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

Activity

A bright yellow tennis ball resting on a white line on a green tennis court, with a blurred background.

Here are measurements for the maximum height of a tennis ball after bouncing several times on a concrete surface.

$n$ Bounce Number	$h$ Height (Centimeters)
0	150
1	80
2	43
3	20
4	11

1. Which is more appropriate for modeling the maximum height $h$ , in centimeters, of the tennis ball after $n$ bounces: A linear function or an exponential function? Use data from the table to support your answer.

Compare your answer:

Exponential. Sample explanations:

Plotting the points in the plane, they are not close to being on a line.

A scatter plot showing the height in centimeters decreasing with each bounce, from about 150 cm before a bounce to about 10 cm at bounce 6. Both axes have gridlines for reference. The line is a curve as it gets closer to zero.

A linear function of the bounce height would decrease by the same amount for each successive bounce. The differences in heights between successive bounces are not close to being the same, as shown in the table. They are decreasing with each bounce, so the function is not linear.

Bounce Number	Difference from Height of Previous Bounce
0	--
1	70
2	37
3	23
4	9

An exponential function of the bounce height would decrease by the same factor for each successive bounce. It looks like each successive bounce is about half as high as the previous bounce, making an exponential model appropriate.

2. Regulations say that a tennis ball, dropped on concrete, should rebound to a height between 53% and 58% of the height from which it is dropped. Does the tennis ball here meet this requirement? Be prepared to show your reasoning.

Compare your answer:

Yes. The first rebound is a little over 53%, the second is almost 54%, and the fourth is 55%. The third rebound is only about 47%, but there could be some measurement error.
No, the third rebound is less than 50% of the height from which the tennis ball fell, falling out of the allowable range.
It is not possible to tell. One of the values is too low, and the others are close enough to 53% that it is impossible to be sure whether the bounces of this ball fall within the regulations.

3. Write an equation that models the bounce height $h$ after $n$ bounces for this tennis ball.

Compare your answer:

$h = 150 \cdot (0.53)^{n}$ (A factor of exactly 0.5 is also acceptable, or any value between 0.53 and 0.58.)

4. About how many bounces will it take before the rebound height of the tennis ball is less than 1 centimeter? Be prepared to show your reasoning.

Compare your answer:

8 bounces. For example:

The height of the ball is decreasing by a factor of about $\frac{1}{2}$ after each bounce. So after 3 more bounces, it will decrease by a factor of about $\frac{1}{8}$ (giving a little more than 1 cm), but after 4 more bounces, it will decrease by a factor of about $\frac{1}{16}$ .

Graphing $h = 150 \cdot (0.53)^{n}$ and $h = 1$ simultaneously using graphing technology shows the bounce height first goes below $h = 1$ on the 8th bounce.

5. Graph the equation that models the bounce height $h$ after $n$ bounces for this tennis ball using the Desmos tool below.

Compare your answer:

A graph with a curve that starts high on the y-axis at higher than 20 and rapidly decreases, approaching the x-axis as it moves right, resembling a reciprocal function. The axes are labeled and the curve is orange. The line is on top of the x axis from 12 to 22. There is no y-intercept.

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6. Use the graph of the equation that models the bounce height $h$ after $n$ bounces for this tennis ball to predict the difference in height after 6 bounces.

Compare your answer:

The difference in height after 6 bounces is 2. Using the equation:

$h = 150 \cdot (0.50)^{n}$ and graphing $h = 150 \cdot (0.50)^{6}$ . The graph shows the height to be 2.344 after 6 bounces $(n = 6)$ .

This is consistent with the reasoning:

The height of the ball is decreasing by a factor of about $\frac{1}{2}$ after each bounce. After the 4th bounce, the height difference is 9. It should decrease by $\frac{1}{2}$ for each bounce after that. So, after 2 more bounces (the 6th bounce), it will decrease by a factor of about $\frac{1}{4}$ . The 6th bounce would be $9 \cdot \frac{1}{4} \approx 2.25$ . Rounding either 2.344 or 2.25 results in the answer 2.

Video: Determining a Model and an Equation from a Real-World Scenario

Watch the following video to learn more about modeling exponential functions:

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Self Check

The table gives predictions of the average costs of a gasoline car after the year 2030.

Which graph best represents the relationship between time and the predicted cost?

Time (years)	Cost (thousand of dollars)
0	35
5	40.2
10	46.3
15	53.2
20	61.2

Gasoline car

Additional Resources

Using an Exponential Model

A lab researcher records the growth of the population of a yeast colony and finds that the population doubles every hour. The table below records the first four hours.

Hours into Study	0	1	2	3	4
Yeast Colony Population (Thousands)	5	10	20	40	80

The starting value is 5 (in thousands) and the population doubles, so the growth factor is 2.

Example

You can use the data you already have to predict the population size of the yeast colony in the future.

1. What will the size of the population be in 5 hours?

The population will be 160,000. You will double the 80 which represents 80,000.

2. What will the size of the population be in 6 hours?

The population will be 320,000. You will double the 160 which represents 160,000.

3. What will the size of the population be in 10 hours?

You may have to use a calculator to figure this one out. You will double it four more times. The population will be 5,120,000.

The equation to model this growth is $P (t) = 5 \cdot 2^{t}$ .

The graph is below:

A graph with an orange curve showing an exponential increase, starting near the origin and rising steeply as x increases. The axes are labeled, and the grid has values marked at intervals of 10.

Try it

Try It: Using an Exponential Model

A ball bounces up and down from a starting height of 65 centimeters. As it bounces, its height decreases to $\frac{1}{3}$ the previous height. Complete a table to represent the first 4 bounces. Predict the height after 5 bounces. Then write a function, $h (x)$ , to determine the height after $x$ bounces.

Compare your answer:

Here is how to find the model for this situation:

Number of Bounces, $x$	0	1	2	3	4
Height, $h (x)$	65	$\frac{65}{3}$	$\frac{65}{9}$	$\frac{65}{27}$	$\frac{65}{81}$

The initial height is 65, and the growth factor is $\frac{1}{3}$ .

The height after the 5th bounce is $\frac{65}{243}$

The maximum height, $h (x)$ , after $x$ bounces can be represented by $h (x) = 65 (\frac{1}{3})^{x}$ .

5.11.2 Choosing an Appropriate Model

Activity

Video: Determining a Model and an Equation from a Real-World Scenario

Additional Resources

Using an Exponential Model

Try It: Using an Exponential Model