I have two sigmoidal functions that I've fitted to some experimental data. The formula that I'm using to fit the function to my data is the following:
$$ f(t) = \frac{A}{1+e^{k(t-t_0)}} $$
Where $k$ is the rate constant, $t_0$ is the midpoint of the curve, and $A$ is the top asymptote.
Given two curves from the formula above, with different $k$, different $t_o$, but equal $A$, how do I prove that one curve starts decreasing earlier than the other?
I ask this because it is possible for two curves to have different $t_0$ and $k$, but still start decreasing roughly at the same location.
Or does the concept of "start decreasing" not apply to these sigmoidal curves since they approach asymptotes in both directions and are always decreasing?
Thank you for the help