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:

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).