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To solve the given problem, we need to transform the graph of the function \( y = f(x) \) based on the given transformation \( y = -3f(2x) + 4 \).

### Step-by-Step Explanation

1. **Understanding the Original Graph:**
   The original graph, \( y = f(x) \), includes the following key points:
   - \( (-2, 2) \)
   - \( (1, 2) \)
   - \( (2, 0) \)
   - \( (3, 0) \)
   - \( (4, 2) \)
   These points define the important features of the graph, including peaks, troughs, and intersections with the x-axis.

2. **Transformation Breakdown:**
   The transformation \( y = -3f(2x) + 4 \) includes several operations:
   
   - **Horizontal Compression (by a factor of 2)**: The term \( f(2x) \) means that all x-coordinates are compressed towards the y-axis by a factor of 2. This halves the x-coordinates of all points.
   
   - **Vertical Stretch (by a factor of 3)**: The term \( -3f(x) \) implies that the graph is stretched vertically by a factor of 3. Additionally, the negative sign reflects the graph about the x-axis.
   
   - **Vertical Shift (upward by 4 units)**: The term \( +4 \) shifts the entire graph upward by 4 units.

### Applying the Transformation to Key Points:
Let's now apply these transformations to each key point on the original graph.

- Point \( (-2, 2) \):
  - Horizontal Compression: New x-coordinate = \( -2/2 = -1 \)
  - Vertical Stretch and Reflection: New y-coordinate = \( -3(2) = -6 \)
  - Vertical Shift: New y-coordinate = \( -6 + 4 = -2 \)
  - Transformed Point: \( (-1, -2) \)

- Point \( (1, 2) \):
  - Horizontal Compression: New x-coordinate = \( 1/2 = 0.5 \)
  - Vertical Stretch and Reflection: New y-coordinate = \( -3(2) = -6 \)
  - Vertical Shift: New y-coordinate = \( -6 + 4 = -2 \)
  - Transformed Point: \( (0.5, -2) \)

- Point \( (2, 0) \):
  - Horizontal Compression: New x-coordinate = \( 2/2 = 1 \)
  - Vertical Stretch and Reflection: New y-coordinate = \( -3(0) = 0 \)
  - Vertical Shift: New y-coordinate = \( 0 + 4 = 4 \)
  - Transformed Point: \( (1, 4) \)

- Point \( (3, 0) \):
  - Horizontal Compression: New x-coordinate = \( 3/2 = 1.5 \)
  - Vertical Stretch and Reflection: New y-coordinate = \( -3(0) = 0 \)
  - Vertical Shift: New y-coordinate = \( 0 + 4 = 4 \)
  - Transformed Point: \( (1.5, 4) \)

- Point \( (4, 2) \):
  - Horizontal Compression: New x-coordinate = \( 4/2 = 2 \)
  - Vertical Stretch and Reflection: New y-coordinate = \( -3(2) = -6 \)
  - Vertical Shift: New y-coordinate = \( -6 + 4 = -2 \)
  - Transformed Point: \( (2, -2) \)

### Final Result:
The transformed graph of \( y = -3f(2x) + 4 \) includes the following points:
- \( (-1, -2) \)
- \( (0.5, -2) \)
- \( (1, 4) \)
- \( (1.5, 4) \)
- \( (2, -2) \)

You can now plot these points and sketch the transformed graph. The graph will be a vertically stretched, reflected, and shifted version of the original, with the same basic shape but modified based on the transformation.

Would you like further details or assistance with this problem?

### Related Questions:
1. How does horizontal compression affect the graph of a function?
2. Why does multiplying the function by a negative constant reflect the graph about the x-axis?
3. How does a vertical stretch differ from a vertical shift in terms of transformations?
4. What are other examples of function transformations that change both the x- and y-coordinates?
5. How would the graph change if the transformation was \( y = 3f(2x) - 4 \) instead?

### Tip:
Always apply transformations in the correct order: handle horizontal changes (like compression/stretch) first, followed by vertical changes (like stretch/reflection), and finish with shifts.

Draw the graph of y = -3f(2x) + 4, with the corresponding points highlighted.

Learn how to transform the graph of y = f(x) using the function transformation y = -3f(2x) + 4. This problem includes steps for applying horizontal compression, vertical stretch, reflection, and vertical shift, with examples of corresponding points. Suitable for high school students.

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