I want the general form of a function $f(x)$ containing two other variables $a, b$
such that the graph of the function resembles the image below:
The function should satisfy the following conditions: $$f(0)=1$$ $$f(\frac{4}{5}b)=c$$ $$f(b)=0$$
Piece-wise defined functions are acceptable.
My first instinct is to fit some modification of $f(x) = x \mathrm{e}^{-x}$. For example the following is a plot of $f(-(x-5))= -(x-5) \mathrm{e}^{x-5}$ on $[0,~5]$:
This fits the general shape quite well, has a maximum at $x=4$. As for your concavity, your data really only looks concave up for $(0,~3b/5)$ and concave down for $(3b/5, ~b)$---note that a local max more or less requires negative local concavity. The above graph has an inflection point at $x=3$, which again seems to fit with your data.