To graph the equation \( y = |x + 1| - 2 \), we can break down the transformations of the basic absolute value function \( y = |x| \).

### Steps:

1. **Basic Absolute Value Function**: Start with \( y = |x| \), which is a V-shaped graph that has its vertex at the origin \( (0, 0) \) and opens upwards.

2. **Horizontal Shift**: The expression \( |x + 1| \) shifts the graph of \( y = |x| \) **1 unit to the left**. This means the new vertex is at \( (-1, 0) \).

3. **Vertical Shift**: The subtraction of 2 from the function, i.e., \( |x + 1| - 2 \), shifts the graph **2 units down**. This moves the vertex to \( (-1, -2) \).

### Key Points:

- The vertex of the graph is at \( (-1, -2) \).
- The graph still maintains its V-shape, with the slope of the arms being 1 (the left arm slopes down at \( -1 \) and the right arm slopes up at \( +1 \)).

### Sketch of the Graph:

- Vertex: \( (-1, -2) \)
- Left side: The slope is \( -1 \) (goes down towards left).
- Right side: The slope is \( +1 \) (goes up towards right).

This is the graph of \( y = |x + 1| - 2 \), a V-shape shifted left and down from the basic \( y = |x| \) graph.

Would you like me to plot this graph or provide more details?

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**Here are 5 questions related to the topic:**
1. What is the domain and range of \( y = |x + 1| - 2 \)?
2. How would the graph change if we had \( y = |x + 1| + 2 \)?
3. What would happen to the graph if the equation was \( y = -|x + 1| - 2 \)?
4. How do you find the intercepts of \( y = |x + 1| - 2 \)?
5. How do you solve the equation \( |x + 1| - 2 = 0 \) for \( x \)?

**Tip**: Horizontal shifts affect the \( x \)-coordinates, while vertical shifts affect the \( y \)-coordinates of the graph.

Learn how to graph the absolute value function y = |x + 1| - 2. This problem involves key algebraic concepts such as horizontal and vertical shifts of absolute value graphs. Follow a step-by-step guide to understand how the vertex is shifted and how the V-shape of the graph changes. Suitable for students in Grades 8-10.

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