Prior Knowledge: Students should be able to transform exponential
data in order to write a linear regression, and then be able to transform the linear regression to fit exponential data. This
experiment is an example of exponential decay. The ideal
equation to describe this situation is:
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!
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Number of Bounces
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0
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(Start Height)
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(Start Height)
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(Start Height)
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(Start Height)
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1
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2
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3
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4
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5
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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?
Does the
graph appear exponential?
How do you know?
7. Since we believe this to be an exponential regression, take the log of each of the Average Heights and enter them into List3.
8. Make a
scatter plot of log (Average Height) against the Number of Bounces. Are the points fairly linear?
9. Find the linear regression between Number of Bounces and log (Average Height).
Linear regression: log = _________
Correlation: _________
10. Graph your linear regression (in Y1) on the
scatter plot of the transformed
data from #8. How well does it fit the data?
11. Construct a residual plot for log. To do this, find the difference between the actual values of log(Average Height) and the predicted values from your regression equation. Enter these values into List4. The command to have your calculator do this for you is: L4 = L3 – Y1(L1). Now make a
scatter plot of L1 and L4. What does this residual plot show?
12. The
inverse operation of the logarithmic
function is raising 10 to a power. To transform the regression
equation to fit your data, raise 10 to the power of the left
side of your
equation and 10 to power of the right
side of your equation. This will eliminate the log on the left
side of your equation.
y = __________
13. Simplify the
equation from #12 so that it is in
y = abx form:
y = __________
How close is a to your start height?
How close is b to the
ratio you found in #5?
14. Enter this
function into Y2 on your calculator (turn off the
equation in Y1). Change your
scatter plot from the residual plot to the original
scatter plot of the actual data. How well does your exponential regression fit your data?
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 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 # 12?