Question

select the choice that is a graph of the function
\\f(x) = \frac{1}{4} \frac{(x + 3)(x - 4)}{x - 3}\\

Explanation:

Step1: Find vertical asymptote

Set the denominator to zero:

$$x - 3 = 0 \implies x = 3$$

Step2: Find x-intercepts

Set the numerator to zero:

$$(x + 3)(x - 4) = 0 \implies x = -3, \quad x = 4$$

Step3: Find y-intercept

Evaluate the function at \(x = 0\):

$$f(0) = \frac{1}{4} \frac{(0 + 3)(0 - 4)}{0 - 3} = 1$$

Step4: Match with graphs

The top graph has a vertical asymptote at \(x = 3\), x-intercepts at \(-3\) and \(4\), and a y-intercept at \(1\).

Answer:

The first graph (top graph) is the correct choice.