Answer
a. a parabola
b. a parabollic cylinder
c. a parabollic cylinder
Work Step by Step
a.
In the xy plane ($\mathbb{R}^{2}$), $y=x^{2}$
It contains the points $(x,x^{2})$
Thus we have a parabola (see image).
b.
We can say for case (a), that the points contained are $(x,x^{2},0).$
In $\mathbb{R}^{3}$ the points contained are $(x,x^{2},z)$, where z can be any value.
The surface obtained by vertically translating the parabola from the xy-plane.
This is a parabolic cylinder (see image).
c.
This surface contains points $(x,y,y^{2})$, where x can be any value.
It is obtained by translating the parabola $z=y^{2}$ (which is in the plane where x=0),
along the x-axis.
This is also a parabolic cylinder (see image).
