Question

follow the steps for graphing a rational function to graph the function ( f(x)=\frac{x^{2}-11 x - 12}{x + 4} ).
determine the behavior of the graph on either side of any vertical asymptotes, if any exist. select the correct choice and, if necessary, fill in the answer box(es) to complete your choice
a. it approaches ( infty ) on one side of the asymptote(s) at ( x=square ) and ( -infty ) on the other. it approaches either ( infty ) or ( -infty ) on both sides of the asymptote(s) at ( x=square )
(type integers or simplified fractions. use a comma to separate answers as needed. type each answer only once)
b. it approaches either ( infty ) or ( -infty ) on both sides of the asymptote(s) at ( x=square )
(type an integer or a simplified fraction. use a comma to separate answers as needed. type each answer only once.)
c. it approaches ( infty ) on one side of the asymptote(s) at ( x=square ) and ( -infty ) on the other
(type an integer or a simplified fraction. use a comma to separate answers as needed. type each answer only once.)
d. there is no vertical asymptote
determine the horizontal asymptote(s), if any exist. select the correct choice and, if necessary, fill in the answer box(es) to complete your choice
a. the function has one horizontal asymptote, ( square )
(type an equation. use integers or fractions for any numbers in the equation)
b. the function has two horizontal asymptotes. the top asymptote is ( square ), and the bottom asymptote is ( square )
(type equations. use integers or fractions for any numbers in the equations.)
c. the function has no horizontal asymptote
determine the oblique asymptote(s), if any exist. select the correct choice and, if necessary, fill in the answer box(es) to complete your choice
a. the function has one oblique asymptote, ( square )
(type an equation. use integers or fractions for any numbers in the equation)
b. the function has two oblique asymptotes. the oblique asymptote with a negative slope is ( square ), and the oblique asymptote with a positive slope is ( square )
(type equations. use integers or fractions for any numbers in the equations.)
c. the function has no oblique asymptote.
determine the points, if any, at which the graph of ( f ) intersects the horizontal or oblique asymptote, if one exists. select the correct choice and, if necessary, fill in the answer box to complete your choice

Explanation:

Step1: Simplify the rational function

First, factor the numerator \(x^{2}-11x - 12=(x + 1)(x-12)\). So \(F(x)=\frac{(x + 1)(x - 12)}{x + 4}\).

Step2: Analyze vertical asymptotes

Set the denominator equal to zero \(x+4=0\), so \(x=-4\).
\(\lim_{x
ightarrow - 4^{-}}\frac{(x + 1)(x - 12)}{x + 4}\), let \(x=-4-\epsilon\) where \(\epsilon
ightarrow0^{+}\). Then \(\frac{(-4-\epsilon + 1)(-4-\epsilon-12)}{-4-\epsilon + 4}=\frac{(-3-\epsilon)(-16-\epsilon)}{-\epsilon}\approx\frac{(-3)(-16)}{-\epsilon}=-\frac{48}{\epsilon}
ightarrow-\infty\)
\(\lim_{x
ightarrow - 4^{+}}\frac{(x + 1)(x - 12)}{x + 4}\), let \(x=-4+\epsilon\) where \(\epsilon
ightarrow0^{+}\). Then \(\frac{(-4+\epsilon + 1)(-4+\epsilon-12)}{-4+\epsilon + 4}=\frac{(-3+\epsilon)(-16+\epsilon)}{\epsilon}\approx\frac{(-3)(-16)}{\epsilon}=\frac{48}{\epsilon}
ightarrow\infty\)
So it approaches \(\infty\) on one side of the asymptote at \(x = - 4\) and \(-\infty\) on the other.

Step3: Analyze horizontal asymptotes

Since the degree of the numerator (\(n = 2\)) is greater than the degree of the denominator (\(m=1\)) (\(n>m\)), there is no horizontal asymptote.

Step4: Analyze oblique asymptotes

Use polynomial long - division: \(\frac{x^{2}-11x - 12}{x + 4}=x-15+\frac{48}{x + 4}\). As \(x
ightarrow\pm\infty\), \(\frac{48}{x + 4}
ightarrow0\). So the oblique asymptote is \(y=x - 15\)

Step5: Find intersection of the graph and the oblique asymptote

Set \(F(x)=x - 15\), \(\frac{x^{2}-11x - 12}{x + 4}=x - 15\)
\(x^{2}-11x-12=(x - 15)(x + 4)\)
\(x^{2}-11x-12=x^{2}-11x-60\)
\(-12=-60\) (no solution)

Answer:

For vertical asymptotes:

• Option C: It approaches \(\infty\) on one side of the asymptote(s) at \(x=-4\) and \(-\infty\) on the other

For horizontal asymptotes:

• Option C: The function has no horizontal asymptote

For oblique asymptotes:

• Option A: The function has one oblique asymptote \(y=x - 15\)

For intersection of the graph and the horizontal/oblique asymptote:

• The graph of \(F(x)\) does not intersect the oblique asymptote \(y=x - 15\)