1. CONFIDENTIAL 1 Algebra1 Rate of Change and Slope

2. CONFIDENTIAL 2 Warm Up Find the domain and range of each relation, and tell whether the relation is a function. 1) {(3, 4), (3, 2), (3, 0), (3, -2)} 2) x024-2-4 Y02424

3. CONFIDENTIAL 3 Rate of Change and Slope 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. Rate of change = Change in dependent variable Change in independent variable Rates of change can be used to find how quickly costs have increased.

4. CONFIDENTIAL 4 The table shows the cost of mailing a 1-ounce letter in different years. Find the rate of change in cost for each time interval. During which time interval did the cost increase at the greatest rate? Step1: Identify the dependent and independent variables. dependent: cost independent: year Consumer Application Year19851988199019912004 Cost (¢)2225 2937 Step2: Find the rates of change. 1985 to 1988 change in cost = _25 -22_ = 3 = 1 change in years 1988 – 1985 3 1 cent/year Next slide 

5. CONFIDENTIAL 5 1988 to 1990 change in cost = _25 -25_ = 0 = 0 change in years 1990 – 1988 2 1990 to 1991 change in cost = _29 -25_ = 4 = 4 change in years 1991 – 1990 1 1991 to 2004 change in cost = _37 -29_ = 8 = 0 change in years 2004 – 1991 13 The cost increased at the greatest rate from 1990 to 1991. 0 cent/year 4 cents/year 8/13 cents/year

6. CONFIDENTIAL 6 Finding Rates of Change from a Graph Graph the data from Example 1 and show the rates of change. Graph the ordered pairs. The vertical blue segments show the changes in the dependent variable, and the horizontal green segments show the changes in the independent variable. Notice that the greatest rate of change is represented by the steepest of the red line segments. Also notice that between 1988 and 1990, when the cost did not change, the red line segment is horizontal.

7. CONFIDENTIAL 7 Now you try! 1) The table shows the balance of a bank account on different days of the month. Find the rate of change during each time interval. During which time interval did the balance decrease at the greatest rate? Day16162230 Balance ($)550285210 175

8. CONFIDENTIAL 8 Now you try! 2) Graph the data from the given table and show the rates of change. Day16162230 Balance ($)550285210 175

9. CONFIDENTIAL 9 If all of the segments have the same rate of change, then they all have the same steepness and together form a straight line. The constant rate of change of a line is called the slope of the line. The rise is the difference in the y-values of two points on a line. The run is the difference in the x-values of two points on a line. The slope of a line is the ratio of rise to run for any two points on the line. Slope of a Line slope = rise = Change in y run Change in x Next slide 

10. CONFIDENTIAL 10 (Remember that y is the dependent variable and x is the independent variable.)

11. CONFIDENTIAL 11 Finding Slope Find the slope of the line. Begin at one point and count vertically to find the rise. Then count horizontally to the second point to find the run. It does not matter which point you start with. The slope is the same. slope = 2 = 2 1 slope = -2 = 2 -1

12. CONFIDENTIAL 12 3) Find the slope of the line that contains (0, -3) and (5, -5). Now you try!

13. CONFIDENTIAL 13 Finding Slopes of Horizontal and Vertical Lines Find the slope of each line. A)B) The slope is 0.The slope is undefined. slope = rise = 0 = 0 run 4 slope = rise = 2 run 0 You cannot divide by 0.

14. CONFIDENTIAL 14 4) Find the slope of each line. Now you try! A)B)

15. CONFIDENTIAL 15 As shown in the previous examples, slope can be positive, negative, zero, or undefined. You can tell which of these is the case by looking at the graph of a line—you do not need to calculate the slope.

16. CONFIDENTIAL 16 Tell whether the slope of each line is positive, negative, zero, or undefined. A)B) The line falls from left to right. The slope is negative. The line is horizontal. The slope is 0. Describing Slope

17. CONFIDENTIAL 17 5) Tell whether the slope of each line is positive, negative, zero, or undefined. Now you try! A)B)

18. CONFIDENTIAL 18 Remember that the slope of a line is its steepness. Some lines are steeper than others. As the absolute value of the slope increases, the line becomes steeper. As the absolute value of the slope decreases, the line becomes less steep. Comparing Slope The line with slope 4 is steeper than the line with slope 1 | 4 | > 1 2 2

19. CONFIDENTIAL 19 The line with slope -2 is steeper than the line with slope -1. | -2 | > | -1 | The line with slope -3 is steeper than the line with slope 3 | -3 | >| 3 | 4 | 4 |

20. CONFIDENTIAL 20 6) What is the rise shown in the graph? What is the run? What is the slope? Now you try!

21. CONFIDENTIAL 21 Assessment 1) The table shows the volume of gasoline in a gas tank at different times. Find the rate of change for each time interval. During which time interval did the volume decrease at the greatest rate? 2) The table shows a person’s heart rate over time. Graph the data and show the rates of change.

22. CONFIDENTIAL 22 Find the slope of each line. 3)4)

23. CONFIDENTIAL 23 5) 6) Tell whether the slope of each line is positive, negative, zero, or undefined.

24. CONFIDENTIAL 24 7) The slope of which line has the greatest absolute value? 8) For which line is the run equal to 0?

25. CONFIDENTIAL 25 Rate of Change and Slope 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. Rate of change = Change in dependent variable Change in independent variable Rates of change can be used to find how quickly costs have increased. Let’s review

26. CONFIDENTIAL 26 The table shows the cost of mailing a 1-ounce letter in different years. Find the rate of change in cost for each time interval. During which time interval did the cost increase at the greatest rate? Step1: Identify the dependent and independent variables. dependent: cost independent: year Consumer Application Year19851988199019912004 Cost (¢)2225 2937 Step2: Find the rates of change. 1985 to 1988 change in cost = _25 -22_ = 3 = 1 change in years 1988 – 1985 3 1 cent/year Next slide 

27. CONFIDENTIAL 27 1988 to 1990 change in cost = _25 -25_ = 0 = 0 change in years 1990 – 1988 2 1990 to 1991 change in cost = _29 -25_ = 4 = 4 change in years 1991 – 1990 1 1991 to 2004 change in cost = _37 -29_ = 8 = 0 change in years 2004 – 1991 13 The cost increased at the greatest rate from 1990 to 1991. 0 cent/year 4 cents/year 8/13 cents/year

28. CONFIDENTIAL 28 Finding Rates of Change from a Graph Graph the data from Example 1 and show the rates of change. Graph the ordered pairs. The vertical blue segments show the changes in the dependent variable, and the horizontal green segments show the changes in the independent variable. Notice that the greatest rate of change is represented by the steepest of the red line segments. Also notice that between 1988 and 1990, when the cost did not change, the red line segment is horizontal.

29. CONFIDENTIAL 29 If all of the segments have the same rate of change, then they all have the same steepness and together form a straight line. The constant rate of change of a line is called the slope of the line. The rise is the difference in the y-values of two points on a line. The run is the difference in the x-values of two points on a line. The slope of a line is the ratio of rise to run for any two points on the line. Slope of a Line slope = rise = Change in y run Change in x Next slide 

30. CONFIDENTIAL 30 (Remember that y is the dependent variable and x is the independent variable.)

31. CONFIDENTIAL 31 Finding Slope Find the slope of the line. Begin at one point and count vertically to find the rise. Then count horizontally to the second point to find the run. It does not matter which point you start with. The slope is the same. slope = 2 = 2 1 slope = -2 = 2 -1

32. CONFIDENTIAL 32 As shown in the previous examples, slope can be positive, negative, zero, or undefined. You can tell which of these is the case by looking at the graph of a line—you do not need to calculate the slope.

33. CONFIDENTIAL 33 Tell whether the slope of each line is positive, negative, zero, or undefined. A)B) The line falls from left to right. The slope is negative. The line is horizontal. The slope is 0. Describing Slope

34. CONFIDENTIAL 34 Remember that the slope of a line is its steepness. Some lines are steeper than others. As the absolute value of the slope increases, the line becomes steeper. As the absolute value of the slope decreases, the line becomes less steep. Comparing Slope The line with slope 4 is steeper than the line with slope 1 | 4 | > 1 2 2

35. CONFIDENTIAL 35 You did a great job today!