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

Question

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

Karson French · Accepted Answer

I would factorize the denominator solving the second degree equation to get:
f(x)=6(x+2)(x−1)
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(x)=2000. This tendency (for f(x) 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]}

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

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