course 2
5-3 Slope and Rates of Change
Tell whether the slope is positive or negative. Then find the slope.
Check It Out: Example 1A Continued
The rise is 0. The run is 2.
slope = riserun
= 02
= 0
M(1, –1) N(3, –1)
2
Course 2
5-3 Slope and Rates of Change
Tell whether the slope is positive or negative. Then find the slope.
Check It Out: Example 1B
Insert Lesson Title Here
(0, –4)
(–2, 4)
8
–2
The rise is 8. The run is –2.
slope = riserun
= 8–2 = –4
Tell whether the slope is positive or negative. Then find the slope.
The line falls from left to right.
The slope is negative.
Course 2
5-3 Slope and Rates of Change
You can graph a line if you know its slope and one of its points.
Course 2
5-3 Slope and Rates of Change
Use the slope and the point (1, –1) to graph the line.
Additional Example 2A: Using Slope and a Point to Graph a Line
2 1
From point (1, 1) move 2units down and 1 unitright, 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
x–4
–4
4
4
0 2
2
–2–2
= orriserun
-2 1
2-1
●
●
Course 2
5-3 Slope and Rates of Change
Course 2
5-3 Slope and Rates of Change
You can write an integer as a fraction by putting the integer in the numerator of the fraction and a 1 in the denominator.
Remember!
Use the slope and the point (–1, –1) to graph the line.
Additional Example 2B: Using Slope and a Point to Graph a Line
1 2
From point (–1, –1) move 1unit up and 2 units right. Mark the point where you end up, and draw a line through the two points.
y
x–4
–4
4
4
0 2
2
–2–2
=riserun
1 2
●
Course 2
5-3 Slope and Rates of Change
Use the slope and the point (2, 0) to graph the line.
Check It Out: Example 2A
2 3
From point (2, 0) move 2units down and 3 unitsright, 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
x–4
–4
4
4
0 2
2
–2–2
= orriserun
-2 3
2-3
●
●
Course 2
5-3 Slope and Rates of Change
Use the slope and the point (–2, 0) to graph the line.
Check It Out: Example 2B
1 4
From point (–2, 0) move 1unit up and 4 units right. Mark the point where you end up, and draw a line through the two points.
y
x–4
–4
4
4
0 2
2
–2–2
=riserun
1 4
●
Course 2
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. 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.
Course 2
5-3 Slope and Rates of Change
Tell whether each graph shows a constant or variable rate of change.
A. B.
Additional Example 3: Identifying Rates of Change in Graphs
The graph is nonlinear,so the rate of change is variable.
The graph is linear, so the rate of change isconstant.
Course 2
5-3 Slope and Rates of Change
Check It Out: Example 3
Insert Lesson Title Here
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 isconstant.
y
x–4
–4
4
4
0 2
2
–2–2
y
x–4
–4
4
4
0 2
2
–2–2
Course 2
5-3 Slope and Rates of Change
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.
Additional Example 4: Using Rate of Change to Solve Problems
Course 2
5-3 Slope and Rates of Change
Additional Example 4 Continued
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.
slope (speed) = rise (distance)run (time) = 20miles
1 hour
The butterfly travels at a rate of 20 miles per hour.
Course 2
5-3 Slope and Rates of Change
Check It Out: Example 4The graph shows the distance a jogger travels over time. Is he traveling at a constant or variable rate. How fast is he traveling?
Insert Lesson Title Here
7 14 21 28 35Time (min)
1
2
3
4
5
6
Dis
tan
ce (
mi)
7 14 21 28 35Time (min)
1
2
3
4
5
6
Dis
tan
ce (
mi)
1
7
7
1
Course 2
5-3 Slope and Rates of Change
Check It Out: Example 4 Continued
Insert Lesson Title Here
The graph is a line, so the jogger is traveling at a constant rate of speed.
slope (speed) = rise (distance)run (time) = 1 mi
7 min
The jogger travels at a rate of 1 mile every 7minutes.
The amount of distance is the rise, and the amount of time is the run. You can find the speed by finding the slope.
Course 2
5-3 Slope and Rates of Change
TOTD
1. Tell whether the slope is positive or negative. Then find the slope.
Negative; -1
Insert Lesson Title Here
Course 2
5-3 Slope and Rates of Change
TOTD
2. Use the slope and the point (–2, –3) to graph
the line.
Insert Lesson Title Here
12
Course 2
5-3 Slope and Rates of Change
TOTD
3. Tell whether the graph shows a constant or
variable rate of change.
Insert Lesson Title Here
Course 2
5-3 Slope and Rates of Change
variable
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