Let's analyze the function \( f(x) = \frac{1}{4} |4 - x| - 3 \).

### 1. **Understanding the Function:**
   - The function includes an absolute value expression \( |4 - x| \), which reflects the graph symmetrically about the line \( x = 4 \).
   - The coefficient \( \frac{1}{4} \) outside the absolute value compresses the graph vertically by a factor of 4.
   - The subtraction of 3 at the end shifts the entire graph downward by 3 units.

### 2. **Key Features of the Function:**

   - **Vertex:** The vertex occurs where the expression inside the absolute value equals zero, i.e., \( 4 - x = 0 \). This gives \( x = 4 \). Plugging \( x = 4 \) into the function:
     \[
     f(4) = \frac{1}{4} |4 - 4| - 3 = 0 - 3 = -3
     \]
     So, the vertex is at \( (4, -3) \).

   - **Behavior on Either Side of the Vertex:**
     - For \( x < 4 \), the expression \( |4 - x| \) becomes \( 4 - x \), so \( f(x) = \frac{1}{4}(4 - x) - 3 \).
     - For \( x > 4 \), the expression \( |4 - x| \) simplifies to \( x - 4 \), so \( f(x) = \frac{1}{4}(x - 4) - 3 \).

   - **Slope Analysis:**
     - For \( x < 4 \), the slope is negative (\( -\frac{1}{4} \)).
     - For \( x > 4 \), the slope is positive (\( \frac{1}{4} \)).

### 3. **Graphical Representation:**

   - The graph is V-shaped, with the vertex at \( (4, -3) \).
   - It opens upwards, but the slopes of the arms of the V are relatively shallow due to the factor \( \frac{1}{4} \).
   - The function decreases linearly with a slope of \( -\frac{1}{4} \) for \( x < 4 \) and increases linearly with a slope of \( \frac{1}{4} \) for \( x > 4 \).

Would you like more details on a specific part of the analysis, or do you have any other questions?

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Here are 5 related questions for further exploration:

1. How would the graph change if the coefficient \( \frac{1}{4} \) were replaced by a larger number?
2. What would happen to the graph if the function was \( f(x) = \frac{1}{4} |4 - x| + 3 \) instead?
3. How would the vertex of the function change if the equation were \( f(x) = \frac{1}{4} |x - 2| - 3 \)?
4. What is the range of the function \( f(x) = \frac{1}{4} |4 - x| - 3 \)?
5. How would the domain and range be affected if the function were defined as \( f(x) = -\frac{1}{4} |4 - x| - 3 \)?

**Tip:** When analyzing absolute value functions, always start by identifying the vertex, as it helps determine the overall shape and direction of the graph.

Explore the absolute value function f(x) = 1/4|4 - x| - 3 with detailed analysis including vertex determination, slope analysis, and graphical representation. Understand how the coefficient and shifts affect the function's graph. Suitable for high school students studying algebra and graphical functions.

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