# Exponential Functions

The "basic" exponential function is the function

$y={a}^{x}$

where $a$ is some positive constant.

For example, the graph of $y={2}^{x}$ looks like this:

Note that:

$1$ ) The $y$ -intercept is $1$ (no matter what the value of $a$ is).

$2$
) The graph approaches the
$x$
-axis asymptotically as
$x$
goes to negative infinity (or as
$x$
goes to
**
positive
**
infinity, if
$0<a<1$
).

$3$
) The graph is
**
always positive
**
(never zero or negative).

The exponential function can be shifted $k$ units upwards and $h$ units to the right with the equation:

$y={a}^{x-h}+k$

**
Example:
**

Graph the equation.

$y={2}^{x-3}+2$

Start with the "basic" exponential graph $y={2}^{x}$ . Then shift the graph three units to the right and two units up.