Help creating a mystery function with these properties


Hello /sci/, basically I'm looking for a function that
>has f(0)=0
>for some integer m , has f(m)=0.5
>is bounded from above by 1
>is roughly this shape

how would i go about getting a function like this? the basic motivation is that i have lots of data points from [0, infinity] with a median of m and i want to assign scores from [0,1) to each of these data points.

let me know if i can give any more info to make the question more clear