Hope this helped!
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 neagtive gives a positive), there is no change
So 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.
So 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.
So y = 1/x has Diagonal Symmetry
Origin 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!
