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To represent the given graph as a piecewise function, let's analyze the different segments of the graph:

### Analysis of the Graph:
1. **First Segment**: From \((-6, 0)\) to \((-2, -2)\), the graph is a straight line.
2. **Second Segment**: From \((-2, -2)\) to \((2, 0)\), the graph is another straight line.
3. **Third Segment**: From \((2, 0)\) to \((6, 4)\), the graph is a straight line.

### Piecewise Function Representation:
To find the equations of each line segment, we'll calculate the slope and the y-intercept for each line.

1. **First Segment**: From \((-6, 0)\) to \((-2, -2)\)
   - Slope \( m = \frac{-2 - 0}{-2 - (-6)} = \frac{-2}{4} = -\frac{1}{2} \)
   - Equation: \( y = -\frac{1}{2}x - 3 \)

2. **Second Segment**: From \((-2, -2)\) to \((2, 0)\)
   - Slope \( m = \frac{0 - (-2)}{2 - (-2)} = \frac{2}{4} = \frac{1}{2} \)
   - Equation: \( y = -\frac{1}{2}x - 3 \)

3. **Third Segment**: From \((2, 0)\) to \((6, 4)\)
   - Slope \( m = \frac{4 - 0}{6 - 2} = \frac{4}{4} = 1 \)
   - Equation: \( y = x-2. \) 

### Piecewise Function Representation
The function is as follows:
\[
f(x) =
\begin{cases}
 -\frac{1}{2}x - 3, & \text{if } -6 < x \leq -2 \\
\frac{1}{2}x - 2, & \text{if } -2 < x \leq 2 \\
   x +2, & \text{if } 2 < x \leq 6 \\
\end{cases}
\]

Would you like more details or have any questions? Here are some questions that might help further expand your understanding: 
1. How can the endpoints of each segment affect the domain restrictions in the piecewise function?
2. How do you determine the slope of a line segment between two points?
3. Can a piecewise function have overlapping domains?
4. How can you graph a piecewise function by hand given its algebraic representation?
5. What changes to the graph would alter the piecewise function representation?

**Tip**: Always check the continuity at the boundary points when analyzing piecewise functions to ensure accuracy in your graph and equations!

Represent the given graph as a piecewise function.

Learn how to represent a graph with multiple linear segments as a piecewise function. This involves understanding the slope of a line, deriving linear equations, and applying domain restrictions for each segment. Suitable for students in Grades 9-11.

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