# How do you graph y=8/(x^2-x-6) using asymptotes, intercepts, end behavior?

Oct 15, 2017

Vertical asymptote: x=3,-2
Horizontal asymptote: $y = 0$
$x$- intercept: none
$y$-intercept: $- \frac{4}{3}$
End behaviour:
As x rarr -∞,y rarr ∞
As x rarr ∞, y rarr ∞

#### Explanation:

Denote the function as $\frac{n \left(x\right)}{d \left(x\right)}$

To find the vertical asymptote,
Find $d \left(x\right) = 0$
$\Rightarrow {x}^{2} - x - 6 = 0$
$\left(x - 3\right) \left(x + 2\right) = 0$

$\therefore$ The vertical asymptotes are at $x = 3 , - 2$

To find the $x$-intercept, plug in $0$ for $y$ and solve for $x$.
$0 = \frac{8}{{x}^{2} - x - 6}$

$\therefore$ There are no $x$-intercepts.

To find the $y$-intercept, plug in $0$ for $x$ and solve for $y$.
y=(8)/(0^2-0-6
$y = - \frac{4}{3}$

$\therefore$ The $y$-intercept is at $- \frac{4}{3}$.

To find the horizontal asymptote,
Compare the leading degrees of the numerator and denominator.
For $n \left(x\right)$, the leading degree is $0$, since $8 \cdot {x}^{0}$ would give $8$. Denote this as $\textcolor{\pi n k}{m}$.

For $d \left(x\right)$, the leading degree is $2$. Denote this as $\textcolor{b r o w n}{n}$.

If $\textcolor{\pi n k}{n} < \textcolor{b r o w n}{m}$, then the horizontal asymptote is $y = 0$.

To find the end behaviour of the graph, plot it in a graphic calculator.
As x rarr -∞,y rarr ∞
As x rarr ∞, y rarr ∞