Exponential Functions  
 
Introduction  
This lesson will inform you of the properties of exponential functions. Here are the sections within this lesson page:

Exponential functions are functions that are in this form.
We refer to the bvalue as the base and the xvalue as the exponent.
 
If the base of the function is greater than one (b > 1), then we say the function is a growth function. If the base of the function is greater than zero but less than one (0 < b < 1), then we say the function is decaying. The distinction is not trivial. The graphs of growth and decay functions look different. Here is a comparison between the graphs of growth and decay functions.
It can be clearly seen that the graphs of growth functions rapidly go up to the right. Conversely, decay functions drop quickly and then level out to the right.
 
The yintercept is an important location within the graphs of exponential functions. To mathematically find them, simply substitute zero for the xvalue within the function. We substitute zero for 'x' because all points on the yaxis have an xvalue equal to zero. For instance, if we needed to locate the yintercept of the function...
...we would substitute zero for the xvalue. Since any number to the zero power is equal to one, we know the yintercept is (0,1). (See our Power Rules lesson for a refresher on this property.) This is confirmed if we examine the graph of the function (see below).
If we examine the general case, we will find the same result. Look at the general form of an exponential function.
If we substitute zero for the xvalue, we will find that any number raised to the zero power (except zero) is equal to one. Which means exponential functions of this form always have a yintercept of (0,1).
Keep in mind, however, that transformations of exponential functions will move the yintercept.
 
Exponential functions have asymptotes. An asymptote is a line (or a curve) that guides the graph of a relation. In the case of an untransformed exponential equation, the asymptote is on the xaxis. Take a look at the graph of the function y = 2^{x}.
As we move on the curve from the right to the left (as x →  ∞), we see that the curve gets closer and closer to the xaxis (f(x) → 0). That is why the xaxis is acting like a guide for the exponential function. This is why the line y = 0 has been highlighted in purple over the xaxis  so that it can easily be seen.
Again, transformations of exponential functions can move the asymptote.
 
Try theses lessons, which are related to the sections above.
esson: Translating Exponential Functions 