Write a piecewise function represented by the graph.

A graph of a piecewise function. One linear piece goes through ordered pair negative 4 comma negative 6 and ends at a closed dot at ordered pair negative 2 comma negative 2. A second linear piece begins at the same closed dot at ordered pair negative 2 comma negative 2 and goes through ordered pair 2 comma 0.

To write the piecewise function represented by the graph, we need to define each linear piece separately.

### Step 1: First Linear Piece (From \(x = -4\) to \(x = -2\))
The first segment goes through the points \((-4, -6)\) and \((-2, -2)\). We will determine the equation of the line using the slope-intercept form \(y = mx + b\).

#### Find the slope \(m\):
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - (-6)}{-2 - (-4)} = \frac{-2 + 6}{-2 + 4} = \frac{4}{2} = 2
\]

So, the slope \(m = 2\).

#### Use point-slope form to find the equation:
Using point \((-4, -6)\) and the slope \(m = 2\), the equation of the line is:
\[
y - y_1 = m(x - x_1)
\]
\[
y - (-6) = 2(x - (-4))
\]
\[
y + 6 = 2(x + 4)
\]
\[
y = 2x + 8 - 6
\]
\[
y = 2x + 2
\]

Thus, the equation for the first piece is \(y = 2x + 2\), valid for \(x \in [-4, -2]\).

### Step 2: Second Linear Piece (From \(x = -2\) to \(x = 2\))
The second segment goes through the points \((-2, -2)\) and \((2, 0)\).

#### Find the slope \(m\):
\[
m = \frac{0 - (-2)}{2 - (-2)} = \frac{0 + 2}{2 + 2} = \frac{2}{4} = \frac{1}{2}
\]

So, the slope \(m = \frac{1}{2}\).

#### Use point-slope form to find the equation:
Using point \((-2, -2)\) and the slope \(m = \frac{1}{2}\), the equation of the line is:
\[
y - (-2) = \frac{1}{2}(x - (-2))
\]
\[
y + 2 = \frac{1}{2}(x + 2)
\]
\[
y + 2 = \frac{1}{2}x + 1
\]
\[
y = \frac{1}{2}x + 1 - 2
\]
\[
y = \frac{1}{2}x - 1
\]

Thus, the equation for the second piece is \(y = \frac{1}{2}x - 1\), valid for \(x \in (-2, 2]\).

### Piecewise Function:
Now, we can write the piecewise function:
\[
f(x) = 
\begin{cases} 
2x + 2 & \text{if } -4 \leq x \leq -2 \\
\frac{1}{2}x - 1 & \text{if } -2 < x \leq 2
\end{cases}
\]

Would you like further details on any part of this? 

Here are some related questions to deepen your understanding:
1. How do you find the slope between two points?
2. What is the point-slope form of a line, and when is it used?
3. Can a piecewise function be continuous? Why or why not?
4. How would you graph this piecewise function?
5. What is the domain of this piecewise function?

**Tip:** When working with piecewise functions, always check if the pieces are connected at the boundary points!

Write a piecewise function represented by the graph. A graph of a piecewise function. One linear piece goes through ordered pair negative 4 comma negative 6 and ends at a closed dot at ordered pair negative 2 comma negative 2. A second linear piece begins at the same closed dot at ordered pair negative 2 comma negative 2 and goes through ordered pair 2 comma 0.

Learn how to write a piecewise function based on a graph with two linear pieces, one through points (-4, -6) to (-2, -2) and another from (-2, -2) to (2, 0). This solution covers key concepts such as slope calculation, point-slope form, and interpreting graph boundaries. Suitable for Grades 9-12.

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