Symmetry In Equations - Math Is Fun

srpe7BAwCuc Symmetry in Equations

Equations can have symmetry:

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

The main types of symmetry are:

• y-axis symmetry: mirror across the y-axis
• x-axis symmetry: mirror across the x-axis
• diagonal symmetry: mirror across the line y = x
• origin symmetry: a half-turn (180° rotation) around the origin

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 = x2 symmetric about the y-axis?

Try to replace x with −x:

y = (−x)2

Since (−x)2 = x2 (multiplying a negative times a negative gives a positive), there's no change

So y = x2 is symmetric about the y-axis

When a graph has y-axis symmetry, then any point (x, y) on the graph has a matching point (−x, y) on the graph.

For Symmetry About X-Axis

For symmetry with respect to the X-Axis, check to see if the equation is the same when we replace 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.

So y = x3 is not symmetric about the x-axis

Example: Is x = y2 symmetric about the x-axis?

(Note: this is not y = x2, but x = y2.)

Try to replace y with −y:

x = (−y)2

Since any negative number squared becomes positive, (−y)2 = y2. The equation becomes:

x = y2

There's no change from the original equation!

So x = y2 is symmetric about the x-axis.

See what it looks like: ../algebra/images/equation-graph.js?fn0=x%3Dy^2&xmin=-2.5&xmax=5.5&ymin=-2.906&ymax=2.9

Note: this equation isn't a genuine function, as it has more than one y-value for a given x-value.

When a graph has x-axis symmetry, then any point (x, y) on the graph has a matching point (x, −y) on the graph.

Diagonal Symmetry

Try swapping y and x (i.e. replace both y with x and x with y).

When a function has diagonal symmetry, it is its own inverse!

Example: does y = 1/x have Diagonal Symmetry?

Start with:

y = 1/x

Try swapping y and 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!

1154, 1155, 2463, 2464, 2465, 4042, 4043, 4044, 4045, 4046 Symmetry Common Functions Reference Algebra Index