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Original post by justkevin Often, I have an idea of the shape of the function's curve, but unless it's simple (e.g., linear or exponential), I don't know how to convert it to a function. For example, with population growth I might want a curve that starts shallow, grows quickly, then levels off, but not know that such a curve is called a "sigmoid function."

Although I admire your wish to find some good mathematics to underlie your work (and I believe it is a worthwhile goal, rather than just doing something random and trying to 'balance' it later), in practice I have never needed much in the way of complication here. Straight lines and x^y+c curves fit 90% of what I need. Tweak the value of y and c to taste. For the other 10%, either sine, cosine, a log curve, or the exp function do the rest. (Although I can't remember exactly what I use for sigmoid curves on the very rare occasion that I've explicitly needed one.)

Wolfram's site has just about anything you can think of, and many I've never heard of. It's worth a second look:

http://mathworld.wolfram.com/SigmoidFunction.html

You can do some pretty cool stuff with their software as well:
http://demonstrations.wolfram.com/DiceProbabilities/