Question

which of the following is the function represented by the graph?

consider the graph below.

\\(y = \frac{1}{(x + 3)} - 5\\)

\\(y = \frac{1}{(x - 3)} + 5\\)

\\(y = \frac{1}{(x + 5)} - 3\\)

\\(y = \frac{1}{(x - 5)} + 3\\)

Explanation:

Identify the vertical asymptote

$$ x = 5 $$

Identify the horizontal asymptote

$$ y = 3 $$

Match asymptotes to the rational function form

$$ y = \frac{1}{x - h} + k \implies y = \frac{1}{x - 5} + 3 $$

Answer:

• (A) \(y = \frac{1}{(x+3)} - 5\)
• (B) \(y = \frac{1}{(x-3)} + 5\)
• (C) \(y = \frac{1}{(x+5)} - 3\)
• (D) \(y = \frac{1}{(x-5)} + 3\) (Correct answer)