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.