I need to define a family (one parameter) of monotonic curves I want to define a function family $f_a(x)$ with a parameter $a$ in $(0,1)$, where:
For any $a$, $f_a(0) = Y_0$ and $f_a(X_0) = 0$ (see image)
For $a = 0.5$, this function is a straight line from $(0,Y_0)$ to $(X_0, 0)$.
For $a < 0.5$, up to zero (asymptotically perhaps), I want $f_a$ to be a curve below, and for $a > 0.5$, the curve should be to the other side. 
I didn't fill the diagram with many examples, but I hope you get the idea. Different values of $a$ always produce a distinct, monotonic curve, below all curves of larger values of $a$, and above all curves for smaller values of $a$. E.g.: when I decrease $a$, the distance of the $(0,0)$ point from the curve decreases, and if I increase $a$, it increases. 
Sorry for the clumsy description but I hope you got the intuition of what I'm trying to define! Any suggestion of how this function $f_a(x)$ could look like? 


How about
$$f_a(x) = y_0\left(1-(x/x_0)^{\frac{a}{1-a}}\right)$$
?
