Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

Activity

Here is a table and a graph that show the number of coffee shops worldwide that a company had in its first 10 years, between 1987 and 1997. The growth in the number of shops was roughly exponential.

A scatter plot showing the number of coffee shops increasing rapidly over 11 years since 1987, from near zero to over 1,400. The growth accelerates significantly after year 5. The graph represents the data in the table. for Year and Number of Shops.

Year	Number Of Shops
1987	17
1988	33
1989	55
1990	84
1991	116
1992	165
1993	272
1994	425
1995	677
1996	1,015
1997	1,412

1. Find the average rate of change for each period of time. Show your reasoning.

a. 1987 to 1990

Compare your answer:

$22 \frac{1}{3}$ stores per year since $\frac{(84 - 17)}{(3 - 0)} = 22 \frac{1}{3}$ .

b. 1987 to 1993

Compare your answer:

42.5 stores per year since $\frac{(272 - 17)}{(6 - 0)} = 42.5$ .

c. 1987 to 1997

Compare your answer:

139.5 stores per year since $\frac{(1412 - 17)}{(10 - 0)} = 139.5$ .

2. Make some observations about the rates of change you calculated. What do these average rates tell us about how the company was growing during this time period?

Compare your answer:

The average rate of change of the first 6 years is almost double that of the first 3 years. The average rate of change of the 10-year period is almost 7 times greater than that of the first 3 years and more than 3 times greater than the first 6 years. The company was growing at an increasing rate over the 10-year time period.

3. Use the graph to support your answers to these questions. How well do the average rates of change describe the growth of the company in:

a. the first 3 years?

Compare your answer:

An average rate of change of $22 \frac{1}{3}$ is a fairly accurate prediction of growth over the first 3 years, but it is a poor prediction of growth much beyond 1992.

b. the first 6 years?

Compare your answer:

An average rate of change of 42.5 is an okay prediction of growth over the first 6 years, but it overestimates early on and greatly underestimates the growth in years after 1994.

c. the entire 10 years?

Compare your answer:

An average rate of change of 139.5 does not accurately predict the actual growth of the company between 1987 and 1997.

4. Let $f$ be the function so that $f (t)$ represents the number of stores $t$ years since 1987. The value of $f (20)$ is 15,011. Find $\frac{f (20) - f (10)}{20 - 10}$ and say what it tells us about the change in the number of stores.

Compare your answer:

The average rate of growth from 1997 to 2007 is 1,359.9 stores per year. This is 60 times greater than the average rate of change over the first 3 years and almost 10 times greater than the average rate of change over the first 10 years of the time period. The number of stores continued to grow at an increasing rate from 1997 to 2007.

Video: Analyzing Average Rate of Change of an Exponential Function

Watch the following video to learn more about the average rate of change of exponential functions:

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Self Check

Suppose a population of cells starts at 500 and triples every day. The number of cells each day can be calculated as follows:

Number Of Days	Number Of Cells
0	500
1	1,500
2	4,500
3	13,500

Find the average rate of change from day 0 to day 2.

-2,000
2,000
4,500
1,000

Additional Resources

Average Rate of Change from a Table

Let’s look at an exponential function we studied earlier. Let $A (t)$ be the function that models the area $A$ , in square yards, of algae covering a pond $t$ weeks after beginning treatment to control the algae bloom. Here is a table showing approximately how many square yards of algae remain during the first 5 weeks of treatment.

$t$	$A (t)$
0	240
1	80
2	27
3	9
4	3

The average rate of change of $A$ from the start of treatment to week 2 is about –107 square yards per week since $\frac{A (2) - A (0)}{2 - 0} \approx - 107$ . The average rate of change of $A$ from week 2 to week 4, however, is only about –12 square yards per week since $\frac{A (4) - A (2)}{4 - 2} \approx - 12$ .

Try it

Try It: Average Rate of Change from a Table

Using the same table from above, what is the average rate of change of $A$ from week 1 to week 3?

$t$	$A (t)$
0	240
1	80
2	27
3	9
4	3

Compare your answer:

Here is how to use the table to find the average rate of change:

Find the points on the table when $t = 1$ and $t = 3$ .	$(1, 80)$ , $(3, 9)$
Substitute into the slope formula to find the rate of change.	$\frac{A (3) - A (1)}{3 - 1} = \frac{9 - 80}{2}$
Simplify to find the rate of change.	$\frac{- 71}{2} = - 35.5$ square yards per week

5.10.2 Explore Average Rates of Change in an Exponential Growth Function

Activity

Video: Analyzing Average Rate of Change of an Exponential Function

Additional Resources

Average Rate of Change from a Table

Try It: Average Rate of Change from a Table