https://file.aiquickdraw.com/mathbot/file/1726701279826bdnh8akl.jpg

Let's analyze the graph and answer the questions provided.

1. **Domain**: 
   The domain refers to the set of all possible x-values (horizontal axis) for which the graph is defined. From the graph, the x-values range from -2 to 4.
   \[
   \text{Domain}: [-2, 4]
   \]

2. **Range**: 
   The range refers to the set of all possible y-values (vertical axis) that the graph attains. From the graph, the lowest point is at y = 0 and the highest is y = 4.
   \[
   \text{Range}: [0, 4]
   \]

3. **Interval where the graph is increasing**: 
   The graph is increasing when it moves upward from left to right. This happens between \(x = -2\) and \(x = 0\).
   \[
   \text{Increasing interval}: (-2, 0)
   \]

4. **Interval where the graph is decreasing**: 
   The graph is decreasing when it moves downward from left to right. This happens between \(x = 0\) and \(x = 2\).
   \[
   \text{Decreasing interval}: (0, 2)
   \]

5. **x-intercept(s)**: 
   The x-intercept(s) are the points where the graph crosses the x-axis. The graph crosses the x-axis at \(x = 0\) and \(x = 2\).
   \[
   \text{x-intercept(s)}: x = 0, x = 2
   \]

6. **y-intercept(s)**: 
   The y-intercept is where the graph crosses the y-axis. This occurs at \(y = 0\).
   \[
   \text{y-intercept(s)}: y = 0
   \]

7. **Average rate of change over the interval [0, 2]**: 
   The average rate of change is calculated by finding the slope between two points. In this case, the interval is from \(x = 0\) to \(x = 2\). At \(x = 0\), \(y = 4\), and at \(x = 2\), \(y = 0\). The slope is:
   \[
   \text{Slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 4}{2 - 0} = \frac{-4}{2} = -2
   \]
   \[
   \text{Average rate of change}: -2
   \]

Would you like to explore any part of this in more detail? Here are some related questions to deepen your understanding:

1. What happens to the graph outside the given domain?
2. How do you interpret the x-intercepts in the context of real-world applications?
3. Can the graph have any maximum or minimum points? Why or why not?
4. How would the graph change if the range were extended?
5. How can the average rate of change help determine whether a function is linear?

**Tip:** The slope or average rate of change helps in identifying whether a function is increasing or decreasing over a given interval.

Given a graph, determine its domain, range, intervals of increase and decrease, intercepts, and the average rate of change over a specific interval.

This problem involves analyzing a piecewise function to determine its domain, range, increasing and decreasing intervals, x- and y-intercepts, and calculating the average rate of change over a specified interval. Ideal for students in grades 9-12 exploring graph interpretation.

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