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.*

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 e^{t}

## 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 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 = ab^{x}. 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*.726657^{x}.

**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.

**CITE THIS AS:**

**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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