Symmetry in Equations
Equations can have symmetry:
Graph of x^{2}  Graph of 1/x 
Symmetry about yaxis  Diagonal symmetry 
In other words, there is a mirrorimage.
Benefits
The benefits of finding symmetry in an equation are:
 we understand the equation better
 it is easier to plot
 it can be easier to solve. When we find a solution on one side, we can then say "also, by symmetry, the (mirrored value)"
How to Check For Symmetry
We can often see symmetry visually, but to be really sure we should check a simple fact:
Is the equation unchanged when using symmetric values?
How we do this depends on the type of symmetry:
For Symmetry About YAxis
For symmetry with respect to the YAxis, check to see if the equation is the same when we replace x with −x:
Example: is y = x^{2} symmetric about the yaxis?
Try to replace x with −x:
y = (−x)^{2}
Since (−x)^{2} = x^{2} (multiplying a negative times a neagtive gives a positive), there is no change
So y = x^{2} is symmetric about the yaxis
For Symmetry About XAxis
Use the same idea as for the YAxis, but try replacing y with −y.
Example: is y = x^{3} symmetric about the xaxis?
Try to replace y with −y:
−y = x^{3}
Now try to get the original equation:
Try multiplying both sides by −1:
y = −x^{3}
It is different.
So y = x^{3} is not symmetric about the yaxis
Diagonal Symmetry
Try swapping y and x (i.e. replace both y with x and x with y).
Example: does y = 1/x have Diagonal Symmetry?
Start with:
y = 1/x
Try swapping y with x:
x = 1/y
Now rearrange that: multiply both sides by y:
xy = 1
Then divide both sides by x:
y = 1/x
And we have the original equation. They are the same.
So y = 1/x has Diagonal Symmetry
Origin Symmetry
Origin Symmetry is when every part has a matching part:

Check to see if the equation is the same when we replace both x with −x and y with −y.
Example: does y = 1/x have Origin Symmetry?
Start with:
y = 1/x
Replace x with −x and y with −y:
(−y) = 1/(−x)
Multiply both sides by −1:
y = 1/x
And we have the original equation.
So y = 1/x has Origin Symmetry
Amazing! y = 1/x has origin symmetry as well as diagonal symmetry!