https://file.aiquickdraw.com/mathbot/file/17256489459867mfi9i80.jpg

The graph in the image represents a piecewise-defined function with different segments. Let's break down the function based on the different sections of the graph.

### 1. **For \( x < -4 \):**
   The graph has a constant value for \( x < -4 \). Observing the graph, the value of \( y \) remains constant at \( -3 \) in this interval. So, the function for this section is:
   \[
   f(x) = -3 \quad \text{for} \quad x < -4
   \]

### 2. **For \( -4 \leq x \leq 4 \):**
   In this interval, the graph forms a straight line with a positive slope. We need to determine the equation of this line. To do so, we can use the slope-intercept form, \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept.
   
   - **Find the slope (m):** The slope is given by:
     \[
     m = \frac{y_2 - y_1}{x_2 - x_1}
     \]
     Picking two points from the line: \( (-4, -3) \) and \( (4, 5) \):
     \[
     m = \frac{5 - (-3)}{4 - (-4)} = \frac{8}{8} = 1
     \]
     So, the slope \( m = 1 \).

   - **Find the y-intercept (b):** Now that we know the slope, we can substitute one of the points, say \( (-4, -3) \), into the equation \( y = mx + b \) to find the intercept \( b \):
     \[
     -3 = 1(-4) + b \quad \Rightarrow \quad b = 1
     \]
     Therefore, the equation of the line is:
     \[
     f(x) = x + 1 \quad \text{for} \quad -4 \leq x \leq 4
     \]

### 3. **For \( x > 4 \):**
   For \( x > 4 \), the graph shows a constant value of \( 5 \). Therefore, the function for this interval is:
   \[
   f(x) = 5 \quad \text{for} \quad x > 4
   \]

### Final Piecewise Function:
\[
f(x) =
\begin{cases}
-3 & \text{if } x < -4 \\
x + 1 & \text{if } -4 \leq x \leq 4 \\
5 & \text{if } x > 4
\end{cases}
\]

Would you like further clarification on any of the steps, or do you have any additional questions?

### Related Questions:
1. How do you determine the slope of a line between two points?
2. What is the importance of the y-intercept in the equation of a line?
3. How do piecewise functions apply to real-world situations?
4. Can a piecewise function have more than three parts?
5. How can discontinuities affect the graph of a piecewise function?

### Tip:
Always pay attention to closed and open intervals in piecewise functions. These dictate where the function includes or excludes specific points.

Find the piecewise function based on the graph provided.

Learn how to define a piecewise function from a given graph. This example explores constant values and linear equations across different intervals of x, including calculating the slope and y-intercept for a straight line. Suitable for Grades 9-11 students.

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