A graph is "symmetric about the orgin" if and only if "whenever (x, y) is on the graph, so is (-x, -y)".

A graph is "symmetric about the y axis" if and only if "whenever (x, y) is on the graph, so is (-x, y)".

If a graph is symmetric about the originandsymmetric about the y axis, then, if (x, y) is on the graph so is (-x, -y) because of the symmetry about the origin, and then so is (x, -y) because of the symmetry about the y-axis.

But that says "whenever (x, y) is on the graph, so is (x, -y)" which is the same as "symmetric about the x axis".