I am trying to understand fibers of morphisms and want to solve the following problem:
Let $X = \operatorname{Spec}(K[x,y]/(y^2-x^3))$ be Neile's parabola over a field $K$. The morphism $\phi: K[x,y]/(y^2-x^3)\rightarrow K[t]$ given by $x\mapsto t^2$ and $y\mapsto t^3$ defines a $K$ - morphism of $K$ - schemes $f=\phi^{a}: \mathbb{A}^1_K\rightarrow X$.
I want to determine the fibers of $f$. I am well aware of the forumla $f^{-1}(x) \simeq \operatorname{Spec}(\kappa(x))\times_X \mathbb{A}_K^1$, where $\kappa(x)$ is the residue field of $x\in X$. But I can't make sense of how to get anything from this here...