Question on the twisted cubic

Question

Suppose we are working over a field $k = \overline{k}$. How do I see that the map $$\mathbb{P}^1 \to \mathbb{P}^3$$given by$$[a, b] \mapsto [a^3, a^2b, ab^2, b^3]$$gives an isomorphism of $\mathbb{P}^1$ with the variety in $\mathbb{P}^3$ cut out by $wz = xy$, $wy = x^2$, $xz = y^2$?