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The graph you uploaded shows a piecewise linear function, with points connected by straight segments. Here’s an analysis of the key points and structure of the graph:

1. **Key Points on the Graph:**
   - The graph starts with an open circle at \( (-6, 2) \), indicating that the value at this point is not included in the function.
   - The line moves downward to the point \( (-2, -3) \).
   - From \( (-2, -3) \), it moves upward to the point \( (1, 2) \).
   - Then, it continues upward to \( (3, 6) \), where there is a closed circle, meaning this point is included in the function.

2. **Slopes of the Line Segments:**
   - Between \( (-6, 2) \) and \( (-2, -3) \), the slope can be calculated as:
     \[
     m = \frac{-3 - 2}{-2 - (-6)} = \frac{-5}{4} = -\frac{5}{4}
     \]
   - Between \( (-2, -3) \) and \( (1, 2) \), the slope is:
     \[
     m = \frac{2 - (-3)}{1 - (-2)} = \frac{5}{3}
     \]
   - Between \( (1, 2) \) and \( (3, 6) \), the slope is:
     \[
     m = \frac{6 - 2}{3 - 1} = \frac{4}{2} = 2
     \]

3. **Piecewise Function Representation:**
   The function can be written as a piecewise function with three distinct intervals:
   \[
   f(x) = \begin{cases} 
   -\frac{5}{4}(x + 6) + 2 & \text{for } -6 < x \leq -2 \\
   \frac{5}{3}(x + 2) - 3 & \text{for } -2 \leq x \leq 1 \\
   2(x - 1) + 2 & \text{for } 1 \leq x \leq 3
   \end{cases}
   \]

Would you like more detailed explanations or clarifications?

Here are 5 related questions:
1. What does the open circle at \( (-6, 2) \) signify in the context of the function?
2. How would the graph look if the circle at \( (-6, 2) \) were closed?
3. Can you express this function in interval notation for its domain and range?
4. How would you find the x-intercepts and y-intercepts of this piecewise function?
5. How does the slope between points affect the steepness of the line segments?

**Tip:** In piecewise functions, open and closed circles are important in determining whether a value at that point is included or excluded from the function!

What is the piecewise function represented by the graph?

Explore the structure of a piecewise linear function with points connected by straight segments. This analysis covers key graph points, slope calculations, and piecewise function representation. Suitable for students in Grades 9-11.

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