Parameterizing cliffs I am looking for a function $f(x; \alpha, X_1, X_2, Y_1, Y_2)$ that has the following property: For $\alpha=0$ it behaves linearly between $(X_1, Y_1)$ and $(X_2, Y_2)$, and as $\alpha$ gets closer to 1, it approximates a sharp cliff, as in the figure below. The function needs not be defined for $\alpha=1$.
Is there a relatively "simple" function (trigonometrics, powers, logarithms and exponentials are fine) that captures this behavior?


The beta distribution is the uniform when $\alpha$ and $\beta$ are both equal to 1, and it gets progressivly "sharper" as the parameters change. Therefore, if you let $B(\alpha, \beta)$ represent the CDF of the beta and let $a$ be the alpha from your equation, to prevent confusion and let $y$ be $\frac{x - X2}{X1 - X2}$, the following may work:
$$
f(y;a, X_1, X_2, Y_1, Y_2) = \left(B\left(1 + \frac{a}{1-a}, 1\right)\biggr\rvert_0^1\cdot\frac{Y_1-Y_2}{X_1-X_2}\right)\cdot\left(X1 - X_2\right) + Y_1
$$
