Symmetry in Equations
Equations can have symmetry:
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| 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:
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
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Origin Symmetry is when every part has a matching part:
- the same distance from the central point
- but in the opposite direction.
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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!
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