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The image you uploaded shows a mathematical problem involving the calculation of a piecewise function based on the graph provided.

### Step-by-Step Solution:

1. **Understanding the Graph**:
   - The graph displays two line segments. Each segment represents a portion of a piecewise function. 
   - The first segment starts at the point \((-4, 0)\) and ends at \((2, 6)\).
   - The second segment starts at \((2, 6)\) and ends at \((6, 2)\).

2. **Defining the Piecewise Function**:
   - For \( x \) values from \(-4\) to \(2\) (inclusive), the function is defined by the first line segment.
   - For \( x \) values from \(2\) to \(6\) (inclusive), the function is defined by the second line segment.

3. **Finding the Equation of Each Line Segment**:
   - **First Segment (from \((-4, 0)\) to \((2, 6)\))**:
     - The slope \( m_1 \) of the line can be calculated as:
       \[
       m_1 = \frac{6 - 0}{2 - (-4)} = \frac{6}{6} = 1
       \]
     - Using the point-slope form \( y - y_1 = m(x - x_1) \), with point \((-4, 0)\):
       \[
       y = 1(x + 4) = x + 4
       \]
     - So, for \( x \) from \(-4\) to \(2\):
       \[
       f(x) = x + 4
       \]

   - **Second Segment (from \((2, 6)\) to \((6, 2)\))**:
     - The slope \( m_2 \) of the line is:
       \[
       m_2 = \frac{2 - 6}{6 - 2} = \frac{-4}{4} = -1
       \]
     - Using the point-slope form \( y - y_1 = m(x - x_1) \), with point \((2, 6)\):
       \[
       y - 6 = -1(x - 2)
       \]
       \[
       y = -x + 2 + 6 = -x + 8
       \]
     - So, for \( x \) from \(2\) to \(6\):
       \[
       f(x) = -x + 8
       \]

### Final Piecewise Function:
The piecewise function based on the given graph is:
\[
f(x) = \begin{cases} 
x + 4 & \text{for } -4 \leq x \leq 2 \\
-x + 8 & \text{for } 2 < x \leq 6 
\end{cases}
\]

Would you like more details on any part of the solution, or do you have any questions?

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### Relative Questions:
1. How do you calculate the slope between two points?
2. What is the point-slope form of a linear equation?
3. How do you determine the equation of a line given two points?
4. What are the general steps to writing a piecewise function from a graph?
5. How does changing the slope affect the graph of a line?

### Tip:
When writing piecewise functions, ensure that each piece covers the correct interval for \(x\) and that the conditions do not overlap.

Write a piecewise function for the given graph.

Learn how to write a piecewise function for a given graph with a detailed step-by-step solution. This problem covers key concepts such as piecewise functions, linear equations, and graphing. Suitable for students in Grades 9-11.

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