I got a call this afternoon from Marc, a friend of mine from college in Minnesota who now lives back East and runs his own business. He was trying to find a model for a collection of functions, but couldn’t figure out what kind of function it would be.

The basic scenario was that the function should start at (0,0), increase rapidly to a point [say, (10,10)], and then slowly decrease. The x-axis could be a practical asymptote, although it didn’t really matter since this would only be looked at in finite time.

My first thought was surge function (something of the form ), and sure enough that works. But I was on Homework Patrol, so I handed it off to TwoPi, and he came up with and . Walphaing these shows that both work well:

This seemed to help. So here’s what I’m wondering — are there any other simple functions that fit the bill? Neither function is too complicated, but it would be fun to be able to share other examples of functions that rise quickly, but then that taper off after a while. [I think the tapering should be more gradual compared to the climb, though I believe that can be controlled with more constants.