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: $\text{Slope} = \frac{0 - 4}{0 - (-2)} = \frac{-4}{2} = -2$ $\text{Rate of change}: -2$

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.