Hi, can anyone please help me with this question? I have no idea how to start.

(a) Consider the map $\displaystyle f(z)=z^2$. Prove that under $\displaystyle f$, lines parallel to the real axis are mapped to parabolas.

(b) Consider the map $\displaystyle g(z)=\sqrt{z}$, for some branch of the square root. Prove that under $\displaystyle g$ lines parallel to the real axis are mapped to hyperbolas.

I did try to do the questions, so I substitute $\displaystyle z=x+iy$, then get $\displaystyle f(x,y)=x^2+y^2-2ixy$, but I really don't know how to show this mapping thing. Please help me. Thanks a lot.