I want to generate a simulator for behavior that is displayed via charts.
The end result will be a behavior that follows some formula like
$$f(x) = \sin(x)$$
which will output a sine behavior.
However, I want to create my own “handmade” graphs to simulate different scenarios. Is there a tool online where I can draw a graph and it will output its formula (in whatever complexity or approximation)?
Yes and no.
There are tools which allow you to design so-called free-form curves obtained by (polynomial or piecewise polynomial) interpolation or by approximation. These will be defined on a finite domain.
https://www.ibiblio.org/e-notes/Splines/Intro.htm
But if you want special features like vertical tangents, cornerpoints, periodicity, specific asymptotic behavior$\ldots$ polynomials won’t suffice.
For a better answer, you need to be more specific.