How to sketch level curves/sections?

Question

I am asked to sketch the level curves and sections of $z = xy$. So first I sketch some level curves by setting $xy=c$ and I get a bunch of curves in all the quadrants (I know what $y=\frac{1}{x}$ looks like, so I repeat this symmetry across all 4 quadrants). So I get all these hyperbolas, and my poor imagination tells me that this graph is gonna end up looking like the edge of 4 hills that never touch each other and the tips of the edges almost meeting at the origin. So for the sections, I first try to set $x=1$ and then $y=1$ so I have $z=y$ and $z=x$, respectively. This gives me 2 planes that look like an "X" with the center at the origin...so now this picture makes completely no sense to me at all because now it seems like 4 hills being cut into pieces by a couple of rectangles up to this point. My first guess would be some sort of hyperbolic paraboloid, but I can't tell from the equation because it's not in the form $x^2 - y^2 = z$, and $z=xy$ is not a form I am familiar with. I checked the plot on the WolframAlpha website, and it looks nothing like my sketches. How do I go about sketching randlm functions like these and even more conceptually difficult ones?

Answer 1

Yes, it is a hyperbolic paraboloid: to get the form you're familiar with, take the change of variables $x=u+v$, $y = u-v$.