graphing-functions
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 = sin(x)
which will output a sin 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)
Two possible guesses.
Probability Density: $\large y=ce^{-\left(\frac{x-k}{h}\right)^2}$
I don't know what a name for this is, but another similar graph is $\displaystyle y=\frac{c}{1+(h(x-k))^2}$.
It would help to know more criteria since there are a lot of possible functions here. The hyperbola has been mentioned. Two more examples with some different behavior are
$$y = 20 - 1000\frac{\exp(-\frac{x}{8})}{x}~ (in~ green~ below)\\
y = \frac{20x^{2}+x-10}{x(x+10)}~ (in~ black~ below)$$
The hyperbola solution is in red below.
Basically $y = 20 - \frac{f(x)}{g(x)}$, up to tweaking signs of terms, will work if $g(0) = 0$ and $g$ has no other positive roots, and $\lim_{x\rightarrow +\infty} f(x) = 0$. Within reason you can get a lot of behaviors.
Edit: it appears the OP has some additional slope constraints. Basically every slope constraint requires a new coefficient. Consider the rational function $$R(x) = \frac{Ax^n + a_1x^{n-1} + \cdots - a_n}{x^n + x^{n-1} + \cdots + dx}, \forall i \in [1,n]~ (a_i \geq 0)$$ Only $a_n$ is marked as negative. Note this will match the criteria in the OP: $$\lim_{x\rightarrow +\infty}R(x) = A \\ \lim_{x\rightarrow 0^+}R(x) = -\infty$$ The derivative of $R(x)$ is relatively straightforward. Each variable we choose to be non-zero will allow us to specify a different slope at a different point; for instance, if we demand that $R'(1) = 1$ and we set $a_n=2$ then we can determine $d$ algebraically. If we instead demand only that $R'(1) \approx 1$ this gives us some more wiggle room to pick "nice" numbers. But in any case additional constraints require additional coefficients and as more conditions are added the exponent will start to rise in order to give us additional coefficients needed to fit our slope parameters.
But since we can always introduce more variables and higher exponents, we can always solve the system of equations demanded by all the required slopes.
Best Answer
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... polynomials won't suffice.
For a better answer, you need to be more specific.