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We are given a piecewise function:

\[
f(x) = 
\begin{cases} 
(x + 3)^2 - 4 & \text{if } x \leq -3 \\
x + 1 & \text{if } -3 < x \leq 1 \\
(x - 2)^2 - 4 & \text{if } x > 1 
\end{cases}
\]

To identify which graph corresponds to this function, let's break down each piece and its behavior:

1. **First piece: \( f(x) = (x + 3)^2 - 4 \), for \( x \leq -3 \):**
   - This is a parabola shifted left by 3 and down by 4. 
   - It opens upwards because of the positive coefficient of \( x^2 \).
   - For \( x = -3 \), the vertex is at \( (-3, -4) \).

2. **Second piece: \( f(x) = x + 1 \), for \( -3 < x \leq 1 \):**
   - This is a straight line with slope 1 and y-intercept at \( 1 \).
   - The line starts from \( (-3, -2) \) (since at \( x = -3 \), \( f(-3) = -3 + 1 = -2 \)) and goes up to \( (1, 2) \).

3. **Third piece: \( f(x) = (x - 2)^2 - 4 \), for \( x > 1 \):**
   - This is a parabola shifted right by 2 and down by 4.
   - It opens upwards as well.
   - For \( x = 2 \), the vertex is at \( (2, -4) \).

### Now, analyzing the graphs:

- **Graph A**: This graph has discontinuities and the second piece doesn’t match the line segment given by \( f(x) = x + 1 \). So, this is not the correct graph.
  
- **Graph B**: This graph appears to show a smooth transition from a parabola to a linear function between \( x = -3 \) and \( x = 1 \), and then to another parabola for \( x > 1 \). The features match the breakdown of the function.

- **Graph C**: The linear part doesn’t connect properly at \( x = -3 \), and the shape of the parabolas doesn't seem correct.

- **Graph D**: The linear piece does not align correctly with the endpoints and the parabola for \( x \leq -3 \) is incorrect.

### Conclusion:
The correct graph is **B**.

Would you like further details on how each function transforms the graph? 

Here are 5 related questions to deepen your understanding:
1. What is the domain and range of the piecewise function?
2. How do you determine if a piecewise function is continuous?
3. Can piecewise functions have different types of functions in each interval, such as a line and a parabola?
4. How do shifts and reflections affect the graphs of functions like parabolas?
5. How can you check if the points where the pieces of the function meet are connected?

**Tip**: When graphing piecewise functions, always pay attention to whether the pieces are defined with strict inequalities (open circles) or inclusive inequalities (closed circles) to see if the endpoints connect.

Which of the following is the graph of the function f(x) = {(x + 3)^2 - 4 ; x ≤ -3, x + 1 ; -3 < x ≤ 1, (x - 2)^2 - 4 ; x > 1}?

Learn how to graph a piecewise function that includes both parabolic and linear components. This problem focuses on graphing transformations, understanding continuity, and analyzing different sections of a function. Suitable for students in Grades 9-12.

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