Graphing Exponential and Logarithmic Functions

Graphing Exponential Functions

A simple exponential function to graph is $y = 2^{x}$ .

$x$	$- 3$	$- 2$	$- 1$	$0$	$1$	$2$	$3$
$y = 2^{x}$	$\frac{1}{8}$	$\frac{1}{4}$	$\frac{1}{2}$	$1$	$2$	$4$	$8$

Math diagram

Notice that the graph has the $x$ -axis as an asymptote on the left, and increases very fast on the right.

Changing the base changes the shape of the graph.

Math diagram

Replacing $x$ with $- x$ reflects the graph across the $y$ -axis; replacing $y$ with $- y$ reflects it across the $x$ -axis.

Math diagram

Replacing $x$ with $x + h$ translates the graph $h$ units to the left.

Math diagram

Replacing $y$ with $y - k$ (which is the same as adding $k$ to the right side) translates the graph $k$ units up.

Math diagram

The function $y = \log_{b} x$ is the inverse function of $y = b^{x}$ . So, it is the reflection of that graph across the diagonal line $y = x$ .

When no base is written, assume that the log is base $10$ .

$x$	$\frac{1}{1000}$	$\frac{1}{100}$	$\frac{1}{10}$	$1$	$10$	$100$	$1000$
$y = \log x$	$- 3$	$- 2$	$- 1$	$0$	$1$	$2$	$3$

Math diagram