Question

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

Answer 1

Horizontal asymptote: y = 0 and vertical ones are `x =+-3`. In `Q_1,` as `x to oo, y to 0 and` as `y to oo, 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,

`(y-3)(x-3)(x+3)=27`

To get asymptotes, See that, `LHS to 0 X (+-oo)` 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.

Answer:

Horizontal asymptote is y = 0 and the vertical ones are `x=+-3`.

In `Q_1,` as `x to oo, y to 0 and` as `y to oo, x to 0`.

In `Q_2`, as `x to -oo, y to 0 and` as `y to oo, x to 0`.
.

In `Q_3 and Q_4`, as `x to 0` , as `y to -oo`