https://file.aiquickdraw.com/mathbot/file/1730328159314mpvvcrf5.jpg

To express the function as a piecewise function, let's examine each segment of the graph and write the corresponding linear equations with their respective intervals.

### Step-by-Step Solution
1. **Identify the line segments and endpoints**:
   - The graph shows two line segments:
     - **Segment 1**: Starts at \( x = -6 \) and ends at \( x = -2 \).
     - **Segment 2**: Starts at \( x = -2 \) and extends downwards, ending at \( x = 6 \).

2. **Find the equations of each segment**:
   - **Segment 1 (from \( x = -6 \) to \( x = -2 \))**:
     - Points on the line: \( (-6, 4) \) and \( (-2, 0) \).
     - Calculate the slope \( m \):
       \[
       m = \frac{0 - 4}{-2 + 6} = \frac{-4}{4} = -1
       \]
     - Using point-slope form with point \( (-6, 4) \):
       \[
       y - 4 = -1(x + 6)
       \]
       Simplify to get the equation:
       \[
       y = -x - 2
       \]
     - This segment is defined for \( -6 \leq x \leq -2 \).

   - **Segment 2 (from \( x = -2 \) to \( x = 6 \))**:
     - Points on the line: \( (-2, 0) \) and \( (6, -8) \).
     - Calculate the slope \( m \):
       \[
       m = \frac{-8 - 0}{6 + 2} = \frac{-8}{8} = -1
       \]
     - Using point-slope form with point \( (-2, 0) \):
       \[
       y - 0 = -1(x + 2)
       \]
       Simplify to get the equation:
       \[
       y = -x - 2
       \]
     - This segment is defined for \( -2 < x \leq 6 \).

3. **Write the piecewise function**:
   \[
   f(x) = \begin{cases} 
      -x - 2 & \text{for } -6 \leq x \leq -2, \\
      -x - 2 & \text{for } -2 < x \leq 6.
   \end{cases}
   \]

This defines the function in piecewise form based on the intervals observed in the graph.

Would you like more details or have any questions?

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### Related Questions
1. How do you determine the slope of a line given two points?
2. How would the piecewise function change if one of the intervals had a different slope?
3. What is the point-slope form, and why is it useful here?
4. How do open and closed intervals affect piecewise functions?
5. Can this function be represented as a single equation instead of piecewise?

**Tip:** In piecewise functions, always pay close attention to open and closed endpoints to match the graph's behavior.

Express the function graphed on the axes below as a piecewise function.

Learn how to express a function as a piecewise function from its graph by identifying segments, calculating slopes, and writing linear equations. This guide covers linear functions and slope calculations, suitable for students in Grades 9-11.

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