Hello, Ideasman!
I use the standard orientation of the three coordinate axes. Code:
z




*       y
/
/
/
x
Consider the following surface: .
What's this surface called? It is a paraboloid.
What coordinate plane does define? is the plane (the "floor" of the graph).
What's the trace of this surface in the plane look like? The plane is the "left wall" of the graph.
Let and we have: .
This is a parabola on the "left wall", vertex at the origin,
. . opening in the positive direction.
What's the trace of this surface in the planes look like?
Let . .We have: .
These are circles: centered on the axis with radius