## Algebraic Geometry: Elliptic Quartic Curve

This is a problem about an elliptic quartic in $P^{3}$

$Y$ is defined by
$x^{2} - xz - yw = 0$ and
$yz - xw -zw = 0$
Let $P$ be the point $(0,0,0,1)$

Let $f$ be the projection from $P$ to the plane $w=0$.

I need to show that $f$ induces an isomorphism of $Y - P$ with the plane cubic curve $K$ defined by:
$zy^{2} - x^{3} + xz^{2}=0$
minus the point $Q:=(1,0,-1)$

The point of the exercise is to prove that $Y$ is irreducible and nonsingular. This follows after showing that $f$ is an isomorphism since then we can show that $K$ is irreducible and nonsingular which is straightforward.

I have been trying to show that $f(Y)$ is contained in $K$ and vice versa.

Am I correct to right the $f(Y)$ as the equations that define $Y$ with $w==0$?

if so $f(Y)= x^{2} - xz = 0 = yz$ (excluding the point $(0,0,0)$)

But the algebra is not working out for me. Is there another approach?