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The task requires graphing the function \( y = -2f(x-1) - 2 \) using the graph of \( y = f(x) \). The points provided are based on transformations applied to the original function \( f(x) \). Let’s break down the transformations:

- The \( (x-1) \) represents a horizontal shift to the **right by 1 unit**.
- The multiplication by \(-2\) reflects the graph vertically (because of the negative sign) and **stretches** it by a factor of 2.
- Finally, the \( -2 \) represents a vertical shift **down by 2 units**.

### Given Points and Transformation:

You are provided with the following points on \( f(x) \):

1. \( (-4, -4) \)
2. \( (-1, -3) \)
3. \( (0, 0) \)
4. \( (1, -4) \)
5. \( (3, 3) \)

Let's apply the transformation \( y = -2f(x-1) - 2 \) to these points:

1. **Point (-4, -4)**:
   \[
   \text{Shift } x = -4 + 1 = -3, \quad y = -2(-4) - 2 = 8 - 2 = 6
   \]
   Transformed point: \( (-3, 6) \)

2. **Point (-1, -3)**:
   \[
   \text{Shift } x = -1 + 1 = 0, \quad y = -2(-3) - 2 = 6 - 2 = 4
   \]
   Transformed point: \( (0, 4) \)

3. **Point (0, 0)**:
   \[
   \text{Shift } x = 0 + 1 = 1, \quad y = -2(0) - 2 = 0 - 2 = -2
   \]
   Transformed point: \( (1, -2) \)

4. **Point (1, -4)**:
   \[
   \text{Shift } x = 1 + 1 = 2, \quad y = -2(-4) - 2 = 8 - 2 = 6
   \]
   Transformed point: \( (2, 6) \)

5. **Point (3, 3)**:
   \[
   \text{Shift } x = 3 + 1 = 4, \quad y = -2(3) - 2 = -6 - 2 = -8
   \]
   Transformed point: \( (4, -8) \)

### Choose the Correct Graph:

Now, compare the transformed points with the graphs (A and B). Based on the transformation:

- The point \( (1, -2) \) suggests the curve will go below the \( x \)-axis.
- Points like \( (0, 4) \) and \( (2, 6) \) suggest the graph will have a stretch above the \( x \)-axis.

From these clues, **Graph B** is the correct representation, as it matches the transformations we’ve computed.

Do you want further details on the transformation process or the points?

### Relative questions:
1. How do horizontal and vertical shifts affect the graph of a function?
2. What is the effect of a vertical stretch by a factor of 2 on a graph?
3. How does a reflection across the x-axis change the graph?
4. How do you determine the correct graph after applying transformations?
5. What would happen if you reflected the graph across the y-axis instead?

**Tip:** Always apply transformations step-by-step, especially when multiple transformations are involved.

Use the graph of y = f(x) to graph the function y = -2f(x - 1) - 2. Label the points on the sketch.

Learn how to apply transformations to graph the function y = -2f(x - 1) - 2 using the graph of y = f(x). This problem involves horizontal and vertical shifts, reflection across the x-axis, and vertical stretching, suitable for students in Grades 9-12.

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