Singularity of Morphism and Its Extension I really need help with this question. 

Assume you have a morphism $\varphi: \mathbb{A}^2 \rightarrow
> \mathbb{A}^1$ such that $\varphi (x,y)= x^2-y^4$.
1) Find all points in $\mathbb{C}$ such that $\varphi^{-1}(a)$ is
  singular. And its type of singularity.
2) If $Y_{a}$ is the closure of curve $\varphi^{-1}(a)$ in
  $\mathbb{P}^{2}$ and L is line at $\infty$ (z=0), find the point
  $Y_{a}\cap L$.
3) Let $\psi: \mathbb{P}^{2}\rightarrow \mathbb{P}^{1}$ be rational
  map that extends $\varphi$, find the domain of $\psi$ and the fiber
  $\psi^{-1}(\infty)$.

This is a reducible curve that I've never seen before. I do not know what can I do!
I found $(0,0)$ is a singular point by using Jacobian. For the second one, I took homogenization of curve but I couldn't how to find the intersection with the line at infinity. For the last one, I have no idea.

About your final remark, If I make blow-up, can I get a such a morphism $f:X\rightarrow \mathbb{P}^{1}$ which an extension of $\varphi$ ? Does it have to be $\mathbb{P}^{2}$ 
