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To solve this question, we analyze how transformations affect the given graph \( f(x) \), as follows:

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### **Transformations Explained**
1. **\( f(x - 3) \) (Option A):**
   - **Horizontal shift to the right** by 3 units.
   - Each \( x \)-coordinate increases by 3, but \( y \)-values remain unchanged.

2. **\( f(x) - 2 \) (Option B):**
   - **Vertical shift downward** by 2 units.
   - Each \( y \)-coordinate decreases by 2, but \( x \)-values remain the same.

3. **\( -f(x) \) (Option C):**
   - **Reflection across the \( x \)-axis**.
   - The \( y \)-coordinates are multiplied by -1, flipping the graph vertically.

4. **\( f(x + 1) \) (Option D):**
   - **Horizontal shift to the left** by 1 unit.
   - Each \( x \)-coordinate decreases by 1, while \( y \)-values stay the same.

5. **\( f(-x) \) (Option E):**
   - **Reflection across the \( y \)-axis**.
   - The \( x \)-coordinates are negated, flipping the graph horizontally.

6. **\( |f(x)| \) (Option F):**
   - All negative \( y \)-values become positive, reflecting any part of the graph **below the \( x \)-axis** upward.

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### **Step-by-Step Summary of Each Option**
1. **Option A: \( f(x - 3) \):**
   - Shift the graph 3 units to the right.

2. **Option B: \( f(x) - 2 \):**
   - Shift the graph 2 units downward.

3. **Option C: \( -f(x) \):**
   - Reflect the graph vertically across the \( x \)-axis.

4. **Option D: \( f(x + 1) \):**
   - Shift the graph 1 unit to the left.

5. **Option E: \( f(-x) \):**
   - Reflect the graph horizontally across the \( y \)-axis.

6. **Option F: \( |f(x)| \):**
   - Reflect all parts of the graph below the \( x \)-axis upward so that \( y \geq 0 \).

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### **Graph Summary**
If you plot each of these transformations, the resulting graphs would look like:
- \( f(x-3) \) shifted right.
- \( f(x)-2 \) shifted down.
- \( -f(x) \) flipped vertically.
- \( f(x+1) \) shifted left.
- \( f(-x) \) flipped horizontally.
- \( |f(x)| \), where negative \( y \)-values are reflected upward.

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Do you want detailed graphs of these transformations? Let me know!

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### **Related Questions**
1. How do you identify horizontal and vertical shifts in a function?
2. What happens when a graph is reflected across the \( x \)-axis?
3. How do you handle \( |f(x)| \) for a given function graph?
4. What is the impact of \( f(-x) \) on the graph of \( f(x) \)?
5. How do you graph \( f(x \pm c) \) transformations step-by-step?

### **Tip:** Always analyze transformations in the order: reflections, stretches/compressions, and then shifts.

Explore graph transformations including horizontal shifts, vertical shifts, reflections across axes, and absolute value transformations. Learn how to modify a graph of f(x) using equations like f(x-3), -f(x), and |f(x)|. Suitable for Grades 9-12.

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