The following graph shows a distribution curve given by a function of the form $f(x)=A/(x+2e^{rx})$

Question

The following graph shows a distribution curve given by a function of the form $$f(x)=\dfrac{A}{x+2e^{rx}}$$

Determine the equation $f(x)$ of the graph.

Answer 1

From the given form $f=\frac{A}{1+2e^{rx}}$ and by checking the points on the graph we get $$\begin{aligned} 10&=f(0)=\frac{A}{0+2e^{r\cdot0}}=\frac{A}{2} \implies A=20 &&(\text{I})\\ 12&=f(1)=\frac{A}{1+2e^{r\cdot1}}\overset{\text{(I)}}{=}\frac{20}{1+2e^r} &&\text{(II)}\\ 9&=f(2)=\frac{A}{2+2e^{r\cdot2}}\overset{\text{(I)}}{=}\frac{10}{1+e^{2r}} &&\text{(III)}\\ \end{aligned}$$ transform (II): $$ 12=\frac{20}{1+2e^r} \iff 12(1+2e^{r})=20 \iff 24e^{r}=8 \iff e^r=\frac{1}{3} \iff r=\ln\left(\frac{1}{3}\right) $$ So $A=20$ and $r=\ln(\frac{1}{3})\approx -1.0986$. (III) is not needed for this.