How do you find all horizontal and vertical asymptotes and all intercepts of #f(x) = (x^2-x-2)/(x^2-x-6)#?

1 Answer
Noah G
May 12, 2016

Vertical asymptotes:

#((x - 2)(x + 1))/((X - 3)(X + 2))#

#x != 3 or -2#, therefore there will be vertical asymptotes at #x = 3 and X = -2#.

Horizontal asymptotes:

When the degree of a function is the same in both the numerator and the denominator, there will be a horizontal asymptote at the ratio of the coefficients of the highest degree.

#= (1x^2...)/(1x^2...)#

#=1/1#

#= 1#

Therefore, there will be an asymptote at #y = 1#

Intercepts:

y intercepts:

#y = (0^2 - 0 - 2)/(0^2 - 0 - 6)#

#y = -2/-6#

#y =1/3#

The y intercept is at #(0, 1/3)#

X intercept:

#0 = (x^2 - x - 2)/(x^2 - x - 6)#

#0 = x^2 - x - 2#

#0 = (x - 2)(x + 1)#

#x = 2 and -1#

There will be x intercepts at #(2, 0)# and #(-1, 0)#

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