Say our question requires us to use traces to sketch the quadric surface with equation $x^2+y^2/9+z^2/4=1$. This is a worked example in our book, but I'm confused by their steps. So first we set $z=k$ to get $x^2+y^2/9=1-k^2/4$. First of all, is $k$ just a constant? And what is the difference between setting $z=k$ rather than $z=0$? Also, if $k$ is a constant (which I'm not sure if it is or not, or what it even represents for that matter), then how come when graphing using Wolfram, if I put in the equation with $k$, I get an ellipsoid, but if I just substitute say $0.5$ for $k^2/4$, I get a circle? And so then we repeat the same steps above with $x$ and $y$, and the same questions apply.

-
Yes, $k$ is a constant. The way you should think of this is that $z=k$ is the plane parallel to the $xy$-coordinate plane at "height" $k$. So $k=0$ corresponds to exactly the $xy$-plane, and as $k$ increases you are getting different planes higher above that one and $k<0$ gives you planes "below" the $xy$-coordinate plane.
Substituting in different numbers gives you a "slice" of the 3D picture, so it is just the curve that you get at that height. This is why when you substitute in a value for $k$ you get just a curve. When you typed it into Wolfram with the $k$, it interpreted $k$ as a variable instead of a constant so it gave you the full 3D ellipsoid. When you just look at a plane section of the ellipsoid you get ellipses (or circles).
And so when I'm graphing this by hand and for the first step where I set z=k, do I just plot that as a one dimensional ellipse? How do I plot that part? How do I treat the $k^2/4$? –  maq Feb 10 '11 at 3:13
I'm not sure what your book does for this, but the main point is to notice that below $z=-2$, you get empty. Nothing is below that. At $z=-2$ you get a point (0,0,2). Then as $k$ goes from $-2$ to $0$, the size of the ellipses gets bigger. Then from $0$ to $2$ they get smaller back to a single point. Then higher up than $2$, is empty again. Maybe on the same 2D, $xy$ graph you could put in $k=-2$, $k=-1$, $k=0$, $k=1$, and $k=2$ and label them like that. –  Matt Feb 10 '11 at 3:21