Parabola in a tilted coordinates

Question

Suppose,we have an equation of a parabola $$y=ax^2+bx$$in $xy$ coordinates. We want to find the equation of this parabola in a coordinate which is tilted at an angle ${\theta}$ with the xy such that $x'$ is below $x$. Is there any easy way to do this ?

Answer 1

Given a point $(x', y')$ in $x'y'$-coordinates, the $xy$-coordinates of that point is $$ (x'\cos\theta + y'\sin\theta, -x'\sin\theta+y'\cos\theta) $$ A point is on the parabola if the first and second components of the $xy$-coordinates of that point fulfills the given equation. Which is to say, if $$ -x'\sin\theta+y'\cos\theta=a(x'\cos\theta + y'\sin\theta)^2+b(x'\cos\theta + y'\sin\theta) $$