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To determine the equation that matches the graph, we need to identify the different parts and behaviors of the function shown. Here's an analysis of the given graph:

1. **For \( x < -2 \)**: The function appears to be decreasing in a parabolic shape.
2. **For \( -2 \leq x < 1 \)**: The function is a continuous curve that increases and then hits a specific point.
3. **For \( 1 \leq x < 2 \)**: The function is constant.
4. **For \( x \geq 2 \)**: The function increases linearly.

Let's write the piecewise function:

### Piecewise function components:
1. **For \( x < -2 \)**: The function is a part of a parabola.
   \[ f(x) = (x + 2)^2 \]
   This matches the shape we see for \( x < -2 \).

2. **For \( -2 \leq x < 1 \)**: This part transitions smoothly from the parabolic section to the point (1,1).
   \[ f(x) = 1 \]
   This represents the constant part of the graph for \( -2 \leq x < 1 \).

3. **For \( x = 1 \)**: The function value is \( f(1) = 2 \).

4. **For \( 1 < x < 2 \)**: The function is again constant but at a different value.
   \[ f(x) = 3 \]

5. **For \( x \geq 2 \)**: The function increases linearly.
   \[ f(x) = x + 1 \]
   This represents the linear increase for \( x \geq 2 \).

### Complete piecewise function:
\[ f(x) = 
\begin{cases} 
(x + 2)^2 & \text{if } x < -2 \\
1 & \text{if } -2 \leq x < 1 \\
2 & \text{if } x = 1 \\
3 & \text{if } 1 < x < 2 \\
x + 1 & \text{if } x \geq 2 
\end{cases} \]

### Verification:
- **For \( x < -2 \)**, \((x + 2)^2\) is the equation of the parabola.
- **For \(-2 \leq x < 1 \)**, \( f(x) = 1 \) is a constant function.
- **At \( x = 1 \)**, \( f(x) = 2 \).
- **For \( 1 < x < 2 \)**, \( f(x) = 3 \) is another constant function.
- **For \( x \geq 2 \)**, \( f(x) = x + 1 \) is a linear function.

This matches the behavior and values shown in the graph.

Would you like more details or have any questions?

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### Questions for Further Exploration:
1. How can we verify the continuity of the function at each boundary point?
2. What are the intercepts of this piecewise function with the x-axis and y-axis?
3. How would you plot this piecewise function on a coordinate plane?
4. Can you find the domain and range of this piecewise function?
5. How does the function behave as \( x \) approaches \(-2\) from the left and the right?
6. How does the function behave as \( x \) approaches 1 from the left and the right?
7. How does the function behave as \( x \) approaches 2 from the left and the right?
8. How would the function change if the linear part for \( x \geq 2 \) was \( 2x - 1 \) instead?

### Tip:
When working with piecewise functions, ensure that you check the continuity and differentiability at the points where the pieces meet. This helps in understanding the overall behavior of the function.

Learn how to understand and write piecewise functions with a detailed step-by-step guide. This problem involves analyzing a graph and identifying different parts of the function, including parabolic, constant, and linear segments. Suitable for high school students in Grades 10-12.

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