What is this line equation?

I am working on a led lighting project that will dim lights. linear dimming isn't working, because the middle values appear to be the same brightness...

The changes happen at the lower and upper values, so I'd like to use a relationship like is pictured in the upper graph of the picture, (not exactly exponential) but I can't remember what how to modify an exponential equation to get that result... I've had to guess, but it's a little off...

Any help would be appreciated?