So here's my problem. Given two parabolic curves (an upper bound and a lower bound), find a similar curve between them from one point of data that lies between the two curves. As a side note, the curves are not guaranteed to be equal to each other, and the point will never be an axial intercept (so just perform all calculations in the first quadrant).
The best example I can give of the problem is throwing a stone. One stone was thrown low, and the other high. I'm then trying to find a function that intersects with a given point (or place the stone at that point in time) and then project the rest of the curve, showing its full path. Note that the stone will not be thrown from the same Y position.
My approach to the problem was to take the integral of the upper and lower bound, then take the difference of those results to get the area between curves. From there I divided the resulting integral by two in order to get a decent approximation of the space the inner curve would occupy.
I'm not sure if I'm on the right track, or if there's a more precise and/or simple approach to the problem. Any advice is appreciated!
If $y=f_1(x)$ and $y=f_2(x)$ are the two given curves, then your new curve could be $$y=\alpha f_1(x)+(1-\alpha)f_2(x),$$ where parameter $\alpha$ is chosen so that the given point $P=(\bar x,\bar y)$ lies on the curve: $$ \bar y=\alpha f_1(\bar x)+(1-\alpha)f_2(\bar x), $$ whence $$ \alpha={\bar y-f_2(\bar x)\over f_1(\bar x)-f_2(\bar x)}. $$