I know there are ways to "restrict" a function, f(x), so that it is forced to go through certain points, I'm not really sure what it's called though.
Simply multiplying it by x will make the function pass through (0, 0), for instance.
$\text{where } f \text{ is the original function;} $
$f_0(x) = xf(x) \text{ passes through (0, 0)}$
$f_1(x) = 1-(f(x)(1-x)) \text{ passes through (1, 1)}$
I've included a diagram to help explain.
Diagram from Desmos Screenshot
Is it possible to do both of these at the same time? I've tried taking averages of the resulting two functions, but that ends up being weird. Preferably, if an input function is symmetrical (ie. the way it moves away from 0,0 should be the same as the way it moves towards 1,1), the final function should also have the same properties.
You could try $$g(x) = f(x) -((1-x)f(0) + x(f(1)-1))$$ This just subtracts the line segment running from $(0,f(0))$ to $(1,f(1)-1)$ from $f$.