I would like to create smooth curves, which have $f(0) = 0$ and $f(1) = 1$.
What I would like to create are curves similar to the gamma curves known from CRT monitors. I don't know any better way to describe it, in computer graphics I used them a lot, but in math I don't know what kind of curves they are. They are defined by the two endpoints and a 3rd point.
What I am looking for is a similar curve, what can be described easily in math. For example with a simple exponential function or power function. Can you tell me what kind of curves these ones are (just by lookin at the image below), and how can I create a function which fits a curve using the 2 endpoints and a value in the middle?
So what I am looking for is some equation or algorithm what takes a midpoint value $f(0.5) = x$, returns me $a, b$ and $c$ for example if the curve can be parameterized like this (just ideas):
$a \exp (bt) + c$ or $a b^t + c$
Update: yes, $x^t$ works like this, but it gets really sharp when $t < 0.1$. I would prefer something with a smooth derivative at all points. Thats why I had exponential functions in mind. (I use smooth here as "not steep")