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

1 Answer
Oct 15, 2017

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 ∞