Lesson 5 I Can See—Can’t You? Practice Understanding

Learning Focus

Understand and find the average rate of change of a function in an interval.

Develop a formula for the average rate of change for any function.

How can I find the average rate of change of a function that is not linear?

What does the average rate of change mean?

Open Up the Math: Launch, Explore, Discuss

Kwan’s parents bought a home for $$50,000$ in 2007 just as real estate values in the area started to rise quickly. Each year, their house was worth more until they sold the home in 2017 for $$309,587$ .

1.

Model the growth of the home’s value from 2007 to 2017 with both a linear and an exponential equation. Then graph the two models on the graph.

Linear model:

Exponential model:

2.

What was the change in the home’s value from 2007 to 2017?

The average rate of change is defined as the change in $y$ (or $f (x)$ ) divided by the change in $x$ .

3.

What was the average rate of change of the linear function from 2007 to 2017?

4.

What is the average rate of change of the exponential function in the interval from 2007 to 2017?

5.

How do the average rates of change from 2007 to 2017 compare for the two functions? Explain.

6.

What was the average rate of change of the linear function from 2007 to 2012?

7.

What is the average rate of change of the exponential function in the interval from 2007 to 2012?

8.

How do the average rates of change from 2007 to 2012 compare for the two functions? Explain.

9.

How can you use the equation of the exponential function to find the average rate of change over a given interval?

How does this process compare to finding the slope of the line through the endpoints of the interval?

Consider the graph:

10.

What is the equation of the graph shown?

11.

What is the average rate of change of this function on the interval from $x = - 3$ to $x = 1$ ?

12.

What is the average rate of change of this function in the interval from $x = - 3$ to $x = 0$ ?

13.

What is the average rate of change of this function in the interval from $x = - 3$ to $x = - 1$ ?

14.

What is the average rate of change of this function in the interval from $x = - 3$ to $x = - 1.5$ ?

15.

Draw the line through the point at the beginning and end of each of the intervals in 11, 12, 13, and 14. What is the slope of each of these lines?

16.

Which of these average rates of change best represents the change at the point $(- 2, 4)$ ?

Explain your answer.

17.

How does the average rate of change compare to the change factor for an exponential function? What is described by each of these quantities?

Ready for More?

It makes sense to some people that the way to find an average rate of change of a function would be to:

find the rate of change at the beginning of the interval,
find the rate of change at the end of the interval,
add the two rates together, and
divide by $2$ .

Does this strategy ever work? Explain why or why not.

Takeaways

Average rate of change of a function over the interval $[a, b]$ :

This formula means:

Secant line:

Vocabulary

average rate of change
secant line
tangent to a curve
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we learned to find the average rate of change of a function in an interval. We learned that the average rate of change is calculated by finding the change in $y$ and dividing by the change in $x$ . Sometimes an equation is used to calculate the $y$ -values at the beginning and end of the interval, and sometimes a graph is used to find the heights at the beginning and end of the interval. In either cases, the $y$ -values are subtracted and then that amount is divided by the width of the interval, or the difference between the $x$ -values.

Retrieval

Use the given information to create a linear equation in slope-intercept form.

1.

$m = - 3$ ; $b = 7$

2.

$m = \frac{2}{3}$ ; $b = - 18$

3.

$m = 8$ through point $(- 2, 4)$

4.

Through points $(2, 8)$ and $(8, - 16)$

5.

Rewrite each of the expressions by combining like terms.

a.

$- 3 (x + 5) - 2$

b.

$7 (x - 4) + 6$

c.

$- 8 (x - 1) - 11$