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?

---

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.