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The problem involves identifying the equation of a transformed function based on the graph of a given parent function. The parent function is \( g(x) = |x| \).

### Analyzing the Transformation
The graph of \( g(x) = |x| \) is a V-shaped graph centered at the origin (0,0) with two arms extending upwards.

In the transformed graph (on the right):
- The V-shape is upside down, indicating a reflection across the x-axis.
- The vertex of the V-shape has moved from the origin to the point \((2, 4)\), indicating a horizontal shift to the right by 2 units and a vertical shift upwards by 4 units.

### Transformations Breakdown
- **Reflection**: The reflection across the x-axis changes the function to \( -|x| \).
- **Horizontal Shift**: Moving the graph right by 2 units modifies the function to \( -|x - 2| \).
- **Vertical Shift**: Moving the graph up by 4 units changes the function to \( -|x - 2| + 4 \).

### Conclusion
The transformed function should be:

\[ h(x) = -|x - 2| + 4 \]

### Answer Choices:
Given the options:
- A. \( h(x) = -4|x - 2| + 4 \)
- B. \( h(x) = |x + 2| + 4 \)
- C. \( h(x) = -|x - 2| + 4 \)
- D. \( h(x) = \frac{1}{2}|x + 2| + 4 \)

**Correct Answer:**
The correct answer is **C**: \( h(x) = -|x - 2| + 4 \).

Would you like to discuss any details, or do you have any other questions?

### Related Questions
1. How does a vertical stretch or compression affect the equation of a function?
2. What would happen if the graph were reflected over the y-axis instead of the x-axis?
3. Can you describe how to determine the equation of a function from its graph?
4. How do transformations affect the domain and range of a function?
5. What is the effect of a horizontal compression on the graph of a function?

**Tip:** When dealing with transformations, always consider the order of operations: reflections, stretches/compressions, then translations.

Determine the equation for the transformed function given the parent function g(x) = |x| and the graph showing the transformation.

Learn how to determine the equation of a transformed absolute value function given the parent function g(x) = |x|. This problem involves reflections, horizontal shifts, and vertical shifts. Suitable for students in Grades 9-11.

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