Level curves and sketch of (x-a)*(y-b) type functions

Question

While doing exercises in level curves I ran across these two functions which I couldn't get my head around. The functions are:

1. $(x^2-1)(y^2-1)=0$

Here showing graphs of YZ-plane and XZ-plane sections is easy, they are just parabolas facing downwards, but obviously this is not a paraboloid, so I'm at a loss here.

2. $(x-1)(y+1)(z+2)=0$

I don't even have an idea how to start this one.

Answer 1

For the first case we have

$$(x^2-1)(y^2-1)=0 \implies (x^2-1)=0 \quad \lor \quad (y^2-1)=0 $$ $$\implies x=1 \quad \lor \quad x=-1 \quad \lor \quad y=1 \quad \lor \quad y=-1$$

which represent for planes (and their intersection lines).

For the second case we have

$$(x-1)(y+1)(z+2)=0 $$ $$\implies x=1 \quad \lor \quad y=-1 \quad \lor \quad z=-2$$

which represent three planes (and their intersection lines).