How graph function $z^2 = xy$ If we want to graph $z^2=xy$ we get this picture.

My question is without using any application how can I deduce that graph of above function will be this? For one-variable we are calculating for example derivatives finding extremas and then we plot using that information.For example it's clear that we only should look where $x>0,y>0$ and $x<0,y<0.$ But how understand that it will be conic shape?  Thank you.
 A: The method I learned long ago was: rotate to eliminate mixed products.  In this case, rotate about the $z$-axis by 45 degrees to eliminate the $xy$ term.  That is, use the substitution
$$
x = \frac{X+Y}{\sqrt{2}},\qquad y = \frac{X-Y}{\sqrt2}
$$
With that substitution, the equation $z^2 = xy$ becomes
$$
z^2 = \frac{X^2}{2}-\frac{Y^2}{2} ,
$$
which (hopefully) you can recognize as a cone.
Rearranging,
$$
Y^2 = X^2+2z^2
$$
I can tell this is a cone, whose axis is the $Y$-axis [in the original
coordinates, the line $x+y=0,z=0$], and whose cross-sections perpendicular to the axis are ellipses.

Colleges and universities in the US used to have a course "Analytic Geometry" which students would take before Calculus (or concurrently).  Some time around the 1960s, there was an "improvement" to the math curriculum.  A new, combined, course called "Calculus and Analytic Geometry" replaced the two old courses.  Unfortunately, some of the topics of the old Analytic Geometry course had to be deleted.

Why is that substitution a 45 degree rotation?
The rotation in the $xy$ plane with angle $\alpha$ is
$$
x = (\cos \alpha) X + (\sin \alpha) Y\\
y = (-\sin\alpha) X + (\cos\alpha)  Y
$$
In this problem, in $xy$ I substitute that in, collect coefficients of $X^2, XY, Y^2$.  Then equate the coefficient of $XY$ to zero to find what angle rotation eliminates that term.  That's the general method.  But
I use rotation $45$ degrees so often that I just recognize what to do.
A: What you have is an equation in the three knowns $x,y$, and $z$, not a "function".
The general topic is called "quadratic surfaces", which is closely related to quadratic forms. For surfaces that are defined by quadratic equations, what you do is
first write it in one of the normal forms by changing coordinates. The classifications are well known.
The general process of getting the normal forms is called orthogonal diagonalization.
GEdgar's calculation shows you that you have an example of elliptic cones.
