I'm trying to apply calculus to distance/time graphs so when differentiated I can show a graph for velocity/time, then acceleration/time, then jerk/time by using higher order differentials. I want the graph to show the object accelerating from zero, reaching a max speed halfway, then decelerating back to zero, hence second derivative of my graph must have points $(0,0)$ and $(a,0)$.
Because I'm relatively new to differentiation, I would like to keep the function relatively simple - thus I have chosen: $$ f\left(x\right)=\frac{1}{1+e^{-x+a}} $$
as my function. It is almost perfect because it is easy to differentiate (I'm not ready to learn differentiation of integrals, smooth-step, logs, or trigonometric functions yet!).
The only issue I have is that the range is $0<f(x)<1$, I however need the range to be $0\le f(x)\le1$ because obviously the object will stop eventually.
Hence I need a function that is of similar form to my current one but has domain $0\le x\le 2a$ and range $0\le f(x)\le1$. Feel free to change the "$+8$" value in the exponent, I simply added that so the point of inflection is positive.
EDIT: To give background I'm in year 12 so I've just started the A Level Maths, Further Maths and Physics course hence I'm interested in learning new things but not learning a completely new topic.