Question

follow the steps for graphing a rational function to graph the function ( r(x)=\frac{x(x - 1)^{2}}{(x + 3)^{3}} ). if necessary, fill in the answer box(es) to complete your choice. a. the function has one horizontal asymptote, (type an equation. use integers or fractions for any numbers in the equation.) b. the function has two horizontal asymptotes. the top asymptote is , and the bottom asymptote is (type equations. use integers or fractions for any numbers in the equations.) c. the function has no horizontal asymptote.

Explanation:

Step1: Determine the degrees of numerator and denominator

The numerator \(N(x)=x(x - 1)^{2}=x(x^{2}-2x + 1)=x^{3}-2x^{2}+x\), so the degree of the numerator \(n = 3\).
The denominator \(D(x)=(x + 3)^{3}=x^{3}+9x^{2}+27x+27\), so the degree of the denominator \(m=3\).

Step2: Find the horizontal asymptote

When \(n=m\), the horizontal asymptote \(y=\frac{a_{n}}{b_{m}}\), where \(a_{n}\) is the leading coefficient of the numerator and \(b_{m}\) is the leading coefficient of the denominator.
For \(N(x)=x^{3}-2x^{2}+x\), \(a_{n}=1\); for \(D(x)=x^{3}+9x^{2}+27x + 27\), \(b_{m}=1\).
So \(y=\frac{1}{1}=1\)

Answer:

A. The function has one horizontal asymptote, \(y = 1\)