For instance the equation is .

Check the symmetry of :

Test algebraically symmetric with respect the -axis:

Substitute for in .

Because  is equivalent to , the graph is symmetric with respect to the -axis.

Test algebraically symmetric with respect to the -axis:

Substitute for in .

Because  is equivalent to , the graph is symmetric with respect to the -axis.

Test algebraically symmetric with respect to the origin:

Substitute for and for in .

Because  is equivalent to , the graph is symmetric with respect to the origin.

Symmetry about the -axis is .

Symmetry about the -axis is .

Symmetry about the origin .

Case 1:

If the point is on the graph of the equation.

If the equation is symmetric with respect to the -axis and -axis:

Then and are passes through the graph.

Thus, are passes through the graph.

This is the origin symmetry.

Case 2:

If the point is on the graph of the equation.

If the equation is symmetric with respect to the -axis and origin:

Then and are passes through the graph.

Thus, are passes through the graph.

This is the -axis symmetry.

Case 3:

If the point is on the graph of the equation.

If the equation is symmetric with respect to the -axis and origin:

Then and are passes through the graph.

Thus, are passes through the graph.

This is the -axis symmetry.

Thus, if two symmetries are present, the remaining one must also be present.

If two symmetries are present, the remaining one must also be present.