Plot of a function

Question

What is the plot of: $$y=\frac{\beta(1-\alpha)x}{\alpha(1-\beta)+(\beta-\alpha)x}$$ with $0<\alpha<\beta<1$.
How do I handle the parameters? How do I compute the derivatives to check for monotonicity and concavity?

Answer 1

These are the things that may help to graph the function.

$1$. Roots

• The function have a root at $x=0$

$2$. Horizontal and Vertical Asymptotes

$$x=-\frac{\alpha(1-\beta)}{\beta-\alpha} \qquad \text{Vertical Asymptote}$$ $$y=\frac{\beta(1-\alpha)}{\beta-\alpha} \qquad \text{Horizontal Asymptote}$$

$3$. The Derivative

$$y'=\frac{\alpha \beta (1-\alpha)(1-\beta)}{[(\beta-\alpha)x+(\alpha+\beta)]^2}$$ and as it is positive so it is increasing in each of its branches.

For example, for the case of $\alpha=0.25$ and $\beta=0.5$, it will look like this.