Slope of a Line

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1. 12-2Slope of a Line
Course 3
Warm UpWarm Up
Problem of the DayProblem of the Day
Lesson PresentationLesson Presentation

2. Warm Up
Evaluate each equation for x = –1, 0,
and 1.
1. y = 3x
2. y = x – 7
3. y = 2x + 5
4. y = 6x – 2
–3, 0, 3
–8, –7, –6
3, 5, 7
Course 3
12-2Slope of a Line
–8, –2, 4

3. Problem of the Day
Write a linear equation that contains
terms with x
2
.
Possible answer: x
2
+ y = x
2
+ x + 4
Course 3
12-2Slope of a Line

4. Learn to find the slope of a line and use
slope to understand and draw graphs.
Course 3
12-2Slope of a Line

5. Insert Lesson Title Here
You looked at slope on the coordinate plane in
Lesson 7-5 (p. 347).
Remember!
Course 3
12-2Slope of a Line

6. Linear equations have constant slope. For
a line on the coordinate plane, slope is
the following ratio:
vertical change
horizontal change
change in y
change in x
=
This ratio is often referred to as , or “rise
over run,” where rise indicates the number of units
moved up or down and run indicates the number of
units moved to the left or right. Slope can be
positive, negative, zero, or undefined. A line with
positive slope goes up from left to right. A line with
negative slope goes down from left to right.
rise
run
Course 3
12-2Slope of a Line

7. Course 3
12-2Slope of a Line

8. Course 3
12-2Slope of a Line

9. If you know any two points on a line, or
two solutions of a linear equation, you
can find the slope of the line without
graphing. The slope of a line through the
points (x
1
, y
1
) and (x
2
, y
2
) is as follows:
yy
22 –– yy
11
xx
22 –– xx
11
Course 3
12-2Slope of a Line

10. Find the slope of the line that passes through
(–2, –3) and (4, 6).
Additional Example 1: Finding Slope, Given Two
Points
Let (x
1
, y
1
) be (–2, –3) and (x
2
, y
2
) be (4, 6).
6 – (–3)
4 – (–2)
Substitute 6 for y
2
, –3 for y
1
,
4 for x
2
, and –2 for x
1
.
9
6
=
The slope of the line that passes through
(–2, –3) and (4, 6) is .
3
2
=
y
2
– y
1
x
2
– x
1
3
2
=
Course 3
12-2Slope of a Line

11. Find the slope of the line that passes through
(–4, –6) and (2, 3).
Check It Out: Example 1
Let (x
1
, y
1
) be (–4, –6) and (x
2
, y
2
) be (2, 3).
3 – (–6)
2 – (–4)
Substitute 3 for y
2
, –6 for y
1
,
2 for x
2
, and –4 for x
1
.
9
6
=
The slope of the line that passes through
(–4, –6) and (2, 3) is .
3
2
=
y
2
– y
1
x
2
– x
1
3
2
=
Course 3
12-2Slope of a Line

12. Nonlinear equations have variable
rates of change. This means that the
rate of change is different between
values. This is shown in a graph by a
curved line.
Course 3
12-2Slope of a Line

13. Determine whether each graph shows a
constant or variable rate of change. Explain
your reasoning.
Additional Example 2A: Identifying Constant and
Variable Rates of Change in Graphs
Course 3
12-2Slope of a Line
The graph shows a
constant rate of
change. The slope
between any two
points is always the
same.

14. Determine whether each graph shows a
constant or variable rate of change. Explain
your reasoning.
Additional Example 2B: Identifying Constant and
Variable Rates of Change in Graphs
Course 3
12-2Slope of a Line
The graph shows a
variable rate of
change. The slope
between any two sets
of points in Quadrant
1 is different.

15. Determine whether each graph shows a
constant or variable rate of change. Explain
your reasoning.
Check It Out: Example 2A
Course 3
12-2Slope of a Line
The graph shows a
constant rate of
change. The slope
between any two
points is always the
same.

16. Determine whether each graph shows a
constant or variable rate of change. Explain
your reasoning.
Check It Out: Example 2B
Course 3
12-2Slope of a Line
The graph shows a
variable rate of
change. The slope is
steeper at the ends
than in the middle.

17. Additional Example 3: 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.
Course 3
12-2Slope of a Line
Graph the data.
Pounds
C
o
s
t
Cost of Fruit

18. You can use any two points to find the
slope of the line.
Course 3
12-2Slope of a Line
Helpful Hint

19. Additional Example 3 Continued
Find the slope of the line:
The slope of the line is 3. This means that for every
pound of fruit, you will pay another $3.
=
y
3
– y
2
x
3
– x
2
15
5
=
30 - 15
10 - 5

= 3
Course 3
12-2Slope of a Line

20. Check It Out: Example 3
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.
Course 3
12-2Slope of a Line
Graph the data.
Cost of Gas
GallonsCost
0 0
3 6
6 12
6
9
9
12
6 0
3
3
x
y
Gallons
Cost of Gas
C
o
s
t

21. Check It Out: Example 3 Continued
Find the slope of the line:
The slope of the line is 2. This means that for every
gallon of gas, you will pay another $2.
=
y
3
– y
2
x
3
– x
2
6
3
=
12 - 6
6 - 3

= 2
Course 3
12-2Slope of a Line

22. Lesson Quiz: Part I
Find the slope of the line passing through
each pair of points.
1. (4, 3) and (–1, 1)
2. (–1, 5) and (4, 2)

Insert Lesson Title Here
2
5
5
3
–
Course 3
12-2Slope of a Line

23. Lesson Quiz: Part II
3. The table shows how much money Susan earned
as a house painter for one afternoon. Use the data
to make a graph. Find the slope of the line and
explain what it shows.
Insert Lesson Title Here
Course 3
12-2Slope of a Line
x
y
64 2 8101214 0
10
20
30
40
50
60
70
80
The slope of the line is 7.
This means Susan earned
$7 for each hour worked.