An expression is "symmetric about the x-axis" if replacing x by -x gives exactly the same result. For example, is symmetric about the x-axis because replacing x by -x you get , the same as before.
An expression is "symmetric about the y-axis" if replacing y by -y gives exactly the same result. For example, |y|= x is symmetric about the y-axis because replacing y by -y gives |-y|= |y|= x.
An expression is "symmetric through the origin" if replacing both x and y by -x and -y, respectively gives the same thing. It is easy to see that any expression that is both "symmetric about the x-axis" and "symmetric about the y-axis" is necessarily "symmetric through the origin. and |x|+ |y|= 1 are examples. However, xy= 1 is symmetric through the origin but not symmetric with respect to the x-axis or y-axis.