https://file.aiquickdraw.com/mathbot/file/1729399299325jfiy8x30.jpg

The image presents a graph of \( y = f(x) \), and the question asks to identify the correct graph of \( y = |f(x)| \) from four options. Let's break down the transformation step-by-step:

1. **Understanding \( y = f(x) \)**: 
   The original graph has positive and negative values. 
   - When \( y = f(x) \) is negative (below the x-axis), these values will be reflected above the x-axis.
   - When \( y = f(x) \) is already positive or zero (on or above the x-axis), the graph will remain unchanged.

2. **Transformation to \( y = |f(x)| \)**:
   - For portions where \( f(x) \geq 0 \), the graph remains the same.
   - For portions where \( f(x) < 0 \), the graph will be reflected upwards, turning negative values positive.

Now, analyzing the graphs in your image:
- In the original graph of \( f(x) \), the portion from \( x = -2 \) to \( x = 0 \) is below the x-axis, meaning this part will be reflected upwards in \( |f(x)| \).
  
- In the options, the correct graph should reflect this behavior, where the negative portion of the graph is flipped.

Looking at the options, the **third graph from the top** seems to match this description. In that graph:
- The segment from \( x = -2 \) to \( x = 0 \), which was below the x-axis in the original graph, is now reflected above the x-axis.
- The rest of the graph remains unchanged where \( f(x) \geq 0 \).

So, the third option is the correct graph for \( y = |f(x)| \).

Would you like further details or have any questions?

Here are five related questions you might consider:
1. What happens to the graph if we apply the transformation \( y = f(|x|) \)?
2. How would the graph change if we transformed it to \( y = -f(x) \)?
3. Can you explain why reflection occurs only for negative values in \( y = |f(x)| \)?
4. How would the behavior change if we worked with a different transformation like \( y = f(x^2) \)?
5. How can you determine the transformation \( y = -|f(x)| \) visually?

**Tip**: Always check whether the transformation affects the x or y values to correctly apply the reflection or distortion!

Below is the graph of y = f(x). Which of the following is the graph of y = |f(x)|?

This problem involves identifying the correct transformation from y = f(x) to y = |f(x)|. Learn how to reflect the negative portions of the graph above the x-axis and apply transformations to find the correct graph. Suitable for high school students studying graph transformations.

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