Question

follow the steps for graphing a rational function to graph the function ( r(x)=\frac{x(x - 2)^{2}}{(x + 3)^{3}} ).

the function has one oblique asymptote,
(type an equation. use integers or fractions for any numbers in the equation.)

the function has two oblique asymptotes. the oblique asymptote with a negative slope is, and the oblique
asymptote with a positive slope is
(type equations. use integers or fractions for any numbers in the equations.)

the function has no oblique asymptote.

determine the points, if any, at which the graph of ( r ) 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.

the graph of ( r ) intersects the horizontal or oblique asymptote at
(simplify your answer. type an ordered pair, using integers or fractions. use a comma to separate answers
as needed.)

the graph of ( r ) intersects the horizontal or oblique asymptote at infinitely many points.

there is no point at which the graph of ( r ) intersects the horizontal or oblique asymptote.

there is no horizontal or oblique asymptote.

Explanation:

Step1: Analyze degrees of numerator and denominator

The degree of the numerator \(n\): Expand \(x(x - 2)^{2}=x(x^{2}-4x + 4)=x^{3}-4x^{2}+4x\), so \(n = 3\).
The degree of the denominator \(m\): Expand \((x + 3)^{3}=x^{3}+9x^{2}+27x + 27\), so \(m=3\).
When \(n=m\), the horizontal asymptote is \(y=\frac{a_{n}}{b_{m}}\), where \(a_{n}\) is the leading coefficient of the numerator and \(b_{m}\) is the leading coefficient of the denominator. Here \(a_{n}=1\) and \(b_{m}=1\), so \(y = 1\) is the horizontal asymptote. There is no oblique asymptote (oblique asymptote occurs when \(n=m + 1\)).

Step2: Check intersection with horizontal asymptote

Set \(R(x)=1\), so \(\frac{x(x - 2)^{2}}{(x + 3)^{3}}=1\).
Cross - multiply: \(x(x^{2}-4x + 4)=x^{3}+9x^{2}+27x + 27\).
Expand: \(x^{3}-4x^{2}+4x=x^{3}+9x^{2}+27x + 27\).
Simplify: \(-4x^{2}+4x-9x^{2}-27x - 27=0\), \(-13x^{2}-23x - 27=0\).
The discriminant \(\Delta=b^{2}-4ac=(-23)^{2}-4\times(-13)\times(-27)=529-1404=- 875<0\). So there is no real solution.

Answer:

C. The function has no oblique asymptote.
D. There is no point at which the graph of \(R\) intersects the horizontal or oblique asymptote.