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Let's analyze the given graph to write the piecewise function.

### Steps to solve:
1. The graph is made of two line segments with different slopes.
   
2. Identify the equations of the two lines:
   - **For \( x \geq -2 \)**: This segment runs from point \((-2, -5)\) to \((5, 2)\). We need to find the slope and the equation.
   - **For \( x < -2 \)**: This segment runs from \((-4, 4)\) to \((-2, -5)\). Similarly, we find the slope and equation.

### Step 1: Equation for \( x \geq -2 \)
The two points on this line are \( (-2, -5) \) and \( (5, 2) \).

- **Slope**:
  \[
  m = \frac{2 - (-5)}{5 - (-2)} = \frac{7}{7} = 1
  \]

- **Equation**:
  Using point-slope form \( y - y_1 = m(x - x_1) \), with point \((-2, -5)\):
  \[
  y + 5 = 1(x + 2)
  \]
  Simplifying:
  \[
  y = x - 3
  \]

Thus, for \( x \geq -2 \), the equation is \( f(x) = x - 3 \).

### Step 2: Equation for \( x < -2 \)
The two points on this line are \( (-4, 4) \) and \( (-2, -5) \).

- **Slope**:
  \[
  m = \frac{-5 - 4}{-2 - (-4)} = \frac{-9}{2}
  \]

- **Equation**:
  Using point-slope form with point \( (-2, -5) \):
  \[
  y + 5 = -\frac{9}{2}(x + 2)
  \]
  Simplifying:
  \[
  y = -\frac{9}{2}(x + 2) - 5
  \]
  Expand:
  \[
  y = -\frac{9}{2}x - 9 - 5
  \]
  \[
  y = -\frac{9}{2}x - 14
  \]

Thus, for \( x < -2 \), the equation is \( f(x) = -\frac{9}{2}x - 14 \).

### Final Piecewise Function:
\[
f(x) = 
\begin{cases}
x - 3 & \text{if } x \geq -2 \\
-\frac{9}{2}x - 14 & \text{if } x < -2
\end{cases}
\]

Would you like further clarification on this? 

Here are some questions to expand your understanding:
1. How is the slope of a line calculated from two points?
2. Why do we use point-slope form to find the equation of a line?
3. What does the piecewise function represent in terms of the graph?
4. How would the function change if the lines were curved instead of straight?
5. What other methods can be used to define a piecewise function?

**Tip:** In piecewise functions, always pay attention to whether the endpoint is included or excluded, as indicated by the inequality.

Write a piecewise function for the given graph.

Learn how to write a piecewise function based on the analysis of a given graph. This problem involves identifying linear equations using the point-slope form and calculating slopes from two points. Suitable for students in Grades 9-12.

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