Determine algebraically whether the graph of the function g is symmetric with respect to the x-axis, y-axis, or origin. g(x)=[(x^2)+(y^2)]^2=2xy

I would plot it. IF you have EXCEL, you can generate a plot.

http://images.google.com/imgres?imgurl=http://www.mccd.edu/faculty/powerd/M4c/M4c_Lab_Surface_files/image002.jpg&imgrefurl=http://www.mccd.edu/faculty/powerd/M4c/M4c_Lab_Surface.htm&h=288&w=388&sz=17&hl=en&start=1&sig2=9dJPuM8wbuHi73vSTKNr9A&um=1&usg=__5P5zcwrFpUZA7xffNrUr2BSqULc=&tbnid=k9n1el8mRJR8EM:&tbnh=91&tbnw=123&ei=bu3GSIj0MpvUMIjojSI&prev=/images%3Fq%3D)%253D%255B(x%255E2)%252B(y%255E2)%255D%255E2%253D2xy%2B%2B%2Bgraph%26um%3D1%26hl%3Den%26client%3Dfirefox-a%26rls%3Dorg.mozilla:en-US:official%26hs%3DYmV%26sa%3DN

In this, z=x^2 +y^2-2xy and you have it.

To determine whether the graph of the function g is symmetric with respect to the x-axis, y-axis, or origin algebraically, we need to check if replacing x with -x, y with -y, or both simultaneously in the function g(x) results in an equivalent equation.

1. Symmetry with respect to the x-axis:
For a function to be symmetric with respect to the x-axis, if we replace y with -y in the equation g(x), the equation should remain the same.
Substituting y with -y in g(x) = [(x^2) + (y^2)]^2 - 2xy, we get g(x) = [(x^2) + ((-y)^2)]^2 - 2x(-y) = [(x^2) + (y^2)]^2 + 2xy.
As you can see, the equation has changed, so the graph of the function g is not symmetric with respect to the x-axis.

2. Symmetry with respect to the y-axis:
For a function to be symmetric with respect to the y-axis, if we replace x with -x in the equation g(x), the equation should remain the same.
Substituting x with -x in g(x) = [(x^2) + (y^2)]^2 - 2xy, we get g(-x) = [((-x)^2) + (y^2)]^2 - 2(-x)y = [(x^2) + (y^2)]^2 + 2xy.
Again, the equation has changed, so the graph of the function g is not symmetric with respect to the y-axis.

3. Symmetry with respect to the origin (origin symmetry):
For a function to be symmetric with respect to the origin, if we replace both x with -x and y with -y in the equation g(x), the equation should remain the same.
Substituting both x with -x and y with -y in g(x) = [(x^2) + (y^2)]^2 - 2xy, we get g(-x, -y) = [((-x)^2) + ((-y)^2)]^2 - 2(-x)(-y) = [(x^2) + (y^2)]^2 - 2xy.
Once again, the equation has changed, so the graph of the function g is not symmetric with respect to the origin.

Thus, the graph of the function g is not symmetric with respect to the x-axis, y-axis, or origin.