https://file.aiquickdraw.com/mathbot/file/17291792305525qbagfq7.jpg

The function provided is \( f(x) = |x + 4| - 3 \). This is a transformation of the basic absolute value function \( f(x) = |x| \).

### Key transformations:
1. **Horizontal shift**: \( |x + 4| \) means a shift 4 units to the left.
2. **Vertical shift**: Subtracting 3 at the end, \( -3 \), shifts the graph down by 3 units.

### Vertex of the absolute value function:
The vertex of \( f(x) = |x + 4| - 3 \) will occur when the expression inside the absolute value equals zero, i.e., when \( x + 4 = 0 \), which gives \( x = -4 \).

Thus, the vertex of the graph is at \( (-4, -3) \).

### Behavior of the graph:
- The graph opens upwards (since the absolute value is positive).
- The slope to the right of the vertex is 1, and to the left, it is -1 (standard V-shape).

### Identifying the correct graph:
From the provided images, we need to look for a graph with a vertex at \( (-4, -3) \).

Upon inspection, the **third graph** (with a vertex at \( (-4, -3) \)) correctly matches the function \( f(x) = |x + 4| - 3 \).

Would you like further explanation or have any questions?

### Related Questions:
1. How do transformations affect the graph of an absolute value function?
2. What would happen to the graph if we change the function to \( f(x) = |x + 4| + 3 \)?
3. How does the slope change in the graph of an absolute value function?
4. What does the absolute value function represent geometrically?
5. How would a vertical stretch or compression impact this graph?

### Tip:
Always check both the vertex position and the direction (opening up or down) when working with transformed absolute value functions.

Graph the function, f(x) = |x + 4| - 3. Identify the correct graph from the options provided.

Learn how to graph the function f(x) = |x + 4| - 3. Understand the key transformations, including horizontal shifts and vertical translations. Suitable for students in Grades 8-10, this example helps reinforce concepts related to absolute value functions.

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