Creation of a General Formula for a family of Curve Shapes

I have a process that requires an operational profile that follows one of the curves in the diagram which follows. I wish to program into the device a general equation that will give me a family of curves of this general shape. The goal is to adjust the parameters of the equation so that I can generate the uppermost curve that rises quickly and at other times generate a curve like the lowest one that is almost approaching a straight line. These curves are not based on pre-existing data so I can't use a curve fitting program to generate an equation to generate additional points on the curve. I need to create a general equation that can then be used to generate any intermediate value along the curve.

In the diagram, x runs from 0 to 100 and y runs from 0 to 100. These represent percentages. So, looking at the diagram, for the uppermost curve at x=25%, y=80% while the lowermost curve is showing that at x=25%, y=50%. Since these curves represent percentages, the y value can never exceed 100. The points (0,0) and (100,100) are fixed points. You can't use a 3 point curve fit where the (25,80) point or (25,50) is the intermediate point, since such a polynomial will generate a parabola whose height exceeds 100. This curve looks more like part of a hyperbola or an arctan function of some sort.

Can anyone give me an idea of how I might find such an equation.

Attachment 23924

There are many functions which follow some shapes like those. One idea would be 100*(x/100)^c with a free parameter c. c=1 gives a straight line, c=1/2 the square root.
If the derivative should vanish at x=100, you can try the circle derivatives y=(100^c - (x-100)^c)^(1/c). c=1 gives a straight line, c=2 gives the usual circle shape.

Just wondering, is the graph somehow related to finding a signal with a lot of background?