1. Introduction
The average running speed of a human being is 5–8 mph. Imagine you set out on an 8-mile run that takes you no longer than 1 hour. You run often, so you run consistently and at the same speed for the entire hour. The rate of change of your position for your runs is always 8 mph and can be modeled linearly, because the rate of change is constant.
5.5.3: Identifying the Average Rate of Change

2. Introduction, continued
However, a friend of yours, averaging the same distance in an hour, hasn’t built up the endurance needed to run 8 miles consistently. Sometimes your friend runs 9 mph, sometimes he stops to rest, sometimes he walks, and then he resumes running at 8 mph. He might do this several times in the hour, sometimes running faster than you, sometimes slower, and sometimes not running at all. The rate of change of your friend's speed is not constant and cannot be modeled linearly.
5.5.3: Identifying the Average Rate of Change

3. Introduction, continued
The average rate of change for your friend’s speed, the ratio of the change in the output of a function to the change in the input for specific intervals, is inconsistent. In this lesson, you will practice calculating the average rate of change of quadratic functions over various intervals.
5.5.3: Identifying the Average Rate of Change

4. Key Concepts
The average rate of change of a function is the rate of change between any two points of a function; it is a measure of how a quantity changes over some interval.
The average can be found by calculating the ratio of the difference of output values to the difference of the corresponding input values, from x = a to x = b. This formula is often referred to as the average rate of change formula.
5.5.3: Identifying the Average Rate of Change

5. Key Concepts, continued
Recall that the slope of a linear function is found using the formula
Although the formula for calculating the average rate of change looks quite different from the formula used to find the slope of a linear function, they are actually quite similar.
Both formulas are used to find the rate of change between two specific points.
5.5.3: Identifying the Average Rate of Change

6. Key Concepts, continued
The rate of change of a linear function is always constant, whereas the average rate of change of a quadratic function is not constant.
Choosing different x-values and their corresponding y-values will result in different rates of change.
5.5.3: Identifying the Average Rate of Change

7. Common Errors/Misconceptions
assuming the rate of change is the same for every interval of a quadratic function
5.5.3: Identifying the Average Rate of Change

8. Guided Practice Example 1
Calculate the average rate of change for the function f(x) = x2 + 6x + 9 between x = 1 and x = 3.
5.5.3: Identifying the Average Rate of Change

9. Guided Practice: Example 1, continued Evaluate the function for x = 3.
f(x) = x2 + 6x + 9
Original function
f(3) = (3)2 + 6(3) + 9
Substitute 3 for x.
f(3) = 36
Simplify.
5.5.3: Identifying the Average Rate of Change

10. Guided Practice: Example 1, continued Evaluate the function for x = 1.
f(x) = x2 + 6x + 9
Original function
f(1) = (1)2 + 6(1) + 9
Substitute 1 for x.
f(1) = 16
Simplify.
5.5.3: Identifying the Average Rate of Change

11. Guided Practice: Example 1, continued
Use the average rate of change formula to determine the average rate of change between x = 1 and x = 3.
5.5.3: Identifying the Average Rate of Change

12. Guided Practice: Example 1, continued
Average rate of change formula
Substitute 1 for a and 3 for b.
Substitute the values for f(3) and f(1).
Simplify.
5.5.3: Identifying the Average Rate of Change

13. Guided Practice: Example 1, continued
The average rate of change of f(x) = x2 + 6x + 9 between x = 1 and x = 3 is 10. ✔
5.5.3: Identifying the Average Rate of Change

14. Guided Practice: Example 1, continued
5.5.3: Identifying the Average Rate of Change

15. Guided Practice Example 2
Use the graph of the function at right to calculate the average rate of change between x = –3 and x = –2.
5.5.3: Identifying the Average Rate of Change

16. Guided Practice: Example 2, continued Use the graph to identify f(–2).
According to the graph, f(–2) = –1.
5.5.3: Identifying the Average Rate of Change

17. Guided Practice: Example 2, continued Use the graph to identify f(–3).
According to the graph, f(–3) = 2.
5.5.3: Identifying the Average Rate of Change

18. Guided Practice: Example 2, continued
Use the average rate of change formula to determine the average rate of change between x = –3 and x = –2.
5.5.3: Identifying the Average Rate of Change

19. Guided Practice: Example 2, continued
Average rate of change formula
Substitute –3 for a and –2 for b.
Substitute the values for f(–3) and f(–2) found from the graph.
Simplify.
5.5.3: Identifying the Average Rate of Change

20. ✔ Guided Practice: Example 2, continued
The average rate of the change of the function between x = –3 and x = –2 is –3. ✔
5.5.3: Identifying the Average Rate of Change

21. Guided Practice: Example 2, continued
5.5.3: Identifying the Average Rate of Change