Question 459152
Change (x,y) to (-x,-y):  {{{(-x)^2 -y -81 = 0}}} <==> {{{x^2 - y - 81 = 0}}}, a different equation from the original.  Hence no symmetry with respect to origin.

Change (x,y) to (x,-y):  {{{x^2 - y - 81 = 0}}} <==> {{{x^2 - y - 81 = 0}}}, a different equation from the original.  Hence no symmetry with respect to x-axis.

Change (x,y) to (-x,y):  {{{(-x)^2 + y - 81 = 0}}} <==> {{{x^2 + y - 81 = 0}}}, the same equation as the original.  Hence there is symmetry with respect to the y-axis.