The function given is \( f(x) = -|x + 1| - 2 \).

### Analysis:

1. **Absolute Value:** The term \( |x + 1| \) represents the absolute value of \( x + 1 \). This means the expression inside the absolute value is always non-negative.

2. **Negative Sign:** The function has a negative sign in front of the absolute value, \( -|x + 1| \), which reflects the graph of \( |x + 1| \) downward.

3. **Vertical Shift:** The \(-2\) at the end shifts the entire graph downward by 2 units.

### Key Points:
- **Vertex:** The vertex of the function occurs where the expression inside the absolute value is zero, i.e., \( x + 1 = 0 \), so \( x = -1 \). The corresponding \( y \)-value is \( f(-1) = -|0| - 2 = -2 \). Thus, the vertex is at \( (-1, -2) \).
  
- **Shape of the Graph:** Since the function involves the negative of an absolute value, the graph is a downward-pointing "V" shape.

- **Behavior:**
  - For \( x > -1 \), \( f(x) = -(x + 1) - 2 \).
  - For \( x < -1 \), \( f(x) = (x + 1) - 2 \).

### Graph of the Function:
The graph has a vertex at \( (-1, -2) \) and opens downward. 

Would you like a detailed explanation on any part of this, or do you have further questions? 

### Follow-up Questions:

1. How does the graph of \( |x + 1| \) compare to the graph of \( -|x + 1| \)?
2. What happens to the function \( f(x) \) if we change \(-2\) to \(+2\)?
3. Can you find the x-intercepts of the function \( f(x) \)?
4. How does the graph of \( f(x) \) shift if the equation is \( f(x) = -|x - 1| - 2 \)?
5. What is the domain and range of the function \( f(x) = -|x + 1| - 2 \)?

### Tip:
When working with absolute value functions, remember that the graph typically forms a "V" shape, and modifications such as a negative sign or vertical shifts will affect its orientation and position on the coordinate plane.

Explore the function f(x) = −|x + 1| − 2, which involves absolute value transformations. This problem demonstrates how to graph the function, identify key points such as the vertex, and understand the effects of vertical shifts and reflections. Suitable for students in Grades 9-10.

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