Question

an equation is being tested for symmetry with respect to the x-axis, the y-axis, and the origin. explain why, if two of these symmetries are present then remaining one must also be present.

Answer 1

For x-axis reflection symmetry,
f(x,y) = f(x,-y)

For y-axis reflection symmetry,
f(x,y) = f(-x,y)

For symmetry about the origin,
f(x,y) = f(-x,-y)

If you reflect about both x and y axes, it is equivalent to reflecting about the origin.

If you reflect about the origin and then one axis, you have reflected about the other axis.