Exponential Functions   

An exponential function has the form

.

The independent variable x is the exponent and the constant b is the base of the exponential function. While the constant a can be any real number, the base, b, must be positive and different from 1.
The domain of an exponential function is the set of all real numbers. The range is the set of all positive real numbers when a>0 and the set of negative numbers when a<0.  The graph  looks like one of those shown below.
Notice that the exponential function is increasing and concave up when a>0 and b>1, and decreasing and concave up when a>0 and 0<b<1. If a<0, the graph is reflected across the x- axis. The graph is decreasing and concave down when a<0 and b>1 and increasing and concave down when a<0 and 0<b<1.
The graph of an exponential function has no x-intercepts and exactly one y-intercept.

Example

Graph the function  and find the y-intercept.

Solution

The graph is shown in Figure 3 below. To find the y-intercept, we replace x with 0 and evaluate f(0),

So, the y-intercept is 1.8. Notice that the y- intercept is the same as the value of the coefficient . This is always the case because