**Types of Symmetry**

When dealing with the graphs of relations, there are times when we recognize certain types of symmetry. There are three types we address here: x-axis symmetry, y-axis symmetry, and origin symmetry.

__x-axis Symmetry__

We say that a graph has “x-axis symmetry”, or is “symmetric about the x-axis”, when its graph would look the same if it were reflected about the x-axis. So a graph is symmetric about the x-axis if whenever the point is on the graph, so is the point . For example, these two graphs have x-axis symmetry.

Here is an example of a graph *without* x-axis symmetry.

(Note that with the exception of , if a graph has x-axis symmetry, it is not a function)

__Y-axis Symmetry__

We say that a graph has “y-axis symmetry”, or is “symmetric about the y-axis”, when its graph would look the same if it were reflected about the y-axis. So, a graph is symmetric about the y-axis if whenever the point is on the graph, so is the point . For example, our standard parabola has y-axis symmetry.

Here is an example of a graph *without* y-axis symmetry:

This doesn’t have y-axis symmetry because, if we reflected it about the y-axis, we would end up with this:

Here’s another example of a graph that doesn’t have y-axis symmetry, and its reflection:

In order to determine if a graph has y-axis symmetry, we must perform one simple test. We need to check that is in the graph whenever is in the graph. In the case for the graph of a function, we need to check if the function has the property that , for __all__ , that is, is an even function.

__Origin Symmetry__

A graph is said to have “origin symmetry”, or is “symmetric around the origin”, when its graph would look the same if it were rotated 180 degrees around the origin. So, a graph is symmetric about the origin if whenever the point is on the graph, so is the point . Our standard parabola does not have this symmetry, since this is what we would get if we rotated it in such a way:

Here are a couple of examples of graphs that are symmetric about the origin:

The test for origin symmetry is very similar to the ones for x-axis symmetry and y-axis symmetry. We need to check that is on the graph whenever is. In the case of a graph of a function we must check that , for __all__ , that is, is an odd function.

__Example 1__

Does the graph of the function have y-axis symmetry?

__Solution 2__

We need to check to see if .

.

Therefore, we see that the function must have y-axis symmetry. Note that f(x) here is an even function.

__Example 3__

Does the graph of the function have y-axis symmetry?

__Solution 3__

Again, we check to see if .

This time, since there are values of for which , we can say that the graph of does __not__ have y-axis symmetry.

__Example 4__

Does the graph of the function have origin symmetry?

__Solution 4__

We must check to see if .

Since for all values of , we know that the graph of the function must be symmetric around the origin.

__Example 5__

Does the graph of the function have origin symmetry?

__Solution 5__

We check to see if .

Since , we know that the graph of does not have origin symmetry.