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To analyze the transformation from the function \( f(x) \) to \( h(x) \), let’s first examine both functions:

1. **Given functions:**
   - \( f(x) = \frac{1}{4}x - 2 \)
   - \( h(x) = -f(x) \)

2. **Step-by-Step Transformation Analysis:**
   - Since \( h(x) = -f(x) \), we are applying a *vertical reflection* of \( f(x) \) across the x-axis.
   - To be explicit, substitute \( f(x) \) into \( h(x) \):
     \[
     h(x) = -\left(\frac{1}{4}x - 2\right) = -\frac{1}{4}x + 2
     \]
   - Therefore, the function \( h(x) = -\frac{1}{4}x + 2 \) represents a reflection of \( f(x) \) across the x-axis.

3. **Summary:**
   - The transformation from the graph of \( f(x) \) to \( h(x) \) is a reflection across the x-axis. This changes the slope's sign and reflects any y-intercepts to the opposite side of the x-axis.

Would you like more details on the effects of this reflection or on the characteristics of the graph?

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Here are some related questions for further exploration:

1. What would be the effect if we reflected \( f(x) \) across the y-axis instead of the x-axis?
2. How would the transformation change if \( h(x) = -f(x) + 3 \) instead of just \( h(x) = -f(x) \)?
3. Can you describe how the slope and y-intercept change under a vertical reflection?
4. What would be the impact of adding a constant to \( f(x) \) before reflecting it?
5. How would we transform \( f(x) \) if we wanted to reflect it across both axes?

**Tip:** When reflecting a function vertically, only the y-values (outputs) of each point change signs, flipping the graph across the x-axis.

Use the graphs of f and h to describe the transformation from the graph of f to the graph of h. Given: f(x) = 1/4x - 2; h(x) = -f(x)

Explore the transformation from the graph of f(x) = 1/4x - 2 to h(x) = -f(x), involving a reflection across the x-axis. This problem covers key concepts in function transformations, focusing on reflections and linear functions. Ideal for students in Grades 9-11.

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