 Solution

A sheaf $\mathcal{F}$ of ${\mathcal{O}}_X$-modules is quasi-coherent when every affine open $U=\mathrm{Spec}A$ has $\mathcal{F}{|}_U\cong \tilde{M}$ for some $A$-module $M$.

Cover the target ${\mathbb{P}}_k^1$ by $D_{+}(y_0)$ and $D_{+}(y_1)$. On the first chart put $s=y_1/y_0$; its inverse image is $D_{+}(x_0)$ with coordinate $t=x_1/x_0$, and the morphism on rings isAs a $k[s]$-module,is free of rank two. The same calculation on the other standard chart uses $s^{-1}\mapsto t^{-2}$. Hence $f_{\ast }{\mathcal{O}}_X$ is locally free of rank two on the target.