The Biology Project > Biomath > Exponential Functions > Graphing Part I

## Exponential Functions

Graphing Part I: Graphs of Basic Exponential Functions

Graphing Exponential Functions

As an experimental scientist you would start with the data, as we did in the examples of cancer incidence and viral decay, determine that the trend is exponential, and then find the best mathematical model to fit your data. However, it is often easier to understand different models if we consider the idealized curves. To better understand the features of exponential growth and decay, we will consider some general exponential functions through a mathematicians eyes.

Consider the following two exponential functions:

Let's construct a table of values for f1 (x) and f2 (x) to use as a guide when plotting.

 x - 4 -3 -2 -1 0 1 2 3 4 16 8 4 2 1 1/2 1/4 1/8 1/16 1/16 1/8 1/4 1/2 1 2 4 8 16

We can then plot the points listed in table above and draw a smooth curve through all of them.

 Let's discuss some of the features of these graphical representations. Exponential Decay 0 < a < 1 Properties As x → ∞,  f1 (x) → 0 This means the curve decrease as you look to the right because   f1 (x) approaches the y = 0 (properly called the horizontal asymptote). As x → - ∞,   f1 (x) → ∞ In other words the curve increases without bound to the left. If x = 0, f1 (x) = 1 The curve intersects the y-axis at the point (0, 1). This point is called the y intercept. Exponential Growth a > 1 Properties As x → ∞,   f2 (x) → ∞ An exponential growth curve increases without bound as you look to the right because the value of f2 (x) increases as x increases. As x → - ∞,  f2 (x) → 0 The curve decreases as you look to left because the value of   f2 (x) approaches the horizontal asymptote, y = 0. If x = 0, f1 (x) = 1 The curve intersects the y-axis at the point (0, 1). This point is called the y intercept. You can verify the accuracy of the curves we have just drawn using a graphing calculator or program.

Notice that  f1 (x) and  f2 (x) exhibit contradictory behavior. In general any exponential function, x, will look similar to either

or     ,

depending on whether a is less than or greater than one respectively.

Constructing graphs of other exponential functions.

Now that we have constructed graphs for  f1 (x) and  f2 (x) and discussed their general characteristics, you may be wondering how to construct graphs of other exponential functions.

When plotting exponential functions it is imperative to determine whether you are working with exponential growth or decay. You can determine this by looking at the base of a function you are given or by the nature of the application. For example, if you a plotting the growth of a culture of bacteria growing under ideal laboratory conditions, you would expect to see an exponential growth curve. However, if you are interested in the decay of a radioisotope, you would expect an exponential decay curve.

When you are confused, it is always useful to construct a table of values for the function in question. This table coupled with your own intuition about the shape of the graph should give you a reasonably accurate sketch of the function in question.

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In the next section we will look at slightly more complicated variants of our basic exponential
function, x.

Graphing Part II

The Biology Project > Biomath > Exponential Functions > Graphing Part I

The Biology Project
Department of Biochemistry and Molecular Biophysics

The University of Arizona
December 2005
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