I have a function in the form: $$ \mathrm{e}^{-t\lambda} \cdot \left[t\lambda - {(t\lambda)^2 \over 2}\right] $$
If one were to plot this for say $\lambda = \frac{2}{3}$ and $t$ from $0$ to $20$, this would produce a graph which has an 'unhappy parabola' shape till $t$ is around $3$, and then a 'happy parabola' shape which asymptotes towards the $x$ axis $0$. Around $t = 20$, the graph is near enough to $0$ that it can be approximated as $0$.
My question is as follows: is there a way in which the functional form can be manipulated so that the 'unhappy parabola' part (after $t=3$) can be streteched out? I want to keep the positive part of the function at $t<3$, but I want the rest of the function to stretch out (e.g. I want the graph to be 'near enough' to $0$ at $t = 40$, while keeping $t<3$ for the 'unhappy parabola').
Is there a way to do this (perhaps by putting some kind of stretch factor at a particular location). Please note: I am not trying to stretch the entire function - which would be trivial. What I want is to stretech one part of a curve.
Any help would be hugely appreciated.