# How do you graph the rational function f(x)=6/(x^2+x-2)?

Feb 8, 2015

I would factorize the denominator solving the second degree equation to get:
$f \left(x\right) = \frac{6}{\left(x + 2\right) \left(x - 1\right)}$
This helps you to "see" the forbidden points, i.e., the points where the denominator becomes zero (you do not want this!!!).
They are:
$x = - 2$
$x = 1$
Excluding these two values of $x$ the other are all allowed.
You can now try to figure out the shape of your graph:
1) for x very big positively or negatively your function gets very small or, better, tends to zero (try to substitute in your function, say, $x = 1000$ or $x = - 1000$ you'll find f(x)~0);
2) getting near to -2 your function gets very big positively (from the left) and negatively (from the right). You can try it by substituting $- 1.999$ (on the right of $x = - 2$) that gives you f(x)~-2000 and (on the left of $x = - 2$) $x = - 2.001$ giving $f \left(x\right) = 2000$. This tendency (for $f \left(x\right)$ to become very big) is repeated as well when you get near $x = 1$ (try it!);
3) setting $x = 0$ gives you $y = - 3$ which is the y-axis intercept;
At the end your graph looks like:

graph{6/(x^2+x-2) [-10, 10, -5, 5]}

hope it helps