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We need to find the corresponding points for the transformation \( y = \frac{1}{2} f(-4 - x) \) based on the given points from the graph of \( y = f(x) \).

### Step-by-Step Transformation:
1. **Horizontal reflection and shift**: The term \( -4 - x \) suggests a reflection across the y-axis and a horizontal shift by 4 units to the left.
2. **Vertical scaling**: The \( \frac{1}{2} \) outside the function scales the output (y-values) by a factor of 0.5.

Let’s apply this transformation to each point:

1. **For \( (-5, -1) \):**
   - New \( x' = -4 - (-5) = 1 \)
   - New \( y' = \frac{1}{2}(-1) = -0.5 \)
   - Transformed point: \( (1, -0.5) \)

2. **For \( (-4, 5) \):**
   - New \( x' = -4 - (-4) = 0 \)
   - New \( y' = \frac{1}{2}(5) = 2.5 \)
   - Transformed point: \( (0, 2.5) \)

3. **For \( (0, 0) \):**
   - New \( x' = -4 - (0) = -4 \)
   - New \( y' = \frac{1}{2}(0) = 0 \)
   - Transformed point: \( (-4, 0) \)

4. **For \( (1, -2) \):**
   - New \( x' = -4 - (1) = -5 \)
   - New \( y' = \frac{1}{2}(-2) = -1 \)
   - Transformed point: \( (-5, -1) \)

5. **For \( (2, 2) \):**
   - New \( x' = -4 - (2) = -6 \)
   - New \( y' = \frac{1}{2}(2) = 1 \)
   - Transformed point: \( (-6, 1) \)

### Completed Table:
| Point on \( f(x) \) | Point on \( \frac{1}{2}f(-4 - x) \) |
|---------------------|------------------------------------|
| (-5, -1)            | (1, -0.5)                         |
| (-4, 5)             | (0, 2.5)                          |
| (0, 0)              | (-4, 0)                           |
| (1, -2)             | (-5, -1)                          |
| (2, 2)              | (-6, 1)                           |

Would you like more details or have any questions on how the transformation works?

Here are five follow-up questions that expand on this concept:
1. What is the effect of a horizontal translation on the graph of a function?
2. How does vertical scaling impact the graph of a function?
3. What happens when a function is reflected across the y-axis?
4. Can you explain the difference between \( f(x + 4) \) and \( f(x - 4) \) in terms of horizontal shift?
5. How would you graph \( f(-x) \) compared to \( f(x) \)?

**Tip**: Always apply transformations in the correct order: first handle the horizontal changes (inside the function), then vertical changes (outside the function).

Use the graph of y = f(x) to graph the function y = (1/2)f(-4 - x). For each point for f(x), determine the corresponding point for y = (1/2)f(-4 - x). Points on f(x) are: (-5, -1), (-4, 5), (0, 0), (1, -2), (2, 2).

This problem involves transforming the graph of y = f(x) to y = (1/2)f(-4 - x) by applying a horizontal reflection, a horizontal shift of 4 units, and vertical scaling. The transformed points are calculated based on given points: (-5, -1), (-4, 5), (0, 0), (1, -2), (2, 2).

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