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The image contains a set of problems asking to graph functions using the given \( f(x) \), and then apply transformations to find \( g(x) = |f(x)| \) and \( h(x) = f(|x|) \).

Here’s how you approach each:

### Problem 41: \( f(x) = \frac{2}{x} \)
- **Graph \( f(x) \)**: This is a hyperbola, with vertical and horizontal asymptotes at \( x = 0 \) and \( y = 0 \).
- **Graph \( g(x) = |f(x)| \)**: This reflects the negative part of \( f(x) \) (where \( f(x) < 0 \)) above the x-axis.
- **Graph \( h(x) = f(|x|) \)**: Reflect the graph of \( f(x) \) to the right of the y-axis onto the left side (making it symmetric).

### Problem 42: \( f(x) = \sqrt{x^2 - 4} \)
- **Graph \( f(x) \)**: This is a square root function with horizontal shifts. It starts at \( x = -2 \) and \( x = 2 \) and goes outward.
- **Graph \( g(x) = |\sqrt{x^2 - 4}| \)**: Since \( f(x) \) is non-negative, this is the same as the original function.
- **Graph \( h(x) = f(|x|) \)**: Reflect the graph over the y-axis to make it symmetric.

### Problem 43: \( f(x) = x^4 - x^3 - 4x^2 \)
- **Graph \( f(x) \)**: This is a quartic function. Identify key features like critical points and inflection points.
- **Graph \( g(x) = |f(x)| \)**: Reflect the parts of the graph where \( f(x) < 0 \) above the x-axis.
- **Graph \( h(x) = f(|x|) \)**: Reflect the right side of the graph onto the left side to make it symmetric.

### Problem 44: \( f(x) = \frac{1}{2}x^3 + 2x^2 - 8x - 2 \)
- **Graph \( f(x) \)**: This is a cubic polynomial. Find the inflection points and turning points for sketching.
- **Graph \( g(x) = |f(x)| \)**: Reflect the negative parts of the graph above the x-axis.
- **Graph \( h(x) = f(|x|) \)**: Reflect the right side of the graph onto the left side for symmetry.

Would you like detailed step-by-step explanations for any of these graphs?

Here are five related questions:
1. How does reflecting across the x-axis differ from reflecting across the y-axis in graphing?
2. How can you determine if a function is even or odd by looking at its graph?
3. What is the effect of the absolute value on the graph of a function?
4. What are some common transformations applied to graphs of functions (shifts, stretches, reflections)?
5. How does graphing cubic and quartic functions differ in terms of identifying key points?

**Tip**: When graphing absolute value transformations, always start by sketching the original function to understand where reflections occur!

Use the graph of f(x) to graph g(x) = |f(x)| and h(x) = f(|x|).

Learn how to graph the transformations of functions using absolute value rules. This covers graphing g(x) = |f(x)| and h(x) = f(|x|) for different functions, including rational, polynomial, and square root functions. Suitable for high school students in Grades 10-12.

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