1. 5-4 Rates of Change and Slope Warm Up Warm Up Lesson Presentation Lesson Presentation Problem of the Day Problem of the Day Lesson Quizzes Lesson Quizzes

2. 5-4 Rates of Change and Slope Warm Up Tell whether y = 2x + 4 is a proportional relationship. Then graph the function. no

3. 5-4 Rates of Change and Slope Problem of the Day What two 3-digit numbers have a product of 19,019? 133 and 143

4. 5-4 Rates of Change and Slope MA.7.A.1.4 Graph proportional relationships and identify the unit rate as the slope of the related linear function. Sunshine State Standards

5. 5-4 Rates of Change and Slope Vocabulary rate of change slope

6. 5-4 Rates of Change and Slope The ratio of two quantities that change, such as distance and time, is a rate of change. In a proportional relationship, you can use two points on a graph to find the rate of change.

7. 5-4 Rates of Change and Slope The graph shows the distance a monarch butterfly travels over time. What is the butterfly’s rate of change? Additional Example 1: Using a Graph to Find Rate of Change The relationship is proportional because the graph is a straight line that goes through the origin. Use two points on the graph to find the rate of change.

8. 5-4 Rates of Change and Slope Additional Example 1 Continued Choose two, such as (1, 20) and (2, 40). rate of change = change in distance ‏ change in time ‏ = (40 – 20) (2 – 1) The rate of change the butterfly travels at is 20 miles per hour. 20 1 =

9. 5-4 Rates of Change and Slope Check It Out: Example 1 The graph shows the distance a loggerhead turtle travels over time. What is the loggerhead turtle’s rate of change? Rate of change = ; The rate of change of the loggerhead turtle is 15 miles per hour. 15 1 y 40 20 x 60 2 4 6 8 0 10 30 50 Distance (mi) Time (hr)

10. 5-4 Rates of Change and Slope 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 variable rate of change is not a straight line.

11. 5-4 Rates of Change and Slope Tell whether each graph shows a constant or variable rate of change. A.B. Additional Example 2: Identifying Rates of Change in Graphs The graph is not a line, so the rate of change is variable. The graph is a line, so the rate of change is constant.

12. 5-4 Rates of Change and Slope Check It Out: Example 2 Tell whether each graph shows a constant or variable rate of change. A.B. The graph is a line so the rate of change is constant. The graph is not a line so The rate of change is variable.

13. 5-4 Rates of Change and Slope The slope of a line is the rate of change between any two points on the line. y The change in y is sometimes called the rise and the change in x is sometimes called the run. If a line rises from left to right, its slope is positive. If a line falls from left to right, its slope negative. In linear functions slope is a constant rate of change. x Run Rise

14. 5-4 Rates of Change and Slope Tell whether the slope is positive or negative. Then find the slope. Additional Example 3A: Identifying the Slope of the Line The line rises from left to right. The slope is positive.

15. 5-4 Rates of Change and Slope Tell whether the slope is positive or negative. Then find the slope. Additional Example 3A Continued The rise is 3. The run is 3. slope = rise run = 3333 =1 3 3

16. 5-4 Rates of Change and Slope Tell whether the slope is positive or negative. Then find the slope. Additional Example 3B: Identifying the Slope of the Line The line falls from right to left. The slope is negative. 0 2 2 –2 y x

17. 5-4 Rates of Change and Slope Tell whether the slope is positive or negative. Then find the slope. Additional Example 3B Continued The rise is 2. The run is -3. slope = rise run = 2 -3 2 0 2 2 –2 y x

18. 5-4 Rates of Change and Slope Check It Out: Example 3 negative; slope = - 4. Tell whether the slope is positive or negative. Then find the slope.

19. 5-4 Rates of Change and Slope Use the slope  and the point (1, –1) to graph the line. Additional Example 4A: Using Slope and a Point to Graph a Line 2121 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. y x –4 4 4 0 2 2 –2 = or rise run -2 1 2 ● ●

20. 5-4 Rates of Change and Slope Use the slope and the point (–1, –1) to graph the line. Additional Example 4B: Using Slope and a Point to Graph a Line 1212 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. y x –4 4 4 0 2 2 –2 = rise run 1 2 ●

21. 5-4 Rates of Change and Slope Use the slope and the point to graph the line on the given coordinate plane. Check It Out: Example 4 y x –4 4 4 0 2 2 –2 ; 1 2 (2, 1) ; 2 3 (3, 0) A. B. 4a 4b

22. 5-4 Rates of Change and Slope Standard Lesson Quiz Lesson Quizzes Lesson Quiz for Student Response Systems

23. 5-4 Rates of Change and Slope Lesson Quiz: Part I 1. Tell whether the graph shows a constant or variable rate of change. variable

24. 5-4 Rates of Change and Slope Lesson Quiz: Part II 2. Tell whether the slope is positive or negative. Then find the slope. negative; -1

25. 5-4 Rates of Change and Slope Lesson Quiz: Part III 3. Use the slope and the point (–2, –3) to graph the line. 1212

26. 5-4 Rates of Change and Slope 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 Lesson Quiz for Student Response Systems

27. 5-4 Rates of Change and Slope 2. Use the slope and the point (–2, –3) to identify the graph of the line. A. B. Lesson Quiz for Student Response Systems 1414

28. 5-4 Rates of Change and Slope 3. Which of the following graphs represents a variable rate of change? A. B. Lesson Quiz for Student Response Systems