I know the basic graph and shifting the graphs. But I have a special equation. I will explain the specialty of the functions later. Okay...let me say the specialty. This function has infinitely many solutions. I would like to know the GRAPH of the function. I can't draw the graph of this function. Please use computer and show me the picture/ graph of the following functions. Once again thank you for this wonderful site and members of this site.
The function is:$$x^y - z^2x + z^2y - y^x = 0.$$
Also, I want to know that, what kind of function it is? I mean, is it elliptic curve? or something else....
$$\begin{align} &&x^y-z^2 x+z^2 y-y^x&=0 \\ &\implies&x^y-y^x-z^2(x-y)&=0 \\ &\implies&x^y-y^x&=z^2(x-y) \\ &\implies&\frac{x^y-y^x}{x-y}&=z^2 \\ &\implies&z=\pm\sqrt{\frac{x^y-y^x}{x-y}}&=\pm\frac{\sqrt{x^y-y^x}}{\sqrt{x-y}} \\ \end{align}$$
Applying WolframAlpha to $z=\frac{\sqrt{x^y-y^x}}{\sqrt{x-y}}$ (just the $+$ part, not the $-$ part) shows several visualizations.