[Math] Creating a function similar to sin(x) with steeper downturns but equal peaks and lows

graphing-functions

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:

enter image description here

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 $H\circ \sin$.

plot of H

modfied sin

Code:

h[x_] := Piecewise[{{-1, x <= -1/3}, {1, x >= 1/3}}, 3 x ]
A = 1/4;    (*parameter for Gaussian;  smaller gives narrower peak *)
G[x_] :=  Exp[-(x/A)^2/2]/(A Sqrt[2 Pi]); 
H[y_] = Convolve[h[x], G[x], x, y]; 
L[x_] :=  H[Sin [x]]
Plot[h[x], {x, -2, 2}]
Plot[H[x], {x, -2, 2}]
Plot[L[x],  {x, -10, 10}]
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