A question about field of fractions of a ring of functions

Question

Let $k$ be a field, $f(x,y) \in \bar{k}[x,y]$ an irreducible polynomial. Define $C_f := \bar{k}[x,y]/(f)$ and $\bar{k}(Z_f) = Frac(C_f)$.

Now, there is an equality and a claim that I can not understand how and any help will be appreciated:

1. $\bar{k}(Z_f) = (\bar{k}(x)[y])/(f)$
2. $\bar{k}(Z_f) $ is an extension of $\bar{k}(x)$ of degree $f$.