1. Preview
Warm Up
Lesson Presentation

2. Warm Up Graph each point on a coordinate plane. 1. (3, 4) 2. (4, 0)
3. (–2, –1)
4. (–3, 2)
5. (0, 1)
6. (2, –4)

3. Vocabulary
rate of change
rise
run
slope

4. change in dependent variable rate of change =
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.

5. 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 1 3 5 8 y 2 6 10 16 +2 +4 +4 +6
Find each ratio of change in y to change in x.
2 1
= 2
4 2
= 2
4 2
= 2
6 3
= 2
The rates of change are constant.

6. 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 2 +2 +4 +0 +3
Find each ratio of change in y to change in x.
2 2
= 1
4 1
= 4
0 2
= 0
3 3
= 1
The rates of change are variable.

7. x 1 2 3 4 y 5 7 11 Partner Share! 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 1 2 3 4 y 5 7 11 +2 +2 +2 +4
Find each ratio of change in y to change in x.
2 1
= 2
2 1
= 2
2 1
= 2
4 1
= 4
The rates of change are variable.

8. x 1 4 6 9 y 2 8 12 18 Partner Share! 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 1 4 6 9 y 2 8 12 18 +2 +6 +4 +6
Find each ratio of change in y to change in x.
2 1
= 2
6 3
= 2
4 2
= 2
6 3
= 2
The rates of change are constant.

9. To show rates of change on a graph, plot the data points and connect them with line segments.
Data in a graph is always read from left to right.
Remember!

10. The constant rate of change of a line is called the slope of the line.

11. 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.
slope = = 2 4 1
The slope of the line is . 1 2

12. Find the slope of the line.
Partner Share! 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)
slope = = 1 4
The slope of the line is 1.

13. Additional Example 3: Finding a Rise or Run
Find the value of a. A. y
slope =
rise
run 2 3
slope = = 2 3 6 a 6 a
2a = 3 ∙ 6
2a = 18
Multiply. x
Divide both sides by 2.
a = 9

14. Additional Example 3: Finding a Rise or Run
Find the value of a. B. y
slope =
rise
run
– = 1 2 a -4 –4 a
-2a = 1 ∙ -4 1 2
slope = -
-2a = -4
Multiply. x
Divide both sides by -2.
a = 2

15. Partner Share! Example 3 Find the value of a. A. slope = rise run 2 5y
slope =
rise
run 2 5
slope = = 2 5 10 a 10
2a = 5 ∙ 10 a
2a = 50
Multiply. x
Divide both sides by 2.
a = 25

16. Partner Share! Example 3 Find the value of a. B. slope = rise run = –1y
slope =
rise
run = –1 3 a –6 –6 a
3a = –1 ∙ –6 1 3
slope = -
3a = 6
Multiply. x
Divide both sides by 3.
a = 2

17. x –1 1 2 4 8 y 6 10 14 x 1 3 4 5 7 y 2 6 Lesson Review: Part I
Determine whether the rates of change are constant or variable. 1. 2. x –1 1 2 4 8 y 6 10 14
variable x 1 3 4 5 7 y 2 6
constant

18. Lesson Review: Part II
3. Find the slope of the line. 3
4. Find the value of a. –4