Help creating a graph?

Question

I need help creating a graph that resembles the attached image. I've tried to do so myself using the arctan and $x^{1/3}$ function but my results have been too steep near the center. If I am being unclear please ask for elaboration in the comments, any help I receive is greatly appreciated. Thanks again.

Answer 1

Use the logistic function as defined here with $x_0=26$ and $L=100$, and tune $k$ to achieve the desired steepness.

Answer 2

Using

\begin{align} f(x)&= \frac{1}{\pi}\left(2\arctan(x)+\arcsin\left(\frac{2x}{1+x^2}\right)\right) = \begin{cases} -1 & x\le -1 \\ -1..1 & -1\le x\le 1 \\ \phantom{-}1 & x\ge 1 \end{cases} \end{align}

and \begin{align} f_d(x)&=f(x)\,\exp(1-|f(x)|), \end{align}

plot $a+b\,f_d((x-d)/d)$.

For example, for $a=50, b=40, d=26$ we have