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The graph provided shows a line with a negative slope, which is an important clue for understanding the transformations involved.

Let's break down the transformations step by step. The parent function \( f(x) = x \) is a line with a slope of 1, passing through the origin (0, 0). The given graph, \( g(x) \), appears to have:

1. **Reflection over the x-axis**: The graph of \( g(x) \) slopes downwards, which indicates the line has been reflected over the x-axis. This changes the slope from positive to negative.
  
2. **Vertical Stretch**: The slope of the transformed line is steeper than that of the parent function. This suggests a vertical stretch has occurred, making the line steeper. From the options, a stretch by a factor of 3 seems reasonable.

3. **Vertical Shift**: It appears that the line has been shifted vertically. The graph seems to have shifted upwards by 6 units from where it would have been after the reflection and stretch.

With this analysis in mind, we can now review the provided choices:

- **Option 2** ("Reflect over the x-axis, vertically stretch by a factor of 3, and then shift up 6 units") matches these transformations exactly.

Thus, the correct sequence of transformations is **Option 2**.

Would you like more detailed explanations or have any further questions?

Here are some related questions for deeper exploration:

1. What happens to the graph of \( f(x) = x \) if it is only reflected over the x-axis?
2. How does a vertical stretch affect the slope of a linear function?
3. Can a line's slope be changed without reflecting or stretching it?
4. What transformations would occur if the graph were shifted left or right instead of up or down?
5. How do transformations affect the intercepts of a line?

**Tip**: When identifying transformations, always check the slope and intercepts to detect reflections and shifts.

Determine which sequences of transformations could be applied to the parent function f(x) = x to obtain the graph of g.

Learn how to apply transformations, including reflection over the x-axis, vertical stretch, and vertical shift, to the parent function f(x) = x in order to match the graph of g(x). This problem focuses on linear functions and transformations, suitable for high school students.

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