I need to modify the equation of the curve $y=a-be^{-cx}$ so that it looks wavy (using sine function), where the amplitude decreases as $x$ increases, and it has an upper bound.
To illustrate this, I just draw how it looks, so excuse me for the imperfect drawing;
I tried to replace $x$ with $sin(x)$, that is, $y=a-be^{-c \sin(x)}$, but did not work.
Also, I tried to add one more constant $d$, so the equation is $y=a-be^{-c \sin(dx)}$ still did not work.
It is okay for me to have any form other than $y=a-be^{-cx}$ as long as the curve will look as illustrated above. One more restriction is that the curve must have finitely many terms, so I can not use series such as Fourier series.
Your help would be appreciated. THANKS.
The function $y=a-(2+\sin x)be^{-cx}$ will work. The $2$ can be changed to any number $\ge1$ and the limiting value will still be an upper bound for the function; if it's changed to exactly $1$, the upper bound will be attained over and over again.