Exponential and Power Functions: which is which?

If you are like me you may find it difficult to remember the distinction between power functions and exponential functions, or at least which is which. After all, they both involve powers/exponents. Here is the short answer:

In a power function the independent variable (x) is raised to a (constant) power (c). In an exponential function the indepent variable is the exponent while the base is a constant.

Here is some more information and illustrations:

1. Power functions. In its most basic form:

y = f(x) = xc

A common variation would be to have a second constant as a coefficient of x:

y = f(x) = axc

And here is an example, plotted. In this case a = 1 (so has no effect) and c=2.5.

power function

By the way, you may be curious about how this power function would look with a negative exponent. The graph below shows the function with c = -2.5 (recall that x-c = 1/xc).

power function with negative constant

2. Exponential functions. Now our independent variable, x, is the exponent and the constant, c, is the base:

y = f(x) = cx

The plot below shows this function for c=2, x = 1-30.

graph of Exponential Function

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