https://file.aiquickdraw.com/mathbot/file/1726430158374ctcy6wet.jpg

The task involves matching transformations of the function \( f(x) \) with their corresponding graphs. Here’s how the transformations generally affect the graph:

1. **\( f(x) - 2 \)**: Shifts the graph 2 units **down**.
2. **\( f(2x) \)**: **Compresses** the graph horizontally by a factor of 2.
3. **\( -f(x+2) \)**: Reflects the graph over the x-axis and shifts it 2 units **left**.
4. **\( 2f(-x) \)**: Reflects the graph over the y-axis and **stretches** it vertically by a factor of 2.
5. **\( f(x)/2 \)**: **Compresses** the graph vertically by a factor of 2.
6. **\( f(x-2) \)**: Shifts the graph 2 units **right**.
7. **\( -f(x/2) \)**: **Stretches** the graph horizontally by a factor of 2 and reflects it over the x-axis.
8. **\( f(-x) + 2 \)**: Reflects the graph over the y-axis and shifts it 2 units **up**.
9. **\( 2f(x) - 2 \)**: **Stretches** the graph vertically by a factor of 2 and shifts it 2 units **down**.

I will now match these transformations with the graphs shown in the image. Here's the matching list:

1. **\( f(x) - 2 \)** → Graph **I** (shifted down)
2. **\( f(2x) \)** → Graph **C** (horizontal compression)
3. **\( -f(x + 2) \)** → Graph **E** (reflection and left shift)
4. **\( 2f(-x) \)** → Graph **D** (reflection over y-axis and vertical stretch)
5. **\( f(x) / 2 \)** → Graph **A** (vertical compression)
6. **\( f(x - 2) \)** → Graph **H** (shifted right)
7. **\( -f(x / 2) \)** → Graph **F** (horizontal stretch and reflection)
8. **\( f(-x) + 2 \)** → Graph **G** (reflection over y-axis and shifted up)
9. **\( 2f(x) - 2 \)** → Graph **B** (vertical stretch and downward shift)

Would you like more details on how each transformation affects the graph? Here are some related questions to explore further:

1. How does a vertical shift differ from a horizontal shift in graph transformations?
2. Why does compressing a function horizontally involve multiplying \(x\) by a constant greater than 1?
3. How does reflecting a graph over the y-axis change the function's equation?
4. What is the effect of stretching or compressing a function vertically?
5. Can you explain how multiple transformations are applied step by step to a function?

**Tip**: When working with transformations, always apply horizontal changes (inside the function) before vertical ones for accurate graphing!

Match each transformation of f(x) with its graph from the options provided.

Learn how to match transformations of a function with their corresponding graphs. This problem involves graph shifting, reflecting, stretching, and compressing, suitable for high school students.

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