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Here is a graph of a function, , defined by .
Suggest a new graphing window that would:
Your teacher will give your group three different kinds of balls.
Your goal is to measure the rebound heights, model the relationship between the number of bounces and the heights, and compare the bounciness of the balls.
| n, number of bounces | a, height for ball 1 (cm) | b, height for ball 2 (cm) | c, height for ball 3 (cm) |
|---|---|---|---|
| 0 | |||
| 1 | |||
| 2 | |||
| 3 | |||
| 4 |
The table shows some heights of a ball after a certain number of bounces.
| bounce number | height in centimeters |
|---|---|
| 0 | |
| 1 | |
| 2 | 73.5 |
| 3 | 51.5 |
| 4 | 36 |
A
B
Sometimes data suggest an exponential relationship. For example, this table shows the bounce heights of a certain ball. We can see that the height decreases with each bounce.
To find out what fraction of the height remains after each bounce, we can divide two consecutive values: is about 0.642, is about 0.639, and is about 0.667.
All of these quotients are close to . This suggests that we could model the relationship with an exponential function, and that the height is decreasing with a factor of about for each successive bounce.
| bounce number | bounce height in centimeters |
|---|---|
| 1 | 95 |
| 2 | 61 |
| 3 | 39 |
| 4 | 26 |
The height, , of the ball, in cm, after bounces can be modeled by the equation:
Here is a graph of the equation.
Graph of a function on grid, origin O. Horizontal axis, number of bounces, from 0 to 12, by 1's. Vertical axis, height in centimeters, from 0 to 150, by 25's. Line of given equation, h equals 142 time 2 thirds to the n, is graphed, passing through 0 comma 142, 1 comma 94 and two thirds, 2 comma 63 and 1 ninth. 3 comma 42 and 2 twenty-sevenths, 4 comma 28 and 4 eighty-firsts. Data points of 1 comma 95, 2 comma 61, 3 comma 39, 4 comma 26 also plotted.
This graph shows both the points from the data and the points generated by the equation, which can give us new insights. For example, the height from which the ball was dropped is not given but can be determined. If of the initial height is about 95 centimeters, then that initial height is about 142.5 centimeters, because . For a second example, we can see that it will take 7 bounces before the rebound height is less than 10 centimeters.