1. Lesson 3.5 Average Rate of Change (exponential functions)
3.3.3: Recognizing Average Rate of Change

2. Introduction
In the previous lesson, we found the slope of linear equations and functions using the slope formula,
The slope of a line is the ratio of the change in y-values to the change in x-values.
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.
3.3.3 Recognizing Average Rate of Change

3. Key Concepts
Finding the rate of change of a non-linear function, in our case, exponential functions, is very similar to that of a linear function. You still use the slope formula to calculate the rate of change. Here are a few differences between the two:
Linear function
you choose your two points for ( 𝑥 1 , 𝑦 1 ) and ( 𝑥 2 , 𝑦 2 )
rate of change is constant
Non-Linear function
you are told by the interval which two points to use for ( 𝑥 1 , 𝑦 1 ) and ( 𝑥 2 , 𝑦 2 )
rate of change is different at different intervals
3.3.3: Recognizing Average Rate of Change

4. Key Concepts, continued
An interval is a continuous portion of a function.
The rate of change of an interval is the average rate of change for that period.
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.
For example, the interval [2, 7] means the portion of the function where x = 2 through x = 7. You would use the points (2, y) & (7, y) in the slope formula when calculating the rate of change for the function.
3.3.3 Recognizing Average Rate of Change

5. Key Concepts, continued
Calculating Average Rate of Change
Identify the interval to be observed.
Identify (x1, y1) as the starting point of the interval.
Identify (x2, y2) as the ending point of the interval.
Substitute (x1, y1) and (x2, y2) into the slope formula to calculate the rate of change. 𝑚= 𝑦 2 − 𝑦 1 𝑥 2 − 𝑥 1
The result is the average rate of change for the interval between the two points identified.
3.3.3: Recognizing Average Rate of Change

6. Households in millions (f(x))
Guided Practice
Example 1
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].
Year (x)
Households in millions (f(x))
2008 66
2009 61
2010 56
2011 51
3.3.3: Recognizing Average Rate of Change

7. Guided Practice: Example 1, continued
Determine the interval to be observed.
The interval to be observed is [2008, 2011], or where 2008 ≤ x ≤ 2011.
Determine (x1, y1).
The initial x-value is 2008 and the corresponding y-value is 66; therefore, (x1, y1) is (2008, 66).
Determine (x2, y2).
The ending x-value is 2011 and the corresponding y-value is 51; therefore, (x2, y2) is (2011, 51).
3.3.3: Recognizing Average Rate of Change

8. Guided Practice: Example 2, continued
Substitute (x1, y1) and (x2, y2) into the slope formula to calculate the rate of change.
Slope formula
Substitute (2008, 66) and (2011, 51) for (x1, y1) and (x2, y2).
Simplify as needed.
= –5
The rate of change for the interval [2008, 2011] is 5 million households per year.
3.3.3: Recognizing Average Rate of Change

9. You Try
The table below represents a type of bacteria that doubles every 36 hours. A Petri dish starts out with 12 of these bacteria.
Calculate the average rate of change over the intervals [1,4] and [2,5].
3.3.3: Recognizing Average Rate of Change

10. Guided Practice Example 2
Jasper has invested an amount of money into a savings account. The graph to the right shows the value of his investment over a period of time. What is the rate of change for the interval [1, 3]?
3.3.3: Recognizing Average Rate of Change

11. Guided Practice: Example 2, continued
Determine the interval to be observed.
The interval to observe is [1, 3], or where 1 ≤ x ≤ 3.
Identify the starting point of the interval.
The x-value of the starting point is 1. The corresponding y- value is approximately 550.
The starting point of the interval is (1, 550).
Identify the ending point of the interval.
The x-value for the ending point is 3. The corresponding y- value is approximately 1,100.
The ending point of the interval is (3, 1100).
3.3.3: Recognizing Average Rate of Change

12. Guided Practice: Example 2, continued
3.3.3: Recognizing Average Rate of Change

13. Guided Practice: Example 2, continued
Substitute (1, 550) and (3, 1100) into the slope formula to calculate the rate of change.
Slope formula
Substitute (1, 550) and (3, 1100) for (x1, y1) and (x2, y2).
Simplify as needed.
= 275
The rate of change for this function over the interval [1, 3] is approximately $275 per year.
3.3.3: Recognizing Average Rate of Change

14. Guided Practice Example 3 Jasper is curious about how
the rate of change differs for
the interval [3, 6]. Calculate
the rate of change using the
graph from Example 2.
3.3.3: Recognizing Average Rate of Change

15. Guided Practice: Example 3, continued
Determine the interval to be observed.
The interval to observe is [3, 6], or where 3 ≤ x ≤ 6.
Identify the starting point of the interval.
The x-value of the starting point is 3. The corresponding y-value is approximately 1,100.
The starting point of the interval is (3, 1100).
Identify the ending point of the interval.
The x-value for the ending point is 6. The corresponding y-value is approximately 3,100. The ending point of the interval is (6, 3100).
3.3.3: Recognizing Average Rate of Change

16. Guided Practice: Example 3, continued
3.3.3: Recognizing Average Rate of Change

17. Guided Practice: Example 3, continued
Substitute (3, 1100) and (6, 3100) into the slope formula to calculate the rate of change.
Slope formula
Substitute (3, 1100) and (6, 3100) for (x1, y1) and (x2, y2).
Simplify as needed.
3.3.3: Recognizing Average Rate of Change

18. ✔ Guided Practice: Example 3, continued
The rate of change for this function over the interval [3, 6] is approximately $ per year.
Notice that the rate of change for the interval [3, 6] is much steeper than that of the interval [1, 3]. ✔
3.3.3: Recognizing Average Rate of Change

19. You Try
Each year, volunteers at a three-day music festival record the number of people who camp on the festival grounds. The graph below shows the number of campers for each of the last 20 years.
Calculate the average rate of change over the intervals [3,9] and [9,16].
3.3.3: Recognizing Average Rate of Change