Answer
The graph will look like this:
Work Step by Step
Sketch $f(x,y)=x^2 + 9y^2$.
We know $f(x,y)=z$ so $z = x^2 + 9y^2$
When looking at this function we know that it will be parabolic.
To find the coordinate where the graph crosses the $z$-axis, fill in x = 0 and y = 0. This gives:
$z = 0^2 + 9*0^2=0$
So it crosses at $z=0$
Make $x$ and $y$ clear:
$x= \sqrt{z-9y^2}$
$y=\sqrt{ \frac{z-x^2}{9}}$
Plugging in 0 for z and y gives $x= \sqrt{0-9\times 0^2}=\sqrt0=0$
Plugging in 0 for z and x gives $y=\sqrt{ \frac{0-0^2}{9}} = \sqrt 0 = 0$
The graph crosses the origin!
By filling in more numbers for $x$, $y$, and $z$, you could plot the whole graph.
The final graph looks like this: