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To help you match the given functions with their corresponding graphs, let's go through the transformations and compare them with the graphs provided.

Here’s a breakdown of each function transformation:

1. **\( y = \frac{1}{2} f(x-1) \)**:
   - **\( f(x-1) \)**: This is a horizontal shift **1 unit to the right**.
   - **\( \frac{1}{2} \)**: This compresses the graph vertically by a factor of \( \frac{1}{2} \).
   
2. **\( y = f\left(\frac{1}{2}x\right) \)**:
   - **\( f\left(\frac{1}{2}x\right) \)**: This is a horizontal **stretch** by a factor of 2 (since \( \frac{1}{2} \) causes a widening along the x-axis).
   
3. **\( y = f(-x) \)**:
   - This reflects the graph across the **y-axis** (a horizontal reflection).
   
4. **\( y = f(x-2) \)**:
   - This is a horizontal shift **2 units to the right**.
   
5. **\( y = 2f(x) \)**:
   - This stretches the graph vertically by a factor of 2 (makes it twice as tall).
   
6. **\( y = f(2x) \)**:
   - This is a horizontal **compression** by a factor of 2 (the graph gets narrower along the x-axis).

### Now, comparing these with the graphs provided (A-F):

- **Graph A**: Appears to have a horizontal stretch, which would correspond to the function \( y = f\left(\frac{1}{2}x\right) \) (Option 2).
- **Graph B**: Looks like it has been horizontally compressed, so it likely corresponds to \( y = f(2x) \) (Option 6).
- **Graph C**: Seems like a horizontal reflection (flipped across the y-axis), which corresponds to \( y = f(-x) \) (Option 3).
- **Graph D**: This one looks vertically stretched, which would match \( y = 2f(x) \) (Option 5).
- **Graph E**: Appears to be shifted 1 unit to the right and vertically compressed, matching \( y = \frac{1}{2} f(x-1) \) (Option 1).
- **Graph F**: Appears to be shifted 2 units to the right, so it corresponds to \( y = f(x-2) \) (Option 4).

### Summary of Matches:
1. \( y = \frac{1}{2}f(x-1) \) → **E**
2. \( y = f\left(\frac{1}{2}x\right) \) → **A**
3. \( y = f(-x) \) → **C**
4. \( y = f(x-2) \) → **F**
5. \( y = 2f(x) \) → **D**
6. \( y = f(2x) \) → **B**

Would you like further details or explanations on any of these transformations?

### Related Questions:
1. How does a vertical compression affect the overall shape of a function?
2. What is the impact of horizontal stretches on periodic functions?
3. How does the reflection \( y = f(-x) \) compare to \( y = -f(x) \)?
4. Why does \( f(x-2) \) shift the graph to the right instead of left?
5. Can a function transformation have both horizontal and vertical changes simultaneously?

### Tip:
When identifying transformations, focus on how the function changes with respect to the x-axis (horizontal shifts, stretches) and y-axis (vertical shifts, stretches).

Match the given functions with their corresponding graphs. The functions involve transformations such as vertical stretching, horizontal stretching, reflections, and translations.

Learn how to match functions to their corresponding graphs based on transformations such as vertical and horizontal stretches, shifts, and reflections. This problem covers key concepts in function transformation suitable for high school students in grades 10-12.

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