What does "4x^2 + y^2 = 16z" look like?

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www dot wolframalpha dot com/input/?i=plot+4x%5E2+%2B+y%5E2+-+16z+%3D+0

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Going a little old school

4x^2+y^2=16z

Start at origin (x,y,z)=(0,0,0)
On the surface

planar projection on y axis (y=0)
4x^2=16z Parabola opening upward

planar projection on x axis (x=0)
y^2=16z Parabola opening upward

so surface point on origin (0,0,0) is a minimum.

Now do some horizontal projections

z=1
4x^2+y^2=16
x^2/4+y^2/16=1
Oval minor axis 2, major axis 4

z=4
4x^2+y^2=64
x^2/16+y^2/64=1
Oval minor axis 4, major axis 8

z=16
4x^2+y^2=256
x^2/64+y^2/256=1
Oval minor axis 8, major axis 16

So 4x^2+y^2=16z

Is an oval shaped paraboloid opening upward from the origin.

Rap