Calculus How To

Exponential Model: Simple Definition, Examples

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An exponential model has a distinctive upward or downward curve that increases (or decreases) sharply and smoothly. If the curve decreases, it’s called exponential decay; If the curve increases, then it’s exponential growth.

exponential decay

An exponential model showing exponential decay.



Many real life data sets follow an exponential pattern, including population growth and decline, environmental concentrations (Ott, 1995) and—oddly—even the amount of revenue collected yearly by the IRS (Larson & Falvo, 2012, p. 380).

Mathematically, exponential models have the form y = A(r)x, where A is the initial value, and r is the rate of increase (or decrease). The image above shows an exponential function N(t) with respect to time, t. The initial value is 5 and the rate of increase is et

Exponential Model Building on a Graphing Calculator

Some graphing calculators (most notably, the TI-89) have an exponential regression features, which allows you to take a set of data and see whether an exponential model would be a good fit.

TI-89 Exponential Regression

exponential regression on the TI-89

The TI-89 has many features to model data, including exponential regression.

Exponential regression fits an exponential function to your data. As an example, let’s say you have the following data:

  • x-values: 1, 2, 3, 4, 5, 6, 7,
  • y-values: 334, 269, 193, 140, 105, 67.

You might notice that the data decreases sharply, so a decreasing exponential function might be a good fit.

Steps

Step 1: Make a scatter plot. Watch the first minute of this video if you don’t know how to create one. This step confirms that the data roughly fits an exponential model. If your data doesn’t fit the model, stop here. You could (theoretically) continue, but your model will be practically useless. Find another model that better fits your data.

Step 2: Press APPS, then scroll to Data/Matrix Editor(using the cursor keys). Press ENTER.

Step 3: Press 1 “Current”.

Step 4: Press F5 “Calc”. A new screen will open.

Step 5: Move the cursor to “Calculation Type”, then press the right-cursor key and choose “4:ExpReg”.

Step 6: Enter your x-values location into the “x” box. For example, if your x-values are in list a1 then type “a1.”

Step 7: Enter the location of your y-values into the “y” box.

Step 8: Move the cursor to the Store ReqEQ line. Press the right cursor key, then move the cursor to y1(x). Press ENTER.

That’s it! A window will appear with a and b . These go into the regression equation y = abx. The same equation will also show in the y1= line of the Y= screen.

If you entered the data in the above example, you should get a solution of y = 490.631792*.726657x.

Tip: Y-values must be greater than zero in order for regression to work prroperly.
Lost your guidebook? You can download a new one from the TI website. The Titanium Guidebook can be found on this page.


References

Larson, R. & Falvo, D. (2012). Algebra and Trigonometry: Real Mathematics, Real People. Cengage Learning.
Ott, W. Environmental Statistics and Data Analysis, Taylor & Francis. 1995.


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Stephanie Glen. "Exponential Model: Simple Definition, Examples" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/exponential-model/
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