I was wondering if there is any systematic way to come up with a function with specific properties.
I want to have a concave function in domain of $[0, 1]$ where $f(0) = 0$ and $f(1) = 1$. In other words it is concave down, but it crosses $f(x) = x$ at $x = 0$ and $x = 1$. I also want to have a parameter that controls the concavity.
I was thinking of $-kx^2 + (k+1)x$ with trial and error.
In general, what is the best way to come up with a function having specific properties?