SLOPE

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SLOPE
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Slope Essential Questions
How can slope be found? How are lines
classified based on their slope? How can slope
be interpreted as a rate of change?
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Vocabulary
rate of change rise run slope
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A rate of change is a ratio that compares the
amount of change in a dependent variable to the
amount of change in an independent variable.
change in dependent variable
rate of change
change in independent variable
The rates of change for a set of data may vary or
they may be constant.
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Additional Example 1 Identifying Constant and
Variable Rates of Change in Data
Determine whether the rates of change are
constant or variable. A.
1
2
2
3
Find the difference between consecutive data
points.
x 0 1 3 5 8
y 0 2 6 10 16
2
4
4
6
Find each ratio of change in y to change in x.
The rates of change are constant.
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Additional Example 1 Identifying Constant and
Variable Rates of Change in Data
Determine whether the rates of change are
constant or variable. B.
2
1
2
3
Find the difference between consecutive data
points.
x 1 3 4 6 9
y 0 2 6 6 3
2
4
0
3
Find each ratio of change in y to change in x.
The rates of change are variable.
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Check It Out! Example 1
Determine whether the rates of change are
constant or variable. A.
1
1
1
1
Find the difference between consecutive data
points.
x 0 1 2 3 4
y 1 3 5 7 11
2
2
2
4
Find each ratio of change in y to change in x.
The rates of change are variable.
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Check It Out! Example 1
Determine whether the rates of change are
constant or variable. B.
1
3
2
3
Find the difference between consecutive data
points.
x 0 1 4 6 9
y 0 2 8 12 18
2
6
4
6
Find each ratio of change in y to change in x.
The rates of change are constant.
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To show rates of change on a graph, plot the data
points and connect them with line segments.
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The constant rate of change of a line is called
the slope of the line.
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Additional Example 2 Finding the Slope of a Line
Find the slope of the line.
(5, 4)
Begin at one point and count vertically to find
the rise.
(1, 2)
Then count horizontally to the second point to
find the run.
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Check It Out! Example 2
Find the slope of the line.
Begin at one point and count vertically to find
the rise.
(3, 2)
Then count horizontally to the second point to
find the run.
(1, 2)
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Additional Example 3 Finding a Rise or Run
Find the value of a. A.
y
slope
6
a
2a 3 6
Multiply.
2a 18
0
x
Divide both sides by 2.
a 9
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Additional Example 3 Finding a Rise or Run
Find the value of a. B.
y
4
a
-2a 1 -4
slope -
Multiply.
-2a -4
0
x
Divide both sides by -2.
a 2
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Check It Out! Example 3
Find the value of a. A.
y
slope
10
2a 5 10
a
Multiply.
2a 50
0
x
Divide both sides by 2.
a 25
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Check It Out! Example 3
Find the value of a. B.
y
6
a
3a 1 6
slope -
Multiply.
3a 6
0
x
Divide both sides by 3.
a 2
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If you know any two points on a line, you can
find the slope of the line without graphing. The
slope of a line through the points (x1, y1) and
(x2, y2) is as follows
slope
When finding slope using the ratio above, it does
not matter which point you choose for (x1, y1)
and which point you choose for (x2, y2).
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Additional Example 1 Finding Slope, Given Two
Points
Find the slope of the line that passes through B.
(1, 3) and (2, 1).
Let (x1, y1) be (1, 3) and (x2, y2) be (2, 1).
Substitute 1 for y2, 3 for y1, 2 for x2, and 1
for x1.
Simplify.
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Additional Example 1 Finding Slope, Given Two
Points
Find the slope of the line that passes through A.
(2, 3) and (4, 6).
Let (x1, y1) be (2, 3) and (x2, y2) be (4, 6).
Substitute 6 for y2, 3 for y1, 4 for x2, and
2 for x1.
Simplify.
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Additional Example 1 Finding Slope, Given Two
Points
Find the slope of the line that passes through C.
(3, 2) and (1, 2).
Let (x1, y1) be (3, 2) and (x2, y2) be (1, 2).
Substitute -2 for y2, -2 for y1, 1 for x2, and
3 for x1.
Rewrite subtraction as addition of the opposite.
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Check It Out! Example 1
Find the slope of the line that passes through A.
(4, 6) and (2, 3).
Let (x1, y1) be (4, 6) and (x2, y2) be (2, 3).
Substitute 3 for y2, 6 for y1, 2 for x2, and
4 for x1.
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Check It Out! Example 1
Find the slope of the line that passes through B.
(2, 4) and (3, 1).
Let (x1, y1) be (2, 4) and (x2, y2) be (3, 1).
Substitute 1 for y2, 4 for y1, 3 for x2, and 2
for x1.
Simplify.
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Check It Out! Example 1
Find the slope of the line that passes through C.
(3, 2) and (1, 4).
Let (x1, y1) be (3, 2) and (x2, y2) be (1, 4).
Substitute -4 for y2, -2 for y1, 1 for x2, and
3 for x1.
Simplify.
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Additional Example 2 Money Application
The table shows the total cost of fruit per pound
purchased at the grocery store. Use the data to
make a graph. Find the slope of the line and
explain what it shows.
Graph the data.
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Helpful Hint
You can use any two points to find the slope of
the line.
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Additional Example 2 Continued
Find the slope of the line
Substitute.
Multiply.
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Check It Out! Example 2
The table shows the total cost of gas per gallon.
Use the data to make a graph. Find the slope of
the line and explain what it shows.
Graph the data.
Cost of Gas Cost of Gas
Gallons Cost
0 0
3 6
6 12
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Check It Out! Example 2 Continued
Find the slope of the line
Substitute.
Multiply.
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The slope of a line may be positive, negative,
zero, or undefined. You can tell which of these
is the case by looking at the graphs of a line
you do not need to calculate the slope.
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