I have edited this question to make it more clear.
I have spent quite some time trying to find this on Google, but haven't succeeded.
I need the formula(s) to determine the intersection between an power curve and a parabola.
The image below shows a curve in black, and a blue power curve (ignore the other curves and the fact that the curves have a somewhat equal x-mirror image). I need to find the point where blue intersects black.
The only other scenario I have is that the arc may be on the "left" of this image, i.e. intersect the opposite shown blue exponential curve.
The power curve may be convex, concave, or flat (linear), and has the following general equation:
$ y = 1 - ( s ( x^p - i ) )^b$
$p, b, s, i$ are known positive constants; $x$ is known.
$x^p > i$
The parabola is derived from 3 known points by using the quadratic equation $ax^2 + bx + c = 0$
The $y$ apex of the parabola is always 1.
Can someone point me to a resource that gives the formula(s)?