# Does anyone know a quick equation for a hyperbolic growth model? **URL:** \ **Category:** Factual Questions\ **Created:** [June 26, 2007, 9:31pm UTC](https://boards.straightdope.com/t/does-anyone-know-a-quick-equation-for-a-hyperbolic-growth-model/409525 "2007-06-26T21:31:48Z")\ **Posts on this page:** 6\ **Page:** 1
**Author:** ![T\_SQUARE](https://avatars.discourse-cdn.com/v4/letter/t/9de053/32.png) [@T\_SQUARE](https://boards.straightdope.com/u/T_SQUARE)\ **Post date:** [June 26, 2007, 9:31pm UTC](https://boards.straightdope.com/t/does-anyone-know-a-quick-equation-for-a-hyperbolic-growth-model/409525/1 "2007-06-26T21:31:48Z")
I need to build a quick curve that models the shape of what I believe is called hyperbolic growth. It should look something like [this](http://www.j-bradford-delong.net/multimedia/animated_gifs/Growth1.GIF) production function or this [curve](http://healthandenergy.com/images/hyperbolic_cream_curve.gif). However, these curves don’t apear to my non-mathmatical eye as the same thing as wiki’s hyperbolic [function](http://en.wikipedia.org/wiki/Hyperbolic_function). Can anyone help? ---
**Author:** ![MissMossie](https://avatars.discourse-cdn.com/v4/letter/m/f6c823/32.png) [@MissMossie](https://boards.straightdope.com/u/MissMossie)\ **Post date:** [June 26, 2007, 9:49pm UTC](https://boards.straightdope.com/t/does-anyone-know-a-quick-equation-for-a-hyperbolic-growth-model/409525/2 "2007-06-26T21:49:53Z")
I’m sure someone with more math skill than myself will be along soon, but, to me, it resembles a transformation done to the function f(x)=ln(x), or the natural log curve. ---
**Author:** ![T\_SQUARE](https://avatars.discourse-cdn.com/v4/letter/t/9de053/32.png) [@T\_SQUARE](https://boards.straightdope.com/u/T_SQUARE)\ **Post date:** [June 26, 2007, 10:03pm UTC](https://boards.straightdope.com/t/does-anyone-know-a-quick-equation-for-a-hyperbolic-growth-model/409525/3 "2007-06-26T22:03:09Z")
I thought about a natural log function, but the log function in unbounded. I think my curve should go to some kind of limit, if that’s the right word. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [June 26, 2007, 10:09pm UTC](https://boards.straightdope.com/t/does-anyone-know-a-quick-equation-for-a-hyperbolic-growth-model/409525/4 "2007-06-26T22:09:38Z")
The curves in the OP look logarithmic to me. But if you need something bounded, try the arctangent or hyperbolic tangent functions. ---
**Author:** ![Omphaloskeptic](https://avatars.discourse-cdn.com/v4/letter/o/bcef8e/32.png) [@Omphaloskeptic](https://boards.straightdope.com/u/Omphaloskeptic)\ **Post date:** [June 26, 2007, 10:15pm UTC](https://boards.straightdope.com/t/does-anyone-know-a-quick-equation-for-a-hyperbolic-growth-model/409525/5 "2007-06-26T22:15:26Z")
“Hyperbolic growth” probably means a function y=-a/(x-x[sub]0[/sub]); as x approaches x[sub]0[/sub] from below, y approaches infinity. This function is a hyperbola, but one rotated so that its asymptotes are vertical and horizontal, not two vertically-mirrored diagonals as in the Wikipedia plot. It might also refer to a model based on one of the hyperbolic trig functions (like tanh), though the models seem to approach the asymptotes more slowly than the hyperbolic functions do. ---
**Author:** ![Bryan\_Ekers](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bryan_ekers/32/183_2.png) [@Bryan\_Ekers](https://boards.straightdope.com/u/Bryan_Ekers)\ **Post date:** [June 26, 2007, 10:22pm UTC](https://boards.straightdope.com/t/does-anyone-know-a-quick-equation-for-a-hyperbolic-growth-model/409525/6 "2007-06-26T22:22:27Z")
[QUOTE=Omphaloskeptic] “Hyperbolic growth” probably means a function y=-a/(x-x[sub]0[/sub]); as x approaches x[sub]0[/sub] from below, y approaches infinity. This function is a hyperbola, but one rotated so that its asymptotes are vertical and horizontal, not two vertically-mirrored diagonals as in the Wikipedia plot. [/QUOTE] You could always write the autonomous first-order equation: y’= a / (50-y) (based loosely on one of the curves linked in the OP) As y-\>50, the derivative of y (i.e. y’) tends to zero and the curve flattens.