To graph the function $f(x) = \log_{\frac{1}{4}}(x - 3) - 3$, we can break it down and analyze each component.

Steps to Graph the Function:

1. Logarithmic Function Properties: The base of the logarithm is $\frac{1}{4}$, which means the logarithm is decreasing (since $0 < \frac{1}{4} < 1$). The function will exhibit a downward curve.

2. Horizontal Shift: The $(x - 3)$ term shifts the graph to the right by 3 units. The vertical asymptote of a logarithmic function is where the argument inside the logarithm equals zero. In this case, $x - 3 = 0$ gives a vertical asymptote at $x = 3$.

3. Vertical Shift: The $-3$ outside the logarithm shifts the entire graph down by 3 units.

4. Asymptote: The function has a vertical asymptote at $x = 3$, which means the graph will approach $x = 3$ from the right, but never touch or cross it.

Key Features:

• Domain: The argument inside the logarithm must be positive. So, $x - 3 > 0$, meaning the domain is $x > 3$.
• Range: Since logarithmic functions have no upper or lower bounds, the range is $(-\infty, \infty)$.
• Asymptote: $x = 3$.
• Behavior: As $x \to 3^+$, $f(x) \to -\infty$, and as $x \to \infty$, $f(x) \to -3$ (since the function approaches $-3$ but never reaches it).

Would you like me to provide a detailed plot of this function?

Follow-Up Questions:

1. What does the base of a logarithm indicate about the shape of the graph?
2. How do horizontal shifts affect the logarithmic function's asymptote?
3. Can logarithmic functions have negative values for $x$?
4. What happens to the graph as $x \to \infty$ for a logarithmic function?
5. How does a vertical shift affect the entire function?

Tip:

Always check for shifts and asymptotes when graphing logarithmic functions as they play a crucial role in defining the graph's shape.