How to construct a sigmoid curve that reaches, not approaches f(x)=0

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.

• isn't smooth-step basically just a spline of polynomials? And if you limit the domain to a closed interval as you suggest, you don't even have to worry about the edges; just take any cubic between its extrema and move it up/down, left/right, stretch as desired; if you want a more interesting third derivative, do the same with an odd quintic. – Nick Pavlov Nov 15 '17 at 20:02
• @NickPavlov How would you construct such a quintic? I'm guessing you need to make (0,0) a minimum, (a,0) a maximum and the middle of the two a point of inflection? I which case wouldn't that just be a quartic? EDIT: I found a forum link with what I might be looking for, though these are two separate functions. Is there a way to make them one whole equation or is there a notation to say, graph y=x and y=2x as f(x) for example? – Adam Bromiley Nov 15 '17 at 20:48
• en.wikipedia.org/wiki/Smoothstep#5th_order_equation – Nick Pavlov Nov 15 '17 at 21:47
• if I understand correctly what you are asking in the second part of your comment, that is exactly what is called a "piece-wise" definition en.wikipedia.org/wiki/Piecewise – Nick Pavlov Nov 15 '17 at 21:52
• @NickPavlov Great! Thank you I'll check it out. Seems to work though! – Adam Bromiley Nov 15 '17 at 22:00