https://file.aiquickdraw.com/mathbot/file/17260740539935uprp32i.jpg

Let's analyze the graph and answer the questions step by step:

1. **Domain**: The graph extends infinitely in both the positive and negative \(x\)-directions, so the domain is all real numbers.
   \[
   \text{Domain}: (-\infty, \infty)
   \]

2. **Range**: The lowest point on the graph is at \(y = 0\), and the graph extends upward infinitely. Hence, the range is all values of \(y \geq 0\).
   \[
   \text{Range}: [0, \infty)
   \]

3. **x-intercepts**: The graph intersects the \(x\)-axis at \(x = -4\) and \(x = 0\).
   \[
   \text{x-intercepts}: -4, 0
   \]

4. **y-intercepts**: The graph intersects the \(y\)-axis at \(y = 0\).
   \[
   \text{y-intercepts}: 0
   \]

5. **Interval positive**: The graph is positive when \(y > 0\), which happens when \(x \in (-4, -\infty) \cup (0, \infty)\).
   \[
   \text{Interval positive}: (-\infty, -4) \cup (0, \infty)
   \]

6. **Interval negative**: There are no intervals where the graph is negative because the graph never goes below the \(x\)-axis.
   \[
   \text{Interval negative}: \text{None}
   \]

7. **Interval increasing**: The graph increases when the slope is positive, which happens for \(x > 0\).
   \[
   \text{Interval increasing}: (0, \infty)
   \]

8. **Interval decreasing**: The graph decreases when the slope is negative, which happens for \(x < 0\).
   \[
   \text{Interval decreasing}: (-\infty, 0)
   \]

9. **Rate of change over the interval \((-2, 0)\)**: The rate of change is the slope of the line segment between \(x = -2\) and \(x = 0\). Using the points \((-2, 4)\) and \((0, 0)\), the slope is:
   \[
   \text{Slope} = \frac{0 - 4}{0 - (-2)} = \frac{-4}{2} = -2
   \]
   \[
   \text{Rate of change}: -2
   \]

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Do you want more details or have any other questions? Here are some related questions:

1. How do you find the rate of change on different intervals of a graph?
2. What does the domain and range tell us about the behavior of a function?
3. How can you tell where a function is increasing or decreasing from a graph?
4. Why is there no negative interval in this particular graph?
5. How do the x- and y-intercepts help in understanding the function's graph?

**Tip**: When analyzing a graph, always check for symmetry, intercepts, and behavior at the edges to understand its full nature.

Determine the domain, range, intercepts, intervals, and rate of change from the given graph.

Learn how to determine the domain, range, intercepts, and intervals of a function's graph, and calculate the rate of change. This problem covers key graph analysis concepts, suitable for students in Grades 9-12.

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