Answer
Elliptic Paraboloid
Work Step by Step
\[x=y^2+4z^2\]
Evaluated at \(x=0\):
\[
y^2+4z^2=0 \\
x=0 \quad y=0
\]
evaluated at \(x=4\):
\[
y^2+4z^2=4 \\
\frac{y^2}{4}+\frac{4z^2}{4} = 1 \\
\frac{y^2}{(2)^2}+\frac{z^2}{(1)^2} = 1 \\ \quad \text{Ellipse with radius 1 on z-axis and 2 on x-axis}
\]
Other way of solving is that it follows the standard formula of the Elliptic Paraboloid:
\[\frac{z}{c}=\frac{x^2}{a^2}+\frac{y^2}{b^2}\]
In this case the paraboloid is oriented along the \(x\)-axis (instead of the \(z\)-axis as in the standard form).
