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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.

Graph the function f(x) = log(1/4)(x - 3) - 3.

Learn how to graph the logarithmic function f(x) = log(1/4)(x - 3) - 3 with a step-by-step breakdown. Explore the properties of logarithmic functions, horizontal and vertical shifts, and asymptotes. Suitable for students in Grades 10-12.

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