Question

choose the graph of (y = log_{\frac{1}{4}} (x - 4))

Explanation:

Step1: Identify the vertical asymptote

The vertical asymptote occurs where the argument is zero:

$$x - 4 = 0 \implies x = 4$$

Step2: Determine the x-intercept

Set \(y = 0\) to find the x-intercept:

$$\log_{\frac{1}{4}}(x - 4) = 0 \implies x - 4 = 1 \implies x = 5$$

Step3: Analyze the function's behavior

Since the base is \(\frac{1}{4} < 1\), the function decreases as \(x\) increases.

Step4: Match with the correct graph

The top-left graph has a vertical asymptote at \(x = 4\), passes through \((5, 0)\), and decreases.

Answer:

The top-left graph.