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I need a function which initially falls slowly, then quickly and then slowly again.

Shape should be like tan but I want to be able to control the gradient

Properties needed:

$x = 0, y=0$

As $x$ increases $y$ decreases

As $x \rightarrow \infty$, $y \rightarrow -1$

(No definition for negative $x$ needed)

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Did you try searching before asking? Even the Wikipedia has a page about similar functions, you would need to do just a small modification. – dtldarek Jan 9 '13 at 19:07
Thanks for pointing to that page . my math skills are non-existent. I know I would need to tweak that sigmoid function, but I have no other option but trial and error :/ – arahant Jan 9 '13 at 19:10
Your description does not match tan. Perhaps you need to post a picture. – Maesumi Jan 9 '13 at 19:18
@Maesumi Perhaps a generic S shape curve is a better description. See the sigmoid function link posted by dtldarek – arahant Jan 9 '13 at 19:24
up vote 1 down vote accepted

$$y = e^{-ax}-1$$ $$y = \dfrac{1+e^{-ab}}{1+e^{a(x-b)}}-1$$ should do the job. You can play with the factor $a$ to control the gradient.

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@RahulNarain Thanks. Updated. – user17762 Jan 9 '13 at 19:28
Thanks, the 2nd one works nicely – arahant Jan 9 '13 at 20:23

You can build on sample formulas on the wiki page you mentioned. For example $y=-kx/\sqrt{1+(kx)^2}$ for different values of $k$.

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