Question

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

Answer 1

Vertical asymptote: x=3,-2
Horizontal asymptote: `y=0`
`x`- intercept: none
`y`-intercept: `-4/3`
End behaviour:
As `x rarr -∞,y rarr ∞`
As `x rarr ∞, y rarr ∞`

Explanation:

Denote the function as `(n(x))/[d(x)]`

To find the vertical asymptote,
Find `d(x)=0`
`rArr x^2-x-6=0`
`(x-3)(x+2)=0`

`:.` The vertical asymptotes are at `x=3,-2`

To find the `x`-intercept, plug in `0` for `y` and solve for `x`.
`0=(8)/(x^2-x-6)`

`:.` 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=-4/3`

`:.` The `y`-intercept is at `-4/3`.

To find the horizontal asymptote,
Compare the leading degrees of the numerator and denominator.
For `n(x)`, the leading degree is `0`, since `8*x^0` would give `8`. Denote this as `color(pink)m`.

For `d(x)`, the leading degree is `2`. Denote this as `color(brown)n`.

If `color(pink)n < color(brown)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 ∞`