I looked at this topic on rotating the parabola and noticed there's a bit of a problem
translation and rotation of a parabola
Suppose for the explicit answer as stated by the user who's answer was voted, the rotation is in fact $$y = -{\alpha\over\beta}x\pm\frac12\sqrt{4(\alpha\delta-\beta\gamma)x-4\beta\epsilon+{\delta^2\over\beta}}\tag{3}$$
such that $ \alpha$ represents $cos( \theta)$ and $ \beta$ represents $ \sin(\theta)$.
Well, if you plug in or take the limit to an angle such as $ \pi$ or $ \frac{ 3 \pi}{2}$, then one can see the right side of the equation can't simplify to any $-x^{2}$ terms as one would expect it to from a 180 degree rotation of the parabola. How is this either explained or resolved?
Despite rotation being a continuous transformation, somehow any infinitesimal rotation of $ax^{2}+bx+c$ past 0 degrees seemingly magically gets rid of the $x^2$ term and imposes this radical term, while rotating to those specific angles of $2 \pi n$ or $ \pi n$ somehow spontaneously restores them. It would seem the implicit form retains more information about the curve and allows the $x^2$ term to remain, but it is still not clear exactly what is lost by taking the explicit form prior to applying the rotation.