We are asked to graph the function `y=(2x)/(x^2-1) ` :

Factoring the denominator we get:

`y=(2x)/((x+1)(x-1)) `

The graph has vertical asymptotes at x=1 and x=-1. The horizontal asymptote is y=0.

The y-intercept is 0 as is the only x-intercept.

The first derivative is ` y'=(-2(x^2+1))/((x^2-1)^2) ` so the function is decreasing on its domain.

The graph: