I'm having trouble finding a reasonable equation for this graph:
http://i58.tinypic.com/15gtrn7.png
The x axis is the horizontal, y-axis is the axis coming out of the screen, the z-axis is vertical.
I don't need an exact equation, but a general, reasonable equation that is possible to make this graph. No specific points given.
It's hard for me to find an equation that dilates the paraboloid to be thinner along the y-axis.
It'd be nice if the equation was in the form of z^2-y^2-z^2=c, for example, where c is a nonzero constant.
This looks like the graph of $$x^2 + y^2 -z^2 = -c^2$$ to me. You can see it by rewriting it as $$x^2 + y^2 =z^2 -c^2$$ and observing the cross sections $z=\text{constant}$ (assume $c\geq 0$):
For $-c<z<c$, there is no solution, so the region between these planes doesn't meet the surface.
For $z=\pm c$, you get $x=y=0$, where the planes just are tangent to the tips of the surface.
For $z<-c$ or $z>c$, you get a circle of radius $\sqrt{z^2-c^2}$.
You can see, too, that sections by vertical planes $x=\text{constant}$ or $y=\text{constant}$ give hyperbolas that open upward and downward. There are no parabolic cross sections by any planes that I have mentioned.
Regarding dilation: just write $$ \left(\frac xa\right)^2 + \left(\frac yb\right)^2 -\left(\frac zc\right)^2 = -1 $$ and you can scale any axis to your heart's content.