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Determine the intervals on which the function is​ increasing, decreasing, and constant.

To determine the intervals where the function is increasing, decreasing, or constant, let's analyze the given graph:

### Intervals of Interest:
1. **Increasing**: A function is increasing where the graph moves upward as \( x \) increases.
2. **Decreasing**: A function is decreasing where the graph moves downward as \( x \) increases.
3. **Constant**: A function is constant where the graph remains horizontal (flat) and does not move up or down.

### Analysis of the Graph:

- From \( x = -8 \) to \( x = -6 \), the graph is moving upward, so the function is **increasing** in this interval.
- From \( x = -6 \) to \( x = -4 \), the graph moves downward, so the function is **decreasing** in this interval.
- From \( x = -4 \) to \( x = -2 \), the graph is flat, so the function is **constant** in this interval.
- From \( x = -2 \) to \( x = 0 \), the graph moves upward again, so the function is **increasing** in this interval.
- From \( x = 0 \) to \( x = 2 \), the graph is constant (flat) again, so the function is **constant** in this interval.
- From \( x = 2 \) onward (as \( x \to \infty \)), the graph remains constant at \( y = 2 \), so the function is **constant** from \( x = 2 \) to infinity.

### Summary:

- **Increasing**: \( (-8, -6) \) and \( (-2, 0) \)
- **Decreasing**: \( (-6, -4) \)
- **Constant**: \( (-4, -2) \), \( (0, 2) \), and \( (2, \infty) \)

Let me know if you need more details or have further questions!

Here are 5 related questions to extend this topic:

1. What is the significance of local maxima and minima in the context of this graph?
2. How would you find the derivative of this piecewise function?
3. Can you identify any points of inflection from the graph?
4. How does the horizontal behavior at \( x = 2 \) relate to limits as \( x \to \infty \)?
5. How do intervals of increasing and decreasing correspond to the derivative's sign?

**Tip**: When analyzing graphs, always look at the slope (whether it's positive, negative, or zero) to determine where the function is increasing, decreasing, or constant.

Determine the intervals on which the function is increasing, decreasing, and constant.

Analyze the given function's graph to determine the intervals where it is increasing, decreasing, or constant. This problem focuses on understanding the behavior of functions through graphical analysis, suitable for Grades 10-12.

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