i believe you set y=0, and then z=0 and then x equal to, say, 1. separately.
so you get x=4z^2
y^2 + 4z^2 = 1
then to draw,
first look at the first equation.
you need to look at the xz plane.
and this will be a narrow parabola (x=4z^2)
(it should be opening up along the x axis)
then look at the next equation.
it will be on the xy plane.
x=y^2 is another parabola (it should be opening up along the x axis again, but this one will probably be flat, unlike the first - depending on how you drew your axes)
now look at your third equation - this is an ellipse. (the equation is similar to that of a circle, but it will be stretched longer in the y direction)
so now look at your graph, and go to x=1. basically, at x=1, you're going to have an ellipse in the yz plane.
connect the "traces" and you'll get something that looks like Paraboloid - Wikipedia, the free encyclopedia in some direction.
i hope that helps!
i only just learned this myself a week or two ago, so i may have made a mistake. but i hope this gives an idea.