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Symmetry in Equations

Equations can have symmetry:

x^2 x^2
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 can make it easier to solve. If you have found a solution on one side, you can then just say "also, by symmetry, the (mirrored value)"
  • it is easier to plot

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 About Y-Axis

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

For Symmetry About X-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?

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.

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?

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.

Hence y = 1/x has Origin Symmetry

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