follow the steps for graphing a rational function to graph the function ( r(x)=\frac{x^{2}+9x + 18}{x + 6}…

follow the steps for graphing a rational function to graph the function ( r(x)=\frac{x^{2}+9x + 18}{x + 6} ).\ndetermine 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) within your choice.\na. it approaches ( infty ) on one side of the asymptote(s) at ( x=) and ( -infty ) on the other. it approaches either ( infty ) or ( -infty ) on both sides of the asymptote(s) at ( x=)\n(type integers or simplified fractions. use a comma to separate answers as needed. type each answer only once.)\nb. it approaches either ( infty ) or ( -infty ) on both sides of the asymptote(s) at ( x=)\n(type an integer or a simplified fraction. use a comma to separate answers as needed. type each answer only once.)\nc. it approaches ( infty ) on one side of the asymptote(s) at ( x=) and ( -infty ) on the other.\n(type an integer or a simplified fraction. use a comma to separate answers as needed. type each answer only once.)\nd. the function has no vertical asymptote\ndetermine the horizontal asymptote(s). if any exist. select the correct choice and, if necessary, fill in the answer box(es) within your choice.\na. the function has one horizontal asymptote,\n(type an equation. use integers or fractions for any numbers in the equation.)\nb. the function has two horizontal asymptotes. the top asymptote is, and the bottom asymptote is\n(type equations. use integers or fractions for any numbers in the equations.)\nc. the function has no horizontal asymptote\ndetermine the oblique asymptote(s). if any exist. select the correct choice and, if necessary, fill in the answer box(es) within your choice.\na. the function has one oblique asymptote,\n(type an equation. use integers or fractions for any numbers in the equation.)\nb. the function has two oblique asymptotes. the oblique asymptote with a negative slope is, and the oblique asymptote with a positive slope is\n(type equations. use integers or fractions for any numbers in the equations.)\nc. the function has no oblique asymptote

follow the steps for graphing a rational function to graph the function ( r(x)=\frac{x^{2}+9x + 18}{x + 6} ).\ndetermine 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) within your choice.\na. it approaches ( infty ) on one side of the asymptote(s) at ( x=) and ( -infty ) on the other. it approaches either ( infty ) or ( -infty ) on both sides of the asymptote(s) at ( x=)\n(type integers or simplified fractions. use a comma to separate answers as needed. type each answer only once.)\nb. it approaches either ( infty ) or ( -infty ) on both sides of the asymptote(s) at ( x=)\n(type an integer or a simplified fraction. use a comma to separate answers as needed. type each answer only once.)\nc. it approaches ( infty ) on one side of the asymptote(s) at ( x=) and ( -infty ) on the other.\n(type an integer or a simplified fraction. use a comma to separate answers as needed. type each answer only once.)\nd. the function has no vertical asymptote\ndetermine the horizontal asymptote(s). if any exist. select the correct choice and, if necessary, fill in the answer box(es) within your choice.\na. the function has one horizontal asymptote,\n(type an equation. use integers or fractions for any numbers in the equation.)\nb. the function has two horizontal asymptotes. the top asymptote is, and the bottom asymptote is\n(type equations. use integers or fractions for any numbers in the equations.)\nc. the function has no horizontal asymptote\ndetermine the oblique asymptote(s). if any exist. select the correct choice and, if necessary, fill in the answer box(es) within your choice.\na. the function has one oblique asymptote,\n(type an equation. use integers or fractions for any numbers in the equation.)\nb. the function has two oblique asymptotes. the oblique asymptote with a negative slope is, and the oblique asymptote with a positive slope is\n(type equations. use integers or fractions for any numbers in the equations.)\nc. the function has no oblique asymptote

Answer

Explanation:

Step1: Simplify the rational function

First, factor the numerator (x^{2}+9x + 18=(x + 3)(x+6)). Then (R(x)=\frac{(x + 3)(x + 6)}{x+6}). For (x\neq - 6), (R(x)=x + 3).

Step2: Analyze vertical asymptotes

Since the simplified function (y=x + 3) (with a hole at (x=-6) because the original function is undefined at (x=-6) but (\lim_{x\rightarrow - 6}\frac{x^{2}+9x + 18}{x + 6}=\lim_{x\rightarrow - 6}(x + 3)=-3)), the function has no vertical asymptote.

Step3: Analyze horizontal asymptotes

The degree of the numerator (n = 2) and the degree of the denominator (m=1). When (n>m), there is no horizontal asymptote.

Step4: Analyze oblique asymptotes

We can also use polynomial long - division. Dividing (x^{2}+9x + 18) by (x + 6): [ \begin{align*} x^{2}+9x+18&=(x + 6)(x+3)+0 \end{align*} ] When we perform polynomial long - division of a degree (n) polynomial by a degree (m) polynomial ((n=m + 1)), the quotient (excluding the remainder) gives the equation of the oblique asymptote. Here, (y=x + 3) is the equation of the non - vertical non - horizontal asymptote. But since the function (R(x)) simplifies to (y=x + 3) (with a hole at (x=-6)), we can also say that in the context of the original rational function (before simplification), if we consider the form of rational functions (y=\frac{f(x)}{g(x)}) with (\text{deg}(f(x))=\text{deg}(g(x))+1), the oblique asymptote is found by polynomial long - division.

Answer:

  • For vertical asymptotes: D. The function has no vertical asymptote
  • For horizontal asymptotes: C. The function has no horizontal asymptote
  • For oblique asymptotes: A. The function has one oblique asymptote, (y=x + 3)