I have a function $g(n)$ which has the following points:
$$\begin{array}{c|c|} n & g(n) \\ \hline \frac{1}{\infty} \text{ or } 0 & 4 \\ \hline \frac{1}{2} & \pi \\ \hline 1 & 2\sqrt{2} \\ \hline 2 & \pi \\ \hline \infty & 4 \end{array} $$
I know that $g(n) = g(\frac{1}{n})$, $g(1)=2\sqrt2$, and $g(n) = $ undefined (or $\infty$) when $n < 0$. I'm trying to figure out how to find the equation of the curve which characterizes these values.
From my understanding the formula for a generalized bell-shaped (-ish) curve is
$$ f(x)=a \cdot e^{-\left(\frac{|(x-b)|^p}{2c^2}\right)}+d $$ Where:
- $a$ is the height parameter (the absolute value of the $y$-value of the asymptote minus the $y$-value of the turning point is equal to $a$),
- $c$ is a spread parameter,
- $b$ is the horizontal shift parameter, i.e. the $x$-value of the turning point (being the maximum point where $a>0$ and the minimum point where $a<0$),
- $d$ is the vertical shift parameter, i.e. the horizontal asymptote is equal to $d$ (unless $p\le0$, then the horizontal asymptote is $1+d$), and
- $p$ is the curve parameter (for lack of a better term), i.e. it reflects the shape of $f'(x)$ or how rapidly $f(x)$ changes - I'm not sure how to best describe the characteristics of $p$.
My question is:
- Is there a way to solve for a specific generalized bell-shaped curve (as per the above $f(x)$), given any set of points?
- What is the smallest number of unique points required to be able to solve for the formula of $f(x)$?
- Is there a required distance between the points in order to be able to solve for $f(x)$? Do I need to know the asymptote and turning point?
- Because I know the asymptote and the turning point, does that change anything?