To determine the graph of the function $f(x) + 3 = |x + 2|$, let's analyze the equation step by step.

Step 1: Rewrite the function

We can express $f(x)$ as: $f(x) = |x + 2| - 3$

Step 2: Understand the transformations

The base function is $g(x) = |x|$, which is a V-shaped graph centered at the origin.

1. Horizontal shift: The term $x + 2$ indicates a shift 2 units to the left.
2. Vertical shift: Subtracting 3 from the absolute value moves the graph 3 units down.

Step 3: Key points and graph shape

1. The vertex of $f(x) = |x + 2| - 3$ is at $(-2, -3)$.
2. The graph opens upwards (retains the V shape).
3. The slope of the arms of the V is 1 and -1, as usual for absolute value functions.

Step 4: Match the graph

Look for a graph with the vertex at $(-2, -3)$ and symmetric arms forming a V-shape. The graph should shift 2 units to the left and 3 units downward compared to $y = |x|$.

If you have a specific set of graph options, feel free to describe or upload them for precise identification.

Would you like a visual representation or detailed exploration of the graph?

Questions to Expand:

1. How do horizontal shifts affect the graph of other functions, like quadratics?
2. Why does the slope of the absolute value arms remain constant at 1 and -1?
3. How does $f(x) + k = |x|$ compare when $k > 0$ or $k < 0$?
4. Can this transformation be applied to non-linear functions like $g(x) = x^2$?
5. How can piecewise functions represent absolute value equations?

Tip: Always identify transformations one step at a time to simplify graph analysis!