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The problem provides a graph showing a function \( y = f(x) \) and two transformations of this function, labeled \( C_1 \) and \( C_2 \). The task is to write down the equations of the transformed curves \( C_1 \) and \( C_2 \) in function form.

### Analysis:

1. **Original Function \( y = f(x) \)**:
   - From the graph, it appears that the original function \( y = f(x) \) is a sinusoidal function, likely a cosine function, given its general shape.

2. **Transformation \( C_1 \)**:
   - \( C_1 \) appears to be a horizontal shift of the original function.
   - The graph shows \( C_1 \) shifted 90° to the right compared to \( y = f(x) \).
   - Therefore, the equation for \( C_1 \) should be: 
     \[
     y = f(x - 90^\circ)
     \]

3. **Transformation \( C_2 \)**:
   - \( C_2 \) seems to be a vertical translation of the original function \( y = f(x) \).
   - The graph shows \( C_2 \) translated 3 units down compared to \( y = f(x) \).
   - Therefore, the equation for \( C_2 \) should be:
     \[
     y = f(x) - 3
     \]

### Final Equations:

- For \( C_1 \): \( y = f(x - 90^\circ) \)
- For \( C_2 \): \( y = f(x) - 3 \)

Would you like further details on how transformations of functions work or have any questions about this solution? Here are some related questions:

1. What are other common transformations of functions besides shifts and translations?
2. How can you identify a vertical shift versus a horizontal shift in a graph?
3. Can you determine the original function \( f(x) \) if only given transformations?
4. How do amplitude and period affect the graph of a sinusoidal function?
5. What are phase shifts, and how do they differ from horizontal shifts?

**Tip:** When analyzing transformations of functions, always compare the graph of the original function to the transformed one to see how the position, shape, or size has changed.

The graph shows the function y = f(x) and two transformations of y = f(x), labeled C1 and C2. Write down the equations of the translated curves C1 and C2 in function form.

Learn how to determine the equations of transformed curves C1 and C2 from the original function y = f(x). This problem involves horizontal and vertical shifts, with a focus on applying the appropriate transformation formulas. Suitable for students in Grades 9-12.

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