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.