Question

which of the following is the graph of this rational function?
\\y = \frac{1}{x} - 3\\

Explanation:

Identify asymptotes of the function

We analyze the rational function:

$$y = \frac{1}{x} - 3$$

The vertical asymptote occurs where the denominator is zero:

$$x = 0$$

The horizontal asymptote is determined by the vertical shift:

$$y = -3$$

Analyze key points on the graph

We evaluate the function at simple test points to find coordinates:
For \(x = 1\):

$$y = \frac{1}{1} - 3 = -2 \implies (1, -2)$$

For \(x = -1\):

$$y = \frac{1}{-1} - 3 = -4 \implies (-1, -4)$$

Match with the given options

We compare our findings with the three graphs:

• First graph: The horizontal asymptote is at \(y = 3\). This is incorrect.
• Second graph: The horizontal asymptote is at \(y = -3\). The vertical asymptote is at \(x = 0\) (the y-axis). It passes through \((1, -2)\) and \((-1, -4)\). This matches our function.
• Third graph: The vertical asymptote is at \(x = -3\). This is incorrect.

Answer:

• (A) Graph with horizontal asymptote at \(y = 3\)
• (B) Graph with horizontal asymptote at \(y = -3\) (Correct answer)
• (C) Graph with vertical asymptote at \(x = -3\)