# Symmetry in Equations

Equations can have symmetry:

Graph of x^{2}

Symmetry about y-axis

Graph of 1/x

Diagonal symmetry

In other words, there is a **mirror-image**.

## 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 Y-Axis

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

### Example: is y = x^{2} symmetric about the y-axis?

Try to replace **x** with **−x**:

y = (−x)^{2}

Since (−x)^{2} = x^{2} (multiplying a negative times a negative gives a positive), there is no change

So y = x^{2} is symmetric about the y-axis

### For Symmetry About X-Axis

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

### Example: is y = x^{3} symmetric about the x-axis?

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 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?

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:

- the
**same distance**from the central point - but in the
**opposite direction.**

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!