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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.

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

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(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.

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

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

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

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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:

 

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

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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 .

 
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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.