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

Step 1: Simplify the equation

The equation can be rewritten as:

$f(x) = |x + 2| - 3$

Step 2: Analyze the components

1. The base graph is $|x + 2|$, which is the absolute value function shifted 2 units to the left compared to $|x|$.
2. Subtracting 3 from $|x + 2|$ shifts the graph down by 3 units.

Step 3: Key points of the graph

• The vertex (where $|x + 2| = 0$) is at $x = -2$, giving $f(-2) = -3$.
• For $x > -2$, $|x + 2| = x + 2$, so $f(x) = (x + 2) - 3 = x - 1$.
• For $x < -2$, $|x + 2| = -(x + 2)$, so $f(x) = -(x + 2) - 3 = -x - 5$.

Step 4: Shape of the graph

• The graph has a V-shape with its vertex at $(-2, -3)$.
• The left branch has slope $-1$, and the right branch has slope $1$.

Step 5: Matching the graph

• Look for a graph with:
  • A vertex at $(-2, -3)$,
  • Slope $-1$ to the left of $x = -2$,
  • Slope $1$ to the right of $x = -2$.

If you provide options or a specific image, I can confirm which graph matches.

Would you like more clarification or have specific options to review?

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Related Questions:

1. What transformations apply to $|x|$ when shifted vertically and horizontally?
2. How do you find the vertex of an absolute value function?
3. What role does the slope play in the branches of an absolute value graph?
4. How can you test points to verify if they lie on the graph?
5. How does the domain and range of $f(x) = |x + 2| - 3$ compare to $|x|$?

Tip: Always identify the vertex and slopes when graphing absolute value functions to ensure accuracy.