# Rotation of Parabolas

What is the maximum angle (degrees or radians) that you can rotate the basic parabola (y=x2) so that it can still be graphed as a function (y=...) with only one possible y-value per x-input.

I think it's 0, because when you include the xy factor, it doesn't become a function anymore.

But more abstractly, I think it's possible to do a slight rotation, but there's an obvious cutoff point. I'm curious where that cutoff point is, it could be zero, I can't quite picture it well enough.

HallsofIvy
Homework Helper
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.

phinds