Answer
Elliptic Paraboloid
Work Step by Step
Rewrite as: $y=4x^{2}+2z^{2} \implies \displaystyle \frac{y}{1}=\frac{x^{2}}{(1/2)^{2}}+\frac{z^{2}}{(\sqrt{2})^{2}}$
or, $\displaystyle \frac{y}{4}=\frac{x^{2}}{1}+\frac{z^{2}}{2}$
On comparing the above form we find that we have an Elliptic Paraboloid
$\displaystyle \frac{z}{c}=\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}$
Thus we have:
Traces in planes y=k are ellipses parallel to the xy-plane and traces in x=k and y=k are parabolas parallel to the yz-plane.
