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} ).\n(type an integer or a simplified fraction. use a comma to separate answers as needed. type each answer only once.)\nd. there is no ( x )-intercept.\ndetermine the vertical 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 vertical asymptote.\n(type an equation. use integers or fractions for any numbers in the equation.)\nb. the function has two vertical asymptotes. the leftmost asymptote is, and the rightmost asymptote is\n(type equations. use integers or fractions for any numbers in the equations.)\nc. the function has three vertical asymptotes. the leftmost asymptote is, the middle asymptote is, and the rightmost asymptote is\n(type equations. use integers or fractions for any numbers in the equations.)\nd. the function has no vertical asymptote.\ndetermine the hole, if it exists. select the correct choice and, if necessary, fill in the answer box to complete your choice.\na. there is a hole in the graph at the point\n(type an ordered pair using integers or fractions.)\nb. there are no holes in the graph.\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.)

follow the steps for graphing a rational function to graph the function ( r(x)=\frac{x^{2}+9x + 18}{x + 6} ).\n(type an integer or a simplified fraction. use a comma to separate answers as needed. type each answer only once.)\nd. there is no ( x )-intercept.\ndetermine the vertical 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 vertical asymptote.\n(type an equation. use integers or fractions for any numbers in the equation.)\nb. the function has two vertical asymptotes. the leftmost asymptote is, and the rightmost asymptote is\n(type equations. use integers or fractions for any numbers in the equations.)\nc. the function has three vertical asymptotes. the leftmost asymptote is, the middle asymptote is, and the rightmost asymptote is\n(type equations. use integers or fractions for any numbers in the equations.)\nd. the function has no vertical asymptote.\ndetermine the hole, if it exists. select the correct choice and, if necessary, fill in the answer box to complete your choice.\na. there is a hole in the graph at the point\n(type an ordered pair using integers or fractions.)\nb. there are no holes in the graph.\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.)

Answer

Explanation:

Step1: Simplify the rational function

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

Step2: Analyze vertical asymptotes

A vertical asymptote occurs when the denominator is zero after simplifying (canceling non - common factors). Since we canceled out (x + 6) (a common factor), there is no vertical asymptote.

Step3: Analyze holes

A hole occurs at the value of (x) that makes the canceled factor zero. Set (x+6 = 0), so (x=-6). Substitute (x=-6) into the simplified function (y=x + 3), (y=-6+3=-3). So the hole is at the point ((-6,-3))

Answer:

  • Vertical asymptote: D. The function has no vertical asymptote.
  • Hole: A. There is a hole in the graph at the point ((-6,-3))