An exponential function is when the independent variable is in the exponent of a constant. The base of the function must be greater than 0 and not equal to 1. Some examples of exponential functions are:

If the base of the exponential function is a fraction, then the graph falls rapidly to the right, as shown here.

Looking for these general shapes when given a graph will indicate whether the graph represents an exponential function.

If you're given a table, to determine whether the function is exponential, check if there's a common multiple difference, meaning you can multiply each of the *y*-values by a number to get to the next *y*-value. This indicates that the function is exponential and, in fact, that that number is your base.

To write an equation for an exponential function, follow these steps:

- Find the common multiple difference (this is your base,
*b*). - Find your
*y*-intercept—this is the coefficient of your exponential function (*a*). - Substitute your values for
*a*and*b*into the general form of an exponential:

For example:

First find whether there's a common multiple difference. It's important to note that the *x*-values are evenly spaced, which allows the common multiple to be identifiable.

You now know that your base is 2. Looking for your *y*-intercept, you see it's (0, 3), so *a* = 3. Now substitute these values into