Modeling Exponential Behavior

Student Summary

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: 6195\frac{61}{95} is about 0.642, 3961\frac{39}{61} is about 0.639, and 2639\frac{26}{39} is about 0.667.

All of these quotients are close to 23\frac{2}{3}. This suggests that we could model the relationship with an exponential function, and that the height is decreasing with a factor of about 23\frac23 for each successive bounce.

bounce number bounce height
in centimeters
1 95
2 61
3 39
4 26

The height, hh, of the ball, in cm, after nn bounces can be modeled by the equation: 

h=142(23)n\displaystyle h = 142 \boldcdot \left(\frac{2}{3}\right)^{n}

Here is a graph of the equation.

<p>Graph of function and data on grid.</p>
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 23\frac23 of the initial height is about 95 centimeters, then that initial height is about 142.5 centimeters, because 95÷23=142.595 \div \frac23 = 142.5. For a second example, we can see that it will take 7 bounces before the rebound height is less than 10 centimeters.

Visual / Anchor Chart

Standards

Addressing
HSN-Q.A.1

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