follow the seven step strategy to graph the following rational function.\n\n$f(x)=\\frac{5x}{x - 2}$\n\na…

follow the seven step strategy to graph the following rational function.\n\n$f(x)=\\frac{5x}{x - 2}$\n\na. the x - intercept(s) is/are 0.\n(type an integer or a simplified fraction. use a comma to separate answers as needed.)\nb. there is no x - intercept.\nfind the vertical asymptote(s). select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the equation(s) of the vertical asymptote(s) is/are $x = 2$.\n(type an equation. use a comma to separate answers as needed.)\nb. there is no vertical asymptote.\nfind the horizontal asymptote(s). select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the equation(s) of the horizontal asymptote(s) is/are $y = 5$.\n(type an equation. use a comma to separate answers as needed.)\nb. there is no horizontal asymptote.\nplot points between and beyond each x - intercept and vertical asymptote. find the value of the function at the given value of x.\n$\\begin{array}{|c|c|c|c|c|c|}\\hline x&-3&-1&1&3&8\\\\\\hline f(x)=\\frac{5x}{x - 2}&3&\\frac{5}{3}&-5&15&\\frac{20}{3}\\\\\\hline\\end{array}$\n(simplify your answers.)\nuse the information obtained in the previous steps to graph the function between and beyond the vertical asymptotes. choose the correct graph below.\na.\nb.\nc.\nd.

follow the seven step strategy to graph the following rational function.\n\n$f(x)=\\frac{5x}{x - 2}$\n\na. the x - intercept(s) is/are 0.\n(type an integer or a simplified fraction. use a comma to separate answers as needed.)\nb. there is no x - intercept.\nfind the vertical asymptote(s). select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the equation(s) of the vertical asymptote(s) is/are $x = 2$.\n(type an equation. use a comma to separate answers as needed.)\nb. there is no vertical asymptote.\nfind the horizontal asymptote(s). select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the equation(s) of the horizontal asymptote(s) is/are $y = 5$.\n(type an equation. use a comma to separate answers as needed.)\nb. there is no horizontal asymptote.\nplot points between and beyond each x - intercept and vertical asymptote. find the value of the function at the given value of x.\n$\\begin{array}{|c|c|c|c|c|c|}\\hline x&-3&-1&1&3&8\\\\\\hline f(x)=\\frac{5x}{x - 2}&3&\\frac{5}{3}&-5&15&\\frac{20}{3}\\\\\\hline\\end{array}$\n(simplify your answers.)\nuse the information obtained in the previous steps to graph the function between and beyond the vertical asymptotes. choose the correct graph below.\na.\nb.\nc.\nd.

Answer

Explanation:

Step1: Analyze the x - intercept

For a rational function (y = \frac{f(x)}{g(x)}), the x - intercepts are found by setting (f(x)=0). Here (f(x) = 5x), so (5x=0) gives (x = 0).

Step2: Analyze the vertical asymptote

The vertical asymptote of a rational function (y=\frac{f(x)}{g(x)}) is found by setting (g(x)=0). Here (g(x)=x - 2), so (x-2=0) gives (x = 2).

Step3: Analyze the horizontal asymptote

For a rational function (y=\frac{ax^{n}+...}{bx^{m}+...}), when (n=m) (here (n = m=1) for (y=\frac{5x}{x - 2})), the horizontal asymptote is (y=\frac{a}{b}). So (y=\frac{5}{1}=5).

Step4: Analyze the function values

  • When (x=-3), (f(-3)=\frac{5\times(-3)}{-3 - 2}=\frac{-15}{-5}=3)
  • When (x=-1), (f(-1)=\frac{5\times(-1)}{-1 - 2}=\frac{-5}{-3}=\frac{5}{3})
  • When (x = 1), (f(1)=\frac{5\times1}{1 - 2}=\frac{5}{-1}=-5)
  • When (x = 3), (f(3)=\frac{5\times3}{3 - 2}=\frac{15}{1}=15)
  • When (x = 8), (f(8)=\frac{5\times8}{8 - 2}=\frac{40}{6}=\frac{20}{3})

Since the x - intercept is (x = 0), vertical asymptote (x = 2), horizontal asymptote (y = 5) and the function values at (x=-3,-1,1,3,8) are (3,\frac{5}{3},-5,15,\frac{20}{3}) respectively, we can graph the function.

Answer:

Based on the x - intercept (x = 0), vertical asymptote (x = 2), horizontal asymptote (y = 5) and the calculated function values, we can determine the correct graph (assuming the graph with these key features is one of the options). But since the actual visual comparison of the graphs (A, B, C, D) is not possible here in a text - based format, if we follow the standard graphing rules for (y=\frac{5x}{x - 2}):

  • It crosses the x - axis at ((0,0))
  • Has a vertical dashed line (x = 2)
  • Has a horizontal dashed line (y = 5)
  • Passes through the points ((-3,3),(-1,\frac{5}{3}),(1, - 5),(3,15),(8,\frac{20}{3}))