Question

follow the seven step strategy to graph the following rational function.

(f(x) = \frac{4x}{x - 2})

\\(\

$$\begin{array}{lccccc} x & -6 & -4 & 1 & 5 & 8 \\\\ f(x) = \\frac{4x}{x - 2} & 3 & \\frac{8}{3} & -4 & \\frac{20}{3} & \\frac{16}{3} \\end{array}$$

\\)

(simplify your answers.)

use the information obtained in the previous steps to graph the function between and beyond the vertical asymptotes. choose the correct graph below.

Explanation:

Find the vertical asymptote

Using the Vertical Asymptotes Determination knowledge point

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Find the horizontal asymptote

Using the Horizontal Asymptotes knowledge point

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Evaluate key points to determine the branches

Using the Rational Function Evaluation knowledge point

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Match asymptotes and points to the correct graph

The vertical asymptote is \(x = 2\) (red dashed vertical line) and the horizontal asymptote is \(y = 4\) (red dashed horizontal line).

• For \(x < 2\), the graph lies below the horizontal asymptote \(y = 4\) (since \(f(1) = -4 < 4\)).
• For \(x > 2\), the graph lies above the horizontal asymptote \(y = 4\) (since \(f(5) = 6.67 > 4\)).

Looking at the choices:

• Graph A shows the vertical asymptote at \(x = -2\) and horizontal asymptote at \(y = -4\).
• Graph B shows the vertical asymptote at \(x = 2\) and horizontal asymptote at \(y = 4\), with the left branch below \(y = 4\) and the right branch above \(y = 4\).
• Graph C shows the vertical asymptote at \(x = -2\) and horizontal asymptote at \(y = 4\).
• Graph D shows the vertical asymptote at \(x = 2\) and horizontal asymptote at \(y = -4\).

Therefore, Graph B is the correct graph.

Answer:

• (A) Graph with asymptotes at x = -2 and y = -4
• (B) Graph with asymptotes at x = 2 and y = 4 (Correct answer)
• (C) Graph with asymptotes at x = -2 and y = 4
• (D) Graph with asymptotes at x = 2 and y = -4