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




*       y
/
/
/
x
Consider the following surface: .$\displaystyle x \:= \:4y^2 + 4z^2$
What's this surface called? It is a paraboloid.
What coordinate plane does $\displaystyle z=0$ define? $\displaystyle z = 0$ is the $\displaystyle xy$plane (the "floor" of the graph).
What's the trace of this surface in the $\displaystyle xz$plane look like? The $\displaystyle xz$plane is the "left wall" of the graph.
Let $\displaystyle y = 0$ and we have: .$\displaystyle x = 4z^2$
This is a parabola on the "left wall", vertex at the origin,
. . opening in the positive $\displaystyle x$direction.
What's the trace of this surface in the planes $\displaystyle x=k$ look like?
Let $\displaystyle x = k$. .We have: .$\displaystyle 4y^2 + 4z^2\:=\:k\quad\Rightarrow\quad y^2 + z^2\:=\:\frac{k}{4}$
These are circles: centered on the $\displaystyle x$axis with radius $\displaystyle \frac{\sqrt{k}}{2}$