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On the Rebound (Algebra 2)
Objective:  Students will find the equation of an exponential function using data from an experimental situation.
Prior Knowledge: The student should be able to recognize exponential data and graphs as well as enter data into a graphing calculator. 
Materials needed:
  • a ball that bounces fairly well (tennis ball, golf ball, or super ball)
  • Meter stick
  • TI-83 (or later) graphing calculator 
Group size: 3 people
Time required: one 50-minute class period
1. Tape a meter stick to a wall so that you can measure the height of your ball at the top of successive bounces.
2. Hold the ball less than a meter above the ground, close to the meter stick and record the starting height (at the bottom of the ball).
3. Drop the ball and record the height of the bottom of the ball on the rebound of at least 5 bounces. Record your data in a table like the one below. Do at least 3 trials, starting from the same height. This is probably going to take several trials to get 3 that work!

Number of Bounces
Height of Ball
Trial 1
Height of Ball
Trial 2
Height of Ball
Trial 3
Average Height
(Start Height)
(Start Height)
(Start Height)
(Start Height)
4. Enter the Number of Bounces in List1 of your calculator and the Average Height of the ball after each bounce in List2.
5. Test to see if this an exponential equation by dividing each of your successive y values by the previous y value. 
Do you get approximately the same ratio? 
If so, what is it?
Does this ratio show exponential growth or decay?

6. Now make a scatter plot of your data in your calculator. 
Does the graph appear exponential? 
How do you know?
We now need to find the equation of the function that fits this data the best.
7. Find the regression equation for the function that best fits your data.
 What type of regression did you use?
 y = ___________
 Correlation (r value): ___
The ideal equation to describe this situation is:
y = abx
a is the starting height of the ball, 
b is percent of the previous height the ball reaches on each bounce,
x is the number of bounces
8. How close is a to your start height?

9. How close is b to the ratio you found in #5?
10. Graph your regression (in Y1) on the scatter plot of the data. How well does it fit the data?

Follow up questions:
1. If your first bounce is from a different height, does this equation still work to predict the rebound heights of each bounce? Try dropping your ball from several different heights (change a in your equation) to support your answer.

2. Will this equation work for another group’s ball? Try it and see.
3. Will this equation work for a different type of ball? Why or why not?
4. Why did we use the average of each height rather than every one of the individual measures?
5. Enter each of the individual measures into your calculator, and graph your equation through the points. How well does your equation from #7 fit the data?
6. Use the calculator to find the exponential regression for all of the data. How close is it to the exponential regression you found in # 7? Which equation fits all of the data better? 

C Adams

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