How do I algebraically rotate a polynomial 90 degrees CW (clockwise) or CCW (counterclockwise) in the $xy$ plane?
For example, rotate $f(x) = x^2$ clockwise $90$ degrees. I understand that a CW rotation of 90 degrees takes a point $(x,y)$ to a point $(y,-x)$, but when I substitute and simplify this, I get $-x = y^2$, then $y = \pm \sqrt{-x}$. I know it should be $y = \pm \sqrt{x}$ and I don't understand how it rotated backwards. I teach algebra II to high schoolers and they do not yet know radians or parametric equations, so I would like a simple explanation if at all possible. Thank you!
Short answer: There is a difference between rotating the plane, and rotating the coordinate grid that you've put on top of the plane. Moving one of them one way yields the same effect as moving the other the exact opposite way.
It's the same reason that $f(x-5)$ moves the graph of $f$ five units to the right compared to $f(x)$, when subtracting ought to move things to the left: it actually moves the coordinate grid to the left, keeping the graph in place. But you don't see that, as the coordinate grid is your only point of reference.