Definitions

At the most basic level, an exponential function is a function in which the variable appears in the exponent. The most basic exponential function is a function of the form $y=b^x$ where $b$ is a positive number.

When $b>1$ the function grows in a manner that is proportional to its original value. This is called exponential growth. 

When $0>b>1$ the function decays in a manner that is proportional to its original value. This is called exponential decay.

Graphing an Exponential Function

Example 1

Let us consider the function $y=2^x$ when $b>1​$. One way to graph this function is to choose values for $x$ and substitute these into the equation to generate values for $y$. Doing so we may obtain the following points:

$(-2,\frac{1}{4})$, $(-1,\frac{1}{2})$, $(0,1)$, $(1,2)$ and $(2,4)$

As you connect the points, you will notice a smooth curve that crosses the $y$-axis at the point $(0,1)$ and is increasing as $x$ takes on larger and larger values. That is, the curve approaches infinity as $x$ approaches infinity. As $x$ takes on smaller and smaller values the curve gets closer and closer to the $x$-axis. That is, the curve approaches zero as $x$ approaches negative infinity making the $x$-axis is a horizontal asymptote of the function. The point $(1,b)$ is on the graph. This is true of the graph of all exponential functions of the form $y=b^x$ for $x>1$.

Graph of $y=2^x$

The graph of this function crosses the $y$-axis at $(0,1)$ and increases as $x$ approaches infinity. The $x$-axis is a horizontal asymptote of the function.

Example 2

Let us consider the function $y=\frac{1}{2}^x$ when $0<b<1$. One way to graph this function is to choose values for $x$ and substitute these into the equation to generate values for $y$. Doing so you can obtain the following points:

$(-2,4)$, $(-1,2)$, $(0,1)$, $(1,\frac{1}{2})$ and $(2,\frac{1}{4})$

As you connect the points you will notice a smooth curve that crosses the y-axis at the point $(0,1)$ and is decreasing as $x$ takes on larger and larger values. The curve approaches infinity zero as approaches infinity. As $x$ takes on smaller and smaller values the curve gets closer and closer to the $x$ -axis. That is, the curve approaches zero as $x$ approaches negative infinity making the $x$-axis a horizontal asymptote of the function. The point $(1,b)$ is on the graph. This is true of the graph of all exponential functions of the form $y=b^x$ for $0<x<1$.

As you can see in the graph below, the graph of $y=\frac{1}{2}^x$ is symmetric to that of $y=2^x$ over the $y$-axis. That is, if the plane were folded over the $y$-axis, the two curves would lie on each other. 

Graph of $y=2^x$ and $y=\frac{1}{2}^x$

The graphs of these functions are symmetric over the $y$-axis.

Why Must $b$ Be a Positive Number?

If $b=1$, then the function becomes $y=1^x$. As $1$ to any power yields $1$, the function is equivalent to $y=1$ which is a horizontal line, not an exponential equation.

If $b$ is negative, then raising $b$ to an even power results in a positive value for $y$ while raising $b$ to an odd power results in a negative value for $y$, making it impossible to join the points obtained an any meaningful way and certainly not in a way that generates a curve as those in the examples above.

Properties of Exponential Graphs

The point $(0,1)$ is always on the graph of an exponential function of the form $y=b^x$ because $b$ is positive and any positive number to the zero power yields $1$.

The point $(1,b)$ is always on the graph of an exponential function of the form $y=b^x$ because any positive number $b$ raised to the first power yields $1$.

The function $y=b^x$ takes on only positive values because any positive number $b$ will yield only positive values when raised to any power.

The function $y=b^x$ has the $x$-axis as a horizontal asymptote because the curve will always approach the $x$-axis as $x$ approaches either positive or negative infinity, but will never cross the axis as it will never be equal to zero.