Exponential Functions - The Basics

Definition and equation

An exponential function is a function of the form

where the term

is called base, with

and

, and

is called exponent, and can be any real number.

The base

must be:

positive: to allow the evaluation in of every real number. In fact, if we had for example , then , but this operation does not yield a real number.
not 0 and not 1: for those values of , the exponential function degenerates to the graph of a horizontal line, respectively and .

The applet below allows you to interact with the graph of an exponential function.

Use the slider that defines the value of the base to view the shape of the graph when or .
Select the Show table checkbox to view a table of values for the displayed function: three of these values are already defined, that is (the inverse value of the base), (the y-intercept) and (the value of the base). These are the three main points that you should always use to draw the graph of an exponential function. Choose the fourth x value at which you want to evaluate the function by dragging the point on the x-axis. (All the values in the table are approximated to 2 decimal places).
Select the Monotonicity checkbox to view and explore the formal definition of increasing or decreasing function applied to the current graph, by dragging the points on the x-axis.
Select the Show checkbox to view the graph of the exponential function with base , that is a mathematical constant: a not terminating decimal number that has a great importance in applied mathematics.

Given an exponential function

, with

and