Sketch the graph and describe the vertical and horizontal traces.

$\displaystyle f(x,y) = y^2$

Is this simply a parabola that touches ONLY the yz-plane in R^3?

It __can't__ be a parabola since it is a 2-variables function and thus is a surface in $\displaystyle \mathbb{R}^3$. Take, for example, the

values $\displaystyle f(0,y)$. Here, you get the usual parabola on the yz plane...but it is THE SAME for ANY value of x, so you actually get

a paraboloid every perpendicular to the xy-plane cut of which looks at the well-known parabola $\displaystyle x=y^2$.

Go to the site Functions of Two Variables to see how this function looks.

Tonio
Would the traces basically be:

Horizontal:

$\displaystyle \pm\sqrt{c} = y } when z = c$

Vertical:

$\displaystyle z = y^2 \} when x = a$

$\displaystyle z = b^2 \} when y = b$