Given $\pi:X\to \mathbb{A}^2$, the blowup of $\mathbb{A}^2$ at the origin, I am trying to calculate the strict transform of $Y=\mathbb{V}(y^2-x^2(x+1))$, which has been defined as the closure of $\pi^{-1}(Y\setminus\{0,0\})$ inside $X$.
What is meant by taking the closure here? If I am not mistaken, the preimage $\pi^{-1}(Y\setminus\{0,0\})$ is contained in a patch of $X$ that is isomorphic to $\mathbb{A}^2$ - can I work in this patch and take an affine closure? Do I have to take some sort of projective closure?