Question

which of the following is the graph of this rational function?
\\y = \frac{1}{x - 4}\\

Explanation:

Identify asymptotes of the rational function

$$ y = \frac{1}{x - 4} $$

• Vertical asymptote: \(x - 4 = 0 \implies x = 4\)
• Horizontal asymptote: Since the degree of the numerator (0) is less than the degree of the denominator (1), \(y = 0\)

Match asymptotes with the given graphs

• First graph: Vertical asymptote is at \(x = 0\), horizontal asymptote is at \(y = 4\).
• Second graph: Vertical asymptote is at \(x = 4\), horizontal asymptote is at \(y = 0\).
• Third graph: Vertical asymptote is at \(x = -4\), horizontal asymptote is at \(y = 0\).

Select the correct graph

• The second graph correctly shows a vertical asymptote at \(x = 4\) and a horizontal asymptote at \(y = 0\).

Answer:

• (A) First graph (vertical asymptote at \(x = 0\), horizontal asymptote at \(y = 4\))
• (B) Second graph (vertical asymptote at \(x = 4\), horizontal asymptote at \(y = 0\)) (Correct answer)
• (C) Third graph (vertical asymptote at \(x = -4\), horizontal asymptote at \(y = 0\))