1. Warm Up
Problem of the Day
Lesson Presentation
Lesson Quizzes

2. Warm Up
Graph the linear function.
y = 2x + 4

3. What two 3-digit numbers have a product of 19,019?
Problem of the Day
What two 3-digit numbers have a product of 19,019?
133 and 143

4. Learn to determine the slope of a line and to recognize constant and variable rates of change.

5. Vocabulary
slope
rate of change

6. The slope of a line is a measure of its steepness and is the ratio of rise to run:y
Run
rise
Rise
slope = x
run
If a line rises from left to right, its slope is positive. If a line falls from left to right, its slope negative.

7. 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.

8. 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
run = 3 1

9. Additional Example 1B: Identifying the Slope of the Line
Tell whether the slope is positive or negative. Then find the slope. 2 –2 y x
The line falls from right to left.
The slope is negative.

10. Additional Example 1B Continued
Tell whether the slope is positive or negative. Then find the slope. 2 –2 y x -3 2
The rise is 2. The run is -3.
slope =
rise
run = 2 -3

11. Check It Out: Example 1A
Tell whether the slope is positive or negative. Then find the slope.
The line does not point upward or downward so it is not positive or negative.

12. Check It Out: Example 1A Continued
Tell whether the slope is positive or negative. Then find the slope.
M(1, –1) N(3, –1)‏ 2
The rise is 0. The run is 2.
slope =
rise
run = 2

13. 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.
The line falls from left to right.
The slope is negative.
(–2, 4)‏ –2 8
The rise is 8. The run is –2.
slope =
rise
run = 8 –2 –4
(0, –4)‏

14. You can graph a line if you know its slope and one of its points.

15. Additional Example 2A: Using Slope and a Point to Graph a Line
2 1
Use the slope  and the point (1, –1) to graph the line. y = or
rise
run -2 1 2 -1 4 2
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. ● x –4 –2 2 4 –2 ● –4

16. 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!

17. Additional Example 2B: Using Slope and a Point to Graph a Line
1 2
Use the slope and the point (–1, –1) to graph the line. y =
rise
run 1 2 4 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. x ● –4 –2 2 4 –2 –4

18. Use the slope – and the point (2, 0) to graph the line.
Check It Out: Example 2A
2 3
Use the slope – and the point (2, 0) to graph the line. y = or
rise
run -2 3 2 -3 4 ● 2
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. x –4 –2 2 4 –2 ● –4

19. Use the slope and the point (–2, 0) to graph the line.
Check It Out: Example 2B
1 4
Use the slope and the point (–2, 0) to graph the line. y =
rise
run 1 4 4 2
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. ● x –4 –2 2 4 –2 –4

20. 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.

21. 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.

22. Tell whether each graph shows a constant or variable rate of change.
Check It Out: Example 3
Tell whether each graph shows a constant or variable rate of change.
A. B. y y 4 4 2 2 x x –4 –2 2 4 –4 –2 2 4 –2 –2 –4 –4
The graph is linear, so
the rate of change is
constant.
The graph is nonlinear,
so the rate of change is variable.

23. Additional Example 4: Using Rate of Change to Solve Problems
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.

24. 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.
rise (distance)‏
run (time)‏
20 miles
1 hour
slope (speed) = =
The butterfly travels at a rate of 20 miles per hour.

25. 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? 7 14 21 28 35
Time (min)‏ 1 2 3 4 5 6
Distance (mi)‏ 6 5 4
Distance (mi)‏ 3 2 1 7 14 21 28 35
Time (min)‏

26. 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.
rise (distance)‏
run (time)‏
1 mi
7 min
slope (speed) = =
The jogger travels at a rate of 1 mile every 7
minutes.

27. Lesson Quizzes
Standard Lesson Quiz
Lesson Quiz for Student Response Systems

28. Lesson Quiz: Part I
1. Tell whether the slope is positive or negative. Then find the slope.
Negative; -1

29. Lesson Quiz: Part II 1 2
2. Use the slope and the point (–2, –3) to graph the line.

30. Lesson Quiz: Part III
3. Tell whether the graph shows a constant or variable rate of change.
variable

31. Lesson Quiz for Student Response Systems
1. Tell whether the slope is positive or negative. Then identify the slope.
A. positive; 1
B. positive; 2
C. negative; –1
D. negative; –2

32. Lesson Quiz for Student Response Systems
1 4
2. Use the slope and the point (–2, –3) to
identify the graph of the line.
A B.

33. Lesson Quiz for Student Response Systems
3. Which of the following graphs represents a variable rate of change?
A B.