Symmetry is when a graph seems to take on a mirror image about a certain line or point. What that means is, take a graph and to see if it is symmetric about the x-axis, mentally fold the graph on the x-axis, and see if it would look the same on the other side of the x-axis. This is a little more easily understood with a few examples. When testing for symmetry it is usually about the x-axis, y-axis, or the origin (0,0).

 

 

Example

 

Is the following graph symmetric in respect to the x-axis, y-axis, origin or none of these?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Take this graph and fold it over the x-axis, notice that the line would lie on top of itself, or in other words, it would look the same. This means there is symmetry about the x-axis.

Now fold the graph along the y-axis and this is what the graph would look like.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This is not the graph first started with, so therefore it is not symmetric about the y-axis. Test and see if it is symmetric in respect to the origin (0,0). To do this, take a two-step approach. First fold it over the y-axis and then over the x-axis. Here is the graph of the result of that.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This graph does not produce the results of the original graph either. Therefore it is not symmetric to the origin. Thus one can conclude that it is only symmetric in respect to the x-axis.

 

 

 

 

 

Example

 

Is the following graph symmetric in respect to the x-axis, y-axis, origin or none of these?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Folding it over the x-axis results in,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This is not the same graph started out with so this is not symmetric about the x-axis. Fold the original graph over the y-axis and test for symmetry about the y-axis.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This is the same graph as the original so it is symmetric about the y-axis. Now test to see if it is symmetric about the origin by first folding it over the x-axis and then folding it over the y-axis. Here is the result of that.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This is not the original graph so it is not symmetric about the origin. This graph is only symmetric about the y-axis.

 

 

 

 

Example

 

Is the following graph symmetric in respect to the x-axis, y-axis, origin or none of these?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

By folding this graph and testing all 3 possibilities one can tell that this would only be symmetric in respect to the origin.

 

Anytime that a function is considered to be symmetric to the x-axis it is called even. If it is symmetric to the origin it is called odd. If it is not symmetric to either one of those it is considered simply neither even nor odd. This would include those that are symmetric to the y-axis. If a function is even (symmetric to the x-axis) then that means f(-x) = f(x). If it is symmetric to the origin then f(-x) = - f(-x).

 

Name	Symmetric to	Function Rule
Even	x-axis	f(-x) = f(x)
Odd	Origin	f(-x) = -f(x)

 

When testing to see if a function is even, odd, or neither, apply the function rules and see what class it falls in. Here are a few examples of that.

 

 

Example

Is f(x)= x² +3 even, odd, or neither.

 

To see if it is an even function, apply the function rule.

 

f(-x) = f(x)

 

This means find out what f(-x) is and compare it to f(x).

 

f(-x)= (-x)² +3

 

Simplify.

 

f(-x)= x² +3

 

This is the same thing as f(x). Therefore this is an even function.

 

Even

 

Answer!

 

 

 

 

 

Example

 

Is f(x)= x³ +3x -3 even, odd, or neither.

Find out what f(-x) is.

f(-x)= (-x)³ +3(-x) –3

 

Simplify.

 

f(-x)= -x³ –3 x –3

 

This is not the same thing as f(x) so the function is not even. Look at the function rule for an odd function.

 

f(-x) = -f(x)

 

f(-x) = -x³ –3 x –3 so find –f(x) and compare the two functions.

-f(x)= -(x³ +3x –3)

 

Distribute out the negative.

-f(x)= -x³ -3x +3

 

Look at the two functions together.

f(-x)= -x³ –3 x –3 and -f(x)= -x³ -3x +3

 

Even though they are close, they are not equal because of the signs in front of the 3. Therefore this function is neither even nor odd.

Neither

 

Answer!

 

 

 

 

 

 

 

 

Example

Is f(x)= x⁵ -2x even, odd, or neither.

Find f(-x).

f(-x)= (-x)⁵ –2(-x)

 

Simplify.

 

f(-x)= -x⁵ + 2x

 

This is not the same as f(x) so it is not even. Find –f(x).

 

-f(x) = -(x⁵ -2x)

 

Distribute out the negative.

 

-f(x) = -x⁵ +2x

 

Now compare f(-x) and –f(x) to see if it is odd.

 

-f(x) = -x⁵ +2x and f(-x)= -x⁵ + 2x

 

These are the same, so this is an odd function.

 

Odd

 

Answer!