1. Introduction In previous lessons, we have found the slope of linear equations and functions using the slope formula,. We have also identified the slope of a line from a given equation by rewriting the equation in slope-intercept form, y = mx + b, where m is the slope of the line. By calculating the slope, we are able to determine the rate of change, or the ratio that describes how much one quantity changes with respect to the change in another quantity of the function. 1 3.3.2: Proving Average Rate of Change

2. Introduction, continued The rate of change can be determined from graphs, tables, and equations themselves. In this lesson, we will extend our understanding of the slope of linear functions to that of intervals of exponential functions. 2 3.3.2: Proving Average Rate of Change

3. Key Concepts The rate of change is a ratio describing how one quantity changes as another quantity changes. Slope can be used to describe the rate of change. The slope of a line is the ratio of the change in y-values to the change in x-values. A positive rate of change expresses an increase over time. A negative rate of change expresses a decrease over time. 3 3.3.2: Proving Average Rate of Change

4. Key Concepts, continued Linear functions have a constant rate of change, meaning values increase or decrease at the same rate over a period of time. Not all functions change at a constant rate. The rate of change of an interval, or a continuous portion of a function, can be calculated. The rate of change of an interval is the average rate of change for that period. 4 3.3.2: Proving Average Rate of Change

5. Key Concepts, continued Intervals can be noted using the format [a, b], where a represents the initial x value of the interval and b represents the final x value of the interval. Another way to state the interval is a ≤ x ≤ b. A function or interval with a rate of change of 0 indicates that the line is horizontal. Vertical lines have an undefined slope. An undefined slope is not the same as a slope of 0. This occurs when the denominator of the ratio is 0. 5 3.3.2: Proving Average Rate of Change

6. Key Concepts, continued The rate of change between any two points of a linear function will be equal. 6 3.3.2: Proving Average Rate of Change Calculating Rate of Change from a Table 1.Choose two points from the table. 2.Assign one point to be (x 1, y 1 ) and the other point to be (x 2, y 2 ). 3.Substitute the values into the slope formula. 4.The result is the rate of change for the interval between the two points chosen.

7. Key Concepts, continued The rate of change between any two points of any other function will not be equal, but will be an average for that interval. 7 3.3.2: Proving Average Rate of Change Calculating Rate of Change from an Equation of a Linear Function 1.Transform the given linear function into slope- intercept form, f(x) = mx + b. 2.Identify the slope of the line as m from the equation. 3.The slope of the linear function is the rate of change for that function.

8. Key Concepts, continued 8 3.3.2: Proving Average Rate of Change Calculating Rate of Change of an Interval from an Equation of an Exponential Function 1.Determine the interval to be observed. 2.Determine (x 1, y 1 ) by identifying the starting x-value of the interval and substituting it into the function. 3.Solve for y. 4.Determine (x 2, y 2 ) by identifying the ending x-value of the interval and substituting it into the function. 5.Solve for y. 6.Substitute (x 1, y 1 ) and (x 2, y 2 ) into the slope formula to calculate the rate of change. 7.The result is the rate of change for the interval between the two points identified.

9. Common Errors/Misconceptions incorrectly choosing the values of the indicated interval to calculate the rate of change substituting incorrect values into the formula for calculating the rate of change assuming the rate of change must remain constant over a period of time regardless of the function interpreting interval notation as coordinates 9 3.3.2: Proving Average Rate of Change

10. Guided Practice Example 2 In 2008, about 66 million U.S. households had both landline phones and cell phones. This number decreased by an average of 5 million households per year. Use the table to the right to calculate the rate of change for the interval [2008, 2011]. 10 3.3.2: Proving Average Rate of Change Year (x) Households in millions (f(x)) 200866 200961 201056 201151

11. Guided Practice: Example 2, continued 1.Determine the interval to be observed. The interval to be observed is [2008, 2011], or where 2008 ≤ x ≤ 2011. 11 3.3.2: Proving Average Rate of Change

12. Guided Practice: Example 2, continued 2.Determine (x 1, y 1 ). The initial x-value is 2008 and the corresponding y-value is 66; therefore, (x 1, y 1 ) is (2008, 66). 12 3.3.2: Proving Average Rate of Change

13. Guided Practice: Example 2, continued 3.Determine (x 2, y 2 ). The ending x-value is 2011 and the corresponding y-value is 51; therefore, (x 2, y 2 ) is (2011, 51). 13 3.3.2: Proving Average Rate of Change

14. Guided Practice: Example 2, continued 4.Substitute (x 1, y 1 ) and (x 2, y 2 ) into the slope formula to calculate the rate of change. Slope formula Substitute (2008, 66) and (2011, 51) for (x 1, y 1 ) and (x 2, y 2 ). Simplify as needed. = –5 14 3.3.2: Proving Average Rate of Change

15. Guided Practice: Example 2, continued The rate of change for the interval [2008, 2011] is 5 million households per year. 15 3.3.2: Proving Average Rate of Change ✔

16. 16 3.3.2: Proving Average Rate of Change Guided Practice: Example 2, continued 16

17. Guided Practice Example 3 A type of bacteria doubles every 36 hours. A Petri dish starts out with 12 of these bacteria. Use the table to the right to calculate the rate of change for the interval [2, 5]. 17 3.3.2: Proving Average Rate of Change Days (x) Amount of bacteria (f(x)) 012 119 230 348 476 5121 6192

18. Guided Practice: Example 3, continued 1.Determine the interval to be observed. The interval to be observed is [2, 5], or where 2 ≤ x ≤ 5. 18 3.3.2: Proving Average Rate of Change

19. Guided Practice: Example 3, continued 2.Determine (x 1, y 1 ). The initial x-value is 2 and the corresponding y-value is 30; therefore, (x 1, y 1 ) is (2, 30). 19 3.3.2: Proving Average Rate of Change

20. Guided Practice: Example 3, continued 3.Determine (x 2, y 2 ). The ending x-value is 5 and the corresponding y-value is 121; therefore, (x 2, y 2 ) is (5, 121). 20 3.3.2: Proving Average Rate of Change

21. Guided Practice: Example 3, continued 4.Substitute (x 1, y 1 ) and (x 2, y 2 ) into the slope formula to calculate the rate of change. Slope formula Substitute (2, 30) and (5, 121) for (x 1, y 1 ) and (x 2, y 2 ). Simplify as needed. 21 3.3.2: Proving Average Rate of Change

22. Guided Practice: Example 3, continued The rate of change for the interval [2, 5] is approximately 30.3 bacteria per day. 22 3.3.2: Proving Average Rate of Change ✔

23. 23 3.3.2: Proving Average Rate of Change Guided Practice: Example 3, continued