Here is an sketch I drew up, I'm not entirely convinced that if I found it's equation we would use it, but it was my first idea.
http://i.imgur.com/57TuSZw.jpg (ignore the scribble
)
@ the posted image:
yes, i have come across those equations in my searches. i would like something that changes concavity once it reaches roughly .5. nothing extremely fancy like my sketch, but something of the likes. my sketch doesn't reflect this paragraph, btw. i'm probably confusing you all. sorry about that.