Q:

does anyone know how to graph $x^2+2y^2+3z^2=12$?

I just can't think of how I should draw this graph in 3 dimensions.
Can anyone draw a graph for this?

A:

As pointed out in the comments, the graph is indeed an ellipsoid. In the event you don't have a 3D graphing program handy, here's one way to think about it: we have the equation
$$x^2+2y^2+3z^2=12.$$
We'll set the coordinates $x,y,z=0$ (one at a time) and this will tell us what the graph looks like in the $yz$, $xz$, and $xy$-planes respectively. For example, if we let $x=0$,
$$0^2+2y^2+3z^2=12\implies \frac{y^2}{6}+\frac{z^2}{4}=1.$$
So in the $yz$-plane, we have an ellipse. Specifically, one that's longer along the $y$ axis than along the $z$ axis. You can check that the other planes look like ellipses too, so graphing one ellipse in each plane and "filling it in" will give a good hand-drawn graph.

