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Given the graph of y equals f left parenthesis x right parenthesis as shown below, sketch the graph of y equals open vertical bar f left parenthesis x right parenthesis close vertical bar.

To sketch the graph of \( y = |f(x)| \) based on the graph provided, follow these steps:

1. **Identify the Portions of the Graph Where \( f(x) \geq 0 \)**:
   - These are the parts of the graph where \( y = f(x) \) is already non-negative. For these portions, the graph of \( y = |f(x)| \) will look exactly the same as the graph of \( y = f(x) \).

2. **Identify the Portions of the Graph Where \( f(x) < 0 \)**:
   - These are the parts where the graph of \( y = f(x) \) is below the x-axis. For these portions, the graph of \( y = |f(x)| \) will be the reflection of the graph of \( y = f(x) \) over the x-axis. This is because taking the absolute value of a negative function value makes it positive.

3. **Sketch the Reflected Portion**:
   - Wherever the original graph is below the x-axis (where \( f(x) < 0 \)), reflect that portion of the graph above the x-axis.

### Application to Your Graph:

- **Left Segment (below x-axis):**
  The part of the graph on the left where \( f(x) \) is negative (below the x-axis) will be reflected above the x-axis. The shape of the curve remains the same, but it will now be above the x-axis.

- **Right Segment (above x-axis):**
  The part of the graph on the right where \( f(x) \) is already positive (above the x-axis) will remain unchanged.

### Sketch:

- **Starting from the left side**:
  - If the graph starts below the x-axis, reflect this portion above the x-axis.
- **Move rightward**:
  - As soon as the graph crosses the x-axis (where \( f(x) = 0 \)), continue with the graph as it is (no changes), since \( f(x) \geq 0 \) here.

Let me know if you want a detailed step-by-step explanation of any part of this or if you have any further questions!

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### Follow-up Questions:
1. How do you determine where \( f(x) \) is negative or positive in a graph?
2. What happens to the turning points of the graph when you apply the absolute value?
3. Can the graph of \( y = |f(x)| \) ever go below the x-axis?
4. How would the graph change if it were \( y = -f(x) \) instead?
5. What would the graph of \( y = |f(x)| \) look like if \( f(x) \) had no negative values?
6. How can you check the accuracy of your reflected graph?
7. What role does symmetry play in reflecting graphs over the x-axis?
8. How would you graph \( y = |f(x) + c| \) for a constant \( c \)?

**Tip:** Reflecting a graph over the x-axis is equivalent to negating the y-values. For absolute value transformations, always focus on flipping negative y-values to positive while keeping the positive values unchanged.

Learn how to sketch the graph of y = |f(x)| based on the given graph of y = f(x). Understand the transformations involved when taking the absolute value of a function and how to apply these transformations to specific graph segments. Suitable for high school students and anyone interested in graph transformations.

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