To sketch the graph of $y = |f(x)|$, follow these steps:

1. Start by sketching $y = f(x)$: Plot the original function as if the absolute value wasn’t present.

2. Identify the negative portions of $f(x)$: Locate where the graph of $y = f(x)$ is below the x-axis (i.e., where $f(x) < 0$).

3. Reflect the negative parts above the x-axis: For any $x$ where $f(x) < 0$, reflect the portion of the graph that is below the x-axis to make it positive. This means flipping it to the other side of the x-axis while maintaining its shape.

4. Keep the positive parts unchanged: For the regions where $f(x) \geq 0$, the graph remains as it is.

5. Erase the parts below the x-axis: After reflecting, remove any parts of the graph that remain below the x-axis, since $|f(x)|$ is always non-negative.

Example

If you have $f(x) = x^2 - 4$, first sketch $f(x) = x^2 - 4$, then reflect the part of the graph below the x-axis, and finally remove the portion that remains negative.

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Do you want to go into more detail or ask any questions about these steps?

Here are 5 related questions to deepen understanding:

1. What happens when the function $f(x)$ is always positive?
2. How does the graph of $|f(x)|$ change if $f(x)$ is a linear function?
3. Can the graph of $y = |f(x)|$ ever touch the x-axis?
4. What are some common mistakes to avoid when sketching $y = |f(x)|$?
5. How would the process differ for a piecewise function?

Tip: Always ensure the points where $f(x) = 0$ remain unchanged when sketching $|f(x)|$.