As the title states, I've been trying to create a function that is identical to $\sin(x)$, except that it has a steeper slope in the middle of the decreasing portion of the curve, but mellows out such that this function and $\sin(x)$ have the same turning point for both peak and bottom.
I've tried to describe it in the following image, if we pretend the top function is $\sin(x)$, I'd like to create something akin to the bottom image. Currently I am trying some variation of $\sin(x-0.5\sin(x))$, which gives me the steeper slope, but this shortens the actual duration and the peaks and lows don't line up.
Any hints on functional forms I could try would be appreciated!
Pardon my extremely poor paint skills:
Best Answer
What do you mean when you say "create"? You can take $H\circ \sin(x)$, where H has the properties: $H(x)$ close to $-1$ for $x \le -1/3$, $H(x)$ close to $1$ for $x \ge 1/3$, $H$ is weakly increasing, $H$ is smooth, and $H(-x) = -H(x)$ You can "create" such an $H$ by taking a piecewise linear function and then convolving with a smoothing function (e.g. normalized Gaussian).
I programmed this in Mathematica. Here's the plot of $H$, followed by the plot of $H\circ \sin$.
Code: