Just a question for fun:
I noted that the curves $y=x^2$ and $y=\sqrt{x}$ are in effect 'rotational transformations' of 90 degrees from each other when plotted on a graph, is there another equation which perhaps models, say a 45 degree rotation of the equation $y=x^2$ or generally any rotation such that the line $y=mx$ acts as a line of symmetry for the equation?
here is quick visualization of the problem: 
Any information on the topic would be greatly appreciated!
On
A parabola is a locus. Specifically, the parabola $y=x^2$ is the set of points that are at the same distance from the point $(0,1/4)$ (the focus) and the line $y=-1/4$ (the directrix).
If you rotate the point and the line, you get the parabola you are asking for. The focus moves to $$\left(\frac1{4\sqrt 2},\frac1{4\sqrt 2}\right)$$ and the directrix becomes $$x+y=-\frac1{2\sqrt 2}$$
There is much information about this topic in the Wikipedia's article about parabolae.
To rotate $y=a x^2+bx+c$ about the origin, you need to replace $x$ and $y$ by $(x \cos (\theta )-y \sin (\theta ))$ and $(x \sin (\theta )+y \cos (\theta ))$, respectively. The replacements are just the coordinates of a 2D rotation matrix with angle $\theta$ times the vector $(x,y)$.