# How to construct a function with desired properties?

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?

If $$f:[0,1] \to \mathbb R$$ is twice differentiable, then we have:
$$f$$ is concave $$\iff f'' \le 0$$ on $$[0,1]$$.
Now let $$f$$ be of the form $$f(x)=ax^2+bx+c$$. Then it is easy to see, that $$f$$ is concave, $$f(0)=0$$ and $$f(1)=1 \iff a \le 0$$ and $$b=1-a.$$