I am attempting to write a function with the following properties:
The highest possible value is 100, and the further you get from a certain value, the lower the result the function will return... I created a simple bell-like curve by subtracting x^2 from an apex of 100:
$$f(x)=100-x^2 $$
I wanted the curve to span from 0 to 100, so I found it necesary to make the apex occur at x=50:
$$f(x)=100 - \frac{1}{25} \cdot (x-50)^2 $$
The function produces exactly what I'm looking for, but only for one value, 50. I would like to shift the apex horizontally while maintaining the x intercepts of 0 and 100. That is, I would like a curve similar to this one, except the apex can be any number greater than 0 and less than 100. Can this be accomplished simply?
One choice is to squash your parabola on one side of the peak and expand it on the other. Say you want the peak at $m$. Let $$g(x)=\begin {cases} 50\frac xm & x \le m \\100-50\cdot \frac {100-x}{100-m}& x \ge m \end {cases}$$ and your function is then $$f(g(x))=100-\frac 1{25}(g(x)-50)^2$$