I was reading some lecture notes online and they mentioned that there are only two plane conics up to affine coordinate change (parabola and hyperbola). However, they cite a book that does not give an explanation of this fact, and I can't figure out how to prove it.
I know this means that every (irreducible) quadratic looks like the zero set of $y-x^2$ or $xy-1$. But why is this true?
Edit: I'm working in projective space with polynomials in $\mathbb{C}[x,y]$.