# Rotation of Parabolas

by Vorde
Tags: parabolas, rotation
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,489 Rotation of Parabolas No, that's not correct. Any rotation at all makes it no longer a function. Start with $y= x^2$. With a rotation through an angle $\theta$ we can write $x= x' cos(\theta)+ y' sin(\theta)$, $y= x' sin(\theta)- y' cos(\theta)$ where x' and y' are the new, tilted coordinates. In this new coordinate system, the parabola becomes $$x'sin(\theta)- y'cos(\theta)= (x'cos(\theta)+ y'sin(\theta))^2= x'^2 cos^2(\theta)+ 2x'y'sin(\theta)cos(\theta)+ y'^2 sin^2(\theta)$$. Now, if we were to fix x' and try to solve for y' we would get, for any non-zero $\theta$, a quadratic equation which would have two values of y for each x.