Symmetry of Graphs

Some graphs of functions are symmetric. In mathematics, there are three kinds of symmetry we observe in graphs of functions.

1. Symmetry with respect to the x-axis.
2. Symmetry with respect to the y-axis.
3. Symmetry with respect to the origin.

Symmetry with Respect to an Axis

Some graphs are symmetric with respect to the x-axis and/or symmetric with respect to the y-axis. Here is a figure from the e-text showing symmetry with respect to each of the axes.

If we have know the equation for a graph, we can test whether or not the graph is symmetric with respect to either of the axes by testing the following:

1. A graph of an equation is symmetric with respect to the y-axis if the replacement of x with -x results in an equivalent equation. A function with y-axis symmetry is also called an even function.
2. A graph of an equation is symmetric with respect to the x-axis if the replacement of y with -y results in an equivalent equation. The only functions that are symmetric with respect to the x-axis are the function $f\left(x\right)=0,\:$which represents the x-axis, or any other function made up entirely by some subset of the points on the x-axis. 

Symmetry with Respect to the Origin

The third type of symmetry involves the origin, which is the point (0,0). A graph is symmetric with respect to the origin if rotating the graph 180 degrees about the origin results in the identical graph.

A graph is symmetric with respect to the origin if replacement of both x with -x and y with -y at the same time results in an equivalent equation. A function that is symmetric about the origin is called an odd function.

Here is a figure from the e-book summarizing the tests for symmetry:

Here is a video that gives a nice introduction to symmetry if you would like some more explanation or visualization of the three types of symmetry

And here is a video showing some examples of testing for symmetry using equations: