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The problem in your image asks for the transformations needed to graph the function \( h(x) = |x + 2| - 5 \) from the absolute value function \( f(x) = |x| \).

To analyze the transformations:

1. **Horizontal Shift**: The term \( |x + 2| \) indicates a horizontal shift. Since \( +2 \) inside the absolute value moves the graph to the left, this is a **horizontal shift 2 units to the left**.

2. **Vertical Shift**: The term \( -5 \) outside the absolute value shifts the graph vertically. This is a **vertical shift 5 units down**.

There are no stretches, shrinks, or reflections in this case because there are no coefficients modifying \( x \) or \( |x| \) directly.

### Correct choices:
- **D. Horizontal shift** (left by 2 units)
- **F. Vertical shift** (down by 5 units)

Let me know if you want more details on this or have any questions!

Here are five follow-up questions to explore this concept further:

1. What is the effect of multiplying the function \( f(x) = |x| \) by a negative constant?
2. How would the graph change if the function were \( h(x) = 2|x + 2| - 5 \)?
3. What transformation would occur if \( h(x) = |x + 2| + 5 \)?
4. How would a reflection over the y-axis be represented in the absolute value function?
5. What happens to the graph of \( f(x) = |x| \) if we add a constant inside the absolute value?

**Tip**: Always remember that transformations inside the absolute value affect the graph horizontally (left/right), and those outside affect it vertically (up/down).

Use transformations of the absolute value function, f(x) = |x|, to graph the function h(x) = |x + 2| - 5. What transformations are needed to obtain the graph of h(x) from the graph of f(x)?

Learn how to graph the function h(x) = |x + 2| - 5 by applying transformations to the absolute value function f(x) = |x|. Understand the horizontal shift 2 units to the left and vertical shift 5 units down. Suitable for high school algebra students.

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