This question is quite general and has been discussed on MSE before, however my case is a little bit different and I'm wondering about the geometric interpretation of a specific example. I think that this example is primarily what I need to complete my intuition for projective spaces in algebraic geometry.
Consider the affine elliptic curve $Y=Z(y^2-x^3+x)\subseteq\mathbb A^2$. The generator $y^2-x^3+x$ of $I(Y)$ can be homogenized by adding the necessary $w$'s, yielding $wy^2-x^3+w^2x$ and, since there is only one generator, we have $I(\overline Y)=(wy^2-x^3+w^2x)$. My two questions:
1) Is it okay to visualize this primarily in the affine chart $w\neq 0$ ($\mathbb A^2$) and view the finite points on the curve as the "same thing" as the regular curve?
2) For the infinite points, if $w=0$ we find $x=0$, yielding one distinguished point $(w,x,y)=(0,0,1)$ on the line at infinity. Why, geometrically, is only one point at infinity needed to projectively close this curve as opposed to two? More generally, how should we think of the line (hyperplane) at infinity in $\mathbb P^2$?