Determining the positioning of rational functions without plotting points When graphing rational functions, how do I determine the orientation of a rational function around the asymptotes without plotting points? For example, is it possible to determine which one of these is the correct graph of $\frac{2x}{x^2-1}$ without plugging in points and checking?


Basically, how would I check which one of these is correct, once I've found the asymptotes, or is plotting points necessary?

You may consider my answer to be essentially plugging in values and checking (because it is), but I'll provide it anyway.
The function
$$
\frac{2x}{x^2-1}
$$
has vertical asymptotes at $-1$ and $1$, so I think about points near those.
An input slightly smaller than $-1$ will give a negative numerator and a positive denominator, so the function value is negative overall. (This is what I mean by "essentially plugging in values", though I'm not actually reaching for my calculator.)
An input slightly bigger than $-1$ will give a negative numerator and a negative denominator, so the function value is positive overall.
An input slightly smaller than $1$ will give a positive numerator and a negative denominator, so the function value is negative overall.
An input slightly bigger than $1$ will give a positive numerator and a positive denominator, so the function value is positive overall.
All this information tells me that the second picture is the correct one.
