I am looking for a way to interpelate between two points given an input from 0 to 1. If the input is 0.5, it becomes a straight line interpolation. If less than 0.5 then it slowly ramps up and then speeds up at the end to meet the second point. More than 0.5 and it's fast ramp up and then slows down.
What is the best function to use? Thank you
EDIT: including a sketch of what I am looking for: sketch
If I understood your question correctly: You want a function $f$ that goes through two points $(x_1,y_1)$, $(x_2,y_2)$ and this $f$ would also depend from an input $t$ (so you would have a family of function $f_t$ where $t$ ranges from 0 to 1? And if $t=0$, the derivative at $x_1$ is "zero", at $x_2$ "infinity" and inversely when $t=1$.
For sake of simplicity, I will start with $(x_1,y_1)=(0,0)$ and $(x_2,y_2)=(1,1)$ The first example I would think of is: $f_t(x)=xe^{(0.5-t)(x-1)}$
If you're happy with that, try to come up with a general case (and come back to me if you didn't find anything :) )
Edit: you can always add a constant in front of the term $(0.5-t)$ in order to balance the derivative at 0/1