# Symmetry in Equations

Equations can have symmetry:

 Graph of x2 Graph of 1/x Symmetry about y-axis Diagonal symmetry

In other words, there is a mirror-image.

## Benefits

The benefits of finding symmetry in an equation are:

• you will understand the equation better
• it is easier to plot
• it can be easier to solve. When you find a solution on one side, you can then say "also, by symmetry, the (mirrored value)"

## How to Check For Symmetry

You can often see symmetry visually, but to be really sure you should check a simple fact:

Is the equation unchanged when using symmetric values?

How you do this depends on the type of symmetry:

For symmetry with respect to the Y-Axis, check to see if the equation is the same when you replace x with -x:

### Example: is y = x2 symmetric about the y-axis?

Try to replace x with -x:

y = (-x)2

Since (-x)2 = x2 (multiplying a negative times a neagtive gives a positive), there will be no change

Hence y = x2 is symmetric about the y-axis

Use the same idea as for the Y-Axis, but try replacing y with -y.

### Example: is y = x3 symmetric about the x-axis?

Try to replace y with -y:

-y = x3

Now try to get the original equation:

Try multiplying both sides by -1:

y = -x3

It is different.

Hence y = x3 is not symmetric about the y-axis

### 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?

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.

Hence y = 1/x has Diagonal Symmetry

### Origin Symmetry

 Origin Symmetry is when every part has a matching part: the same distance from the central point but in the opposite direction.

Check to see if the equation is the same when you replace both x with -x and y with -y.

### Example: does y = 1/x have Origin Symmetry?

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.

Hence y = 1/x has Origin Symmetry

Amazing! y = 1/x has origin symmetry as well as diagonal symmetry!