I'm looking for the equation of a family of functions that roughly resembles the sketch below (with apologies for the crudeness of said sketch):
Properties I'm looking for:
- $\lim_{x\to-\infty}f(x)=y_1$ (i.e. approaches asymptote y=y1)
- $\lim_{x\to+\infty}f(x)=y_2$ (i.e. approaches asymptote y=y2)
- $|f'(x)|$ is at a maximum at x=0
I remember something involving ex/(??), but I can't remember exactly and can't seem to come up with the formula. I want to use it to weight tags in a tag cloud (i.e. the more recently a tag was used (smaller x), the more weight (larger y) the tag gets). The whole graph would be shifted up and to the right of what is shown in the sketch, but I know how to do that part once I have a formula.