The Graph of an Exponential Function
The graph of 
Domain and Range
For 
, we may use any real number as an exponent on a. Therefore, the domain of 
is all real numbers. As we’ve observed, 
for all x, but the function f(x) approaches 0 as x approaches negative infinity. As x approaches positive infinity, f(x) increases without bound. Thus the range of f(x) is all positive real numbers.
Intercepts
For all a > 1, 
. The graph has an y-intercept at (0, 1). Since the function is never zero, there is no x-intercept.
Asymptotes
The domain of the function is all real numbers, so there are no vertical asymptotes. As x approaches negative infinity, the value of f(x) approaches 0. Thus the line y = 0 is a horizontal asymptote. (As x approaches positive infinity, f(x) approaches positive infinity. This end behavior yields no additional asymptotes).
Shape of the graph
The function 
is continuous on its domain. Further, the function is strictly increasing; thus it contains no maximum or minimum points. The graph will always contain the points (1, a) and 
. With these, you should be able to create a reasonable sketch of the graph.
When 0 < a < 1
We may again use any real number as the exponent on a. For these values of a, the function will be strictly decreasing. As x approaches positive infinity, f(x) approaches 0; as x approaches negative infinity, the function approaches positive infinity. All other attributes remain the same. Note that the graph of y = ax is a “y-axis reflection” of the graph y = bx if a and b are reciprocals.
