How can I rotate a parabola.

[RusselMorty]

How can I "rotate" a parabola.

I want to take an image and make it kind of wavy to distort it (like in a capcha form the letters are distorted sometimes with a wave) . So i figure that I could translate the pixels along a parabola. So the function where y=X$^{2}$ would make the image curve downwards. If i want to curve it in other directions I would need to somehow rotate that parabola. I am not an expert at math so I have no idea how to do this. I am sure it is a simple thing but It would really help me a lot. Thanks for any advice.

 

[Muphrid]

Write the parabola in parametric form:

$$y = t^2, \quad x = t$$

Pick an angle $\theta$ to rotate by. The general equations governing the new coordinates are

$$x'(t) = x(t) \cos \theta - y(t) \sin \theta \\<br /> y'(t) = y(t) \cos \theta + x(t) \sin \theta$$

(It's not strictly necessary to write in parametric, but it's usually easy to plot that way rather than go back and resolve for $y(x)$).

 

[RusselMorty]

Thank you very much for your answer. It also gives me some more insight on how to rotate other things too. It will be very helpful.