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!