Write a piecewise function represented by the graph.

A graph of a piecewise function. One linear piece goes through ordered pair negative 2 comma 2 and ends at a closed dot at ordered pair negative 1 comma 3. A second linear piece begins at an open dot at ordered pair negative 1 comma 0 and ends at an open dot at ordered pair 3 comma negative 1. A third linear piece begins at a closed dot at ordered pair 3 comma negative 3 and goes through ordered pair 4 comma negative 3.

To represent the given graph as a piecewise function, we need to identify the equation of each linear segment and the domain over which each segment is defined.

### 1. First Segment:
- **Line Segment:** Passes through the points \((-2, 2)\) and \((-1, 3)\).
- **Equation of the line:**

  Find the slope (\(m\)) of the line:
  \[
  m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 2}{-1 - (-2)} = \frac{1}{1} = 1
  \]

  Using the point-slope form:
  \[
  y - y_1 = m(x - x_1)
  \]
  Substitute \((x_1, y_1) = (-2, 2)\) and \(m = 1\):
  \[
  y - 2 = 1(x + 2)
  \]
  Simplifying:
  \[
  y = x + 4
  \]

  **Domain:** \(-2 \leq x \leq -1\)

### 2. Second Segment:
- **Line Segment:** Passes through the points \((-1, 0)\) and \((3, -1)\).
- **Equation of the line:**

  Find the slope (\(m\)) of the line:
  \[
  m = \frac{-1 - 0}{3 - (-1)} = \frac{-1}{4} = -\frac{1}{4}
  \]

  Using the point-slope form:
  \[
  y - 0 = -\frac{1}{4}(x + 1)
  \]
  Simplifying:
  \[
  y = -\frac{1}{4}x - \frac{1}{4}
  \]

  **Domain:** \(-1 < x < 3\) (Note that \((-1, 0)\) and \((3, -1)\) are not included)

### 3. Third Segment:
- **Line Segment:** Passes through the points \((3, -3)\) and \((4, -3)\).
- **Equation of the line:**

  The line is horizontal, so the equation is:
  \[
  y = -3
  \]

  **Domain:** \(3 \leq x \leq 4\)

### Piecewise Function:
Combining all the segments, the piecewise function is:
\[
f(x) = \begin{cases} 
x + 4 & \text{if } -2 \leq x \leq -1 \\
-\frac{1}{4}x - \frac{1}{4} & \text{if } -1 < x < 3 \\
-3 & \text{if } 3 \leq x \leq 4 
\end{cases}
\]

Would you like more details or have any questions?

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**Related Questions:**
1. How do you find the equation of a line given two points?
2. What are the steps to determine the slope of a line?
3. How do you represent a horizontal line as a function?
4. What does an open dot versus a closed dot signify on a graph?
5. How do you determine the domain for each piece of a piecewise function?

**Tip:** When working with piecewise functions, carefully check each interval to ensure that it correctly represents the function's graph.

Write a piecewise function represented by the graph. A graph of a piecewise function. One linear piece goes through ordered pair negative 2 comma 2 and ends at a closed dot at ordered pair negative 1 comma 3. A second linear piece begins at an open dot at ordered pair negative 1 comma 0 and ends at an open dot at ordered pair 3 comma negative 1. A third linear piece begins at a closed dot at ordered pair 3 comma negative 3 and goes through ordered pair 4 comma negative 3.

Explore the step-by-step process of writing a piecewise function from a graph with three linear segments. Learn how to find the equation of each line using the slope formula and point-slope form. This problem is suitable for students in Grades 9-11, covering algebra and linear equations.

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