describe the transformation from the parent function |x| of y=|x+4|-2

The transformation from the parent function \( y = |x| \) to the function \( y = |x+4| - 2 \) involves three distinct steps:

1. **Horizontal Shift**:
   - The expression \( |x + 4| \) inside the absolute value indicates a **horizontal shift**.
   - Since it's \( x + 4 \), this represents a **leftward shift** of 4 units. The general rule is that \( y = |x - h| \) shifts the graph \( h \) units horizontally. Here, \( h = -4 \), so the graph shifts 4 units to the left.

2. **Vertical Shift**:
   - The **"-2"** outside the absolute value affects the vertical position of the graph.
   - This represents a **downward shift** of 2 units. The general rule is that \( y = |x| + k \) shifts the graph \( k \) units vertically. In this case, \( k = -2 \), so the graph shifts down 2 units.

3. **No Stretch or Reflection**:
   - There are no multiplicative factors affecting the steepness or orientation of the graph, so there is **no vertical stretch or reflection**.

### Summary of Transformations:
- Shift the graph of \( y = |x| \) **4 units to the left**.
- Then, shift the resulting graph **2 units down**.

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Do you have any questions or want further details?

### Related Questions:
1. What would happen if the function was \( y = |x-4| + 2 \)?
2. How do vertical stretches or compressions affect the graph of \( y = |x| \)?
3. What if there were a negative sign outside the absolute value, like \( y = -|x+4| - 2 \)?
4. Can you explain how to find the vertex of an absolute value function from its equation?
5. How do you solve for \( x \) when given an equation in the form \( y = |x+4| - 2 \)?

### Tip:
For absolute value transformations, focus first on what's inside (horizontal shifts) and then what's outside (vertical shifts and stretches).

Understand the transformations of the absolute value function from y = |x| to y = |x+4| - 2. This problem covers horizontal and vertical shifts, providing a step-by-step explanation of how the graph moves left and down. Ideal for students in Grades 9-11 studying graph transformations.

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