How do you graph f(x)=(x^2−2x−8)/(x^2−9) using holes, vertical and horizontal asymptotes, x and y intercepts?

Jazmyn Pugh 2022-09-26 Answered
How do you graph f ( x ) = x 2 - 2 x - 8 x 2 - 9 using holes, vertical and horizontal asymptotes, x and y intercepts?
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Answers (1)

lufi8c
Answered 2022-09-27 Author has 11 answers
Asymptotes

The denominator of f(x) cannot be zero as this would make f(x) undefined. Equating the denominator to zero and solving gives the values that x cannot be and if the numerator is non-zero for these values then they are vertical asymptotes.

solve : x 2 - 9 = 0 x 2 = 9 x = ± 3

x = - 3 and x = 3 are the asymptotes

Horizontal asymptotes occur as

lim x ± , f ( x ) c ( a constant)

divide numerator/denominator by the highest power of x, that is x 2

f ( x ) = x 2 x 2 - 2 x x 2 - 8 x 2 x 2 x 2 - 9 x 2 = 1 - 2 x - 8 x 2 1 - 9 x 2

as x ± , f ( x ) 1 - 0 - 0 1 - 0

y = 1 is the asymptote

Holes occur when there is a duplicate factor on the numerator/denominator. This is not the case here, hence there are no holes.

Intercepts

x = 0 y = - 8 - 9 = 8 9

y-intercept at ( 0 , 8 9 )

y = 0 x 2 - 2 x - 8 = 0 ( x - 4 ) ( x + 2 ) = 0

x-intercepts at ( - 2 , 0 ) and ( 4 , 0 )
graph{(x^2-2x-8)/(x^2-9) [-10, 10, -5, 5]}
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