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Alrighty, math noob here, so be nice :P. We're building an app and need an exponential function that exists between zero and one, and, depending on the importance we give it, will fluctuate between rising near the beginning or end of the graph.

Lemme know if you have any ideas or need clarification :P

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You need to clarify your question, it doesn't make much sense (at least to me). –  Olivier Bégassat Jul 16 '11 at 1:51
    
You mean like $1-\exp(-x)$? –  Emre Jul 16 '11 at 2:00
    
something like this: plus.google.com/104554751324609579088/posts/6RZbHExUmF2 –  Chris Bolton Jul 16 '11 at 2:19
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@Chris: It is if $x$ lies between $0$ and $1$, as your picture suggests. If in fact $x$ ranges over an interval $[a,b]$, use $\left(\frac{x-a}{b-a} \right)^\alpha$ for $\alpha>0$. $\alpha$ close to $0$ gives you a high importance curve, and large $\alpha$ gives you a low importance curve. You can think of $1/\alpha$ as a measure of the importance. –  Brian M. Scott Jul 16 '11 at 2:42
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"You need to clarify your question, it doesn't make much sense" - even a crude sketch of the curve you'd like to see would be awfully nice... –  J. M. Jul 16 '11 at 4:17
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up vote 1 down vote accepted

Maybe you mean power law rather than exponential. In that case @fedja's suggestion is great: $x^a$. $a<1$ for low importance and $a>1$ for high importance.

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Sounds like you want the Logistic function. The values are between 0 and 1, though the growth is centered around $0$ and it has an exponential in it. You can shift it horizontally by adding a constant to $x$ and change the steepness by changing $e^{-x}$ to $e^{-ax}$ for your choice of $a$. Does that work?

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yeah.. similar but not quite there. this looks to be an s-curve, I'm looking for more of what was in the graph I linked above. –  Chris Bolton Jul 16 '11 at 8:01
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