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?

Since the rational expression factors as $\dfrac{2x}{(x-1)(x+1)}$, and the power of each factor of the numerator/denominator is odd, the sign definitely changes at the zeros of the numerator and the denominator (in contrast to a situation like $x^4$ or $1/x^2$). Therefore, one of the two pictures is correct (as opposed to pictures where it approaches $\infty$ on both sides of $x=1$, say). Since extremely large $x$ makes all the factors of the numerator and denominator positive, it must be the picture on the right.
The point of the above is that looking at whether you're dealing with odd/even powers saves you from looking at every interval as in Austin Mohr's answer, and this is a general technique.
