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The problem provided asks to map the point \( (4, 1) \) on the graph of \( f(x) \) to its corresponding point on the graph of \( p(x) \), where \( p(x) = -2f(x - 3) + 4 \).

### Step-by-Step Solution:

1. **Transformation Formula**: The transformation involves:
   - A **horizontal shift** of \( 3 \) units to the **right** because of \( (x - 3) \),
   - A **vertical stretch** by a factor of \( -2 \) (reflection and scaling),
   - A **vertical shift** up by \( 4 \) units because of \( +4 \).

2. **Original Point**: Start with \( (4, 1) \) from the graph of \( f(x) \).

3. **Horizontal Shift**: The transformation \( f(x - 3) \) means shifting the \( x \)-coordinate 3 units to the **right**:
   \[
   x = 4 \implies x - 3 = 1
   \]
   So, the new point becomes \( (1, 1) \).

4. **Vertical Stretch and Reflection**: Apply the vertical transformation \( -2 \). This means multiplying the \( y \)-coordinate by \( -2 \):
   \[
   y = 1 \implies y \times (-2) = -2
   \]
   Now, the point is \( (1, -2) \).

5. **Vertical Shift**: Finally, add 4 to the \( y \)-coordinate because of the \( +4 \):
   \[
   y = -2 \implies y + 4 = 2
   \]
   So, the final transformed point is \( (1, 2) \).

### Final Answer:
The point \( (4, 1) \) on the graph of \( f(x) \) maps to \( (1, 2) \) on the graph of \( p(x) \).

Would you like further explanation or any details on the steps?

### Follow-up Questions:
1. How does a vertical stretch affect the graph of a function?
2. What would happen if we had a transformation like \( f(x + 3) \) instead of \( f(x - 3) \)?
3. How does the reflection caused by a negative coefficient affect the shape of the graph?
4. Can you describe what a horizontal shift does to the domain of a function?
5. What happens to a point if only a vertical shift is applied without any stretch or reflection?

### Tip:
When working with transformations, always apply horizontal shifts before vertical transformations (stretches and shifts) for clarity!

The graph of y = p(x) is the result of the transformation p(x) = -2f(x - 3) + 4. The point (4, 1) on the graph of f maps to which of the following points on the graph of p?

Learn how to map the point (4, 1) from the graph of f(x) under the transformation p(x) = -2f(x - 3) + 4. This solution covers function transformations including horizontal shifts, vertical stretching, reflection, and vertical shifts. Suitable for Grades 9-11.

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