# How do you graph y=(3x^2)/(x^2-9) using asymptotes, intercepts, end behavior?

Nov 23, 2016

Horizontal asymptote: y = 0 and vertical ones are $x = \pm 3$. In ${Q}_{1} ,$ as $x \to \infty , y \to 0 \mathmr{and}$ as $y \to \infty , x \to 0$. See explanation, for continuation on end behavior.

#### Explanation:

graph{y(x^2-9)-3x^2=0 [-40, 40, -20, 20]}

By actual division and rearrangement,

$\left(y - 3\right) \left(x - 3\right) \left(x + 3\right) = 27$

To get asymptotes, See that, $L H S \to 0 X \left(\pm \infty\right)$ indeterminate

form, so that the limit exists as 27.

Easily, you could sort ouy the equations to the ayymptotes by setting

the factors on the LHS to 0.

Horizontal asymptote is y = 0 and the vertical ones are $x = \pm 3$.
In ${Q}_{1} ,$ as $x \to \infty , y \to 0 \mathmr{and}$ as $y \to \infty , x \to 0$.
In ${Q}_{2}$, as $x \to - \infty , y \to 0 \mathmr{and}$ as $y \to \infty , x \to 0$.
In ${Q}_{3} \mathmr{and} {Q}_{4}$, as $x \to 0$ , as $y \to - \infty$
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