Answer
Please see the figure attached.
Notes:
- Red dashed lines are the vertical traces parallel to the $xz$-plane
- Black dashed lines are the horizontal traces
Work Step by Step
We have $f\left( {x,y} \right) = {y^2}$. By setting $y=a$ we fix the $y$-coordinate and obtain the line (red dashed line in the figure) $f\left( {x,a} \right) = z = {a^2}$ as vertical trace that lies in the plane parallel to the $xz$-plane.
The horizontal traces can be obtained by setting $c = {y^2}$. Thus, the horizontal traces are the lines $y = \pm \sqrt c $ that lie in the plane $z=c$ (black dashed line in the figure).
