I'm searching for a function that looks somewhat like a shifted
$-\tanh(x)$-function
Through some searching and playing with Wolfram Alpha I managed to shift it in the x-direction, which is partly what I want. Now the plot looks like that and the function like that:
$-\tanh(x-t)$
I also found out that there is an (at least similar) exponential form of this, perhaps it might help anybody: $-\frac{(e^{x-2t}-e^{-x})}{(e^{x-2t}+e^{-x})}$ What I can't figure out at the moment is how to make the function fulfil the following requirements:
Edited requirements, sorry!
- $f(0) \approx 1$
- "stretch" and "compress" the function in x-direction. What I try to reach is a function with a variable $t$ that also fulfils the following conditions:
- $f(t)=0$
- $f(2t)\approx-1$
By $\approx$ I mean something near $1$ and $-1$. Perhaps as near as $0.9$ or even as near as I want it to be?
I hope I made myself clear, it's a long time since I actually used mathematics...
Instead of $-\tanh(x-t)$, try $-\tanh(c(x-t))$ for $c > 1$. That will cause the function to get close to $-1$ faster.