For Symmetry About Y-Axis, let's try to replace x with −x, and check if the equations still equal:
-x - y² = 0
The equation changed, so there is no symmetry about Y-Axis.
For Symmetry About X-Axis, let's try to replace y with −y, and check if the equations still equal:
x - (-y)² = 0
x - y² = 0
The equation stayed the same, so there is a symmetry in X-Axis.
Lets try Origin Symmetry Replacing x with −x and y with −y:
-x - (-y)² = 0
-x - y² = 0
The equation changed, so there is no Origin Symmetry.