To determine over which interval the function has a negative average rate of change, we need to analyze the graph by focusing on the intervals between the points where the function decreases. A negative average rate of change means that the function is decreasing over that interval.

Looking at the graph:

• From $x = 0$ to $x = 1$, the function increases.
• From $x = 1$ to $x = 2$, the function is constant (horizontal line).
• From $x = 2$ to $x = 3$, the function decreases from $f(x) = 4$ to $f(x) = 0$, indicating a negative average rate of change.
• From $x = 3$ to $x = 4$, the function increases.
• From $x = 4$ to $x = 5$, the function is constant again.
• From $x = 5$ to $x = 6$, the function decreases from $f(x) = 4$ to $f(x) = 0$, indicating another negative average rate of change.

Intervals where the average rate of change is negative:

• $[2, 3]$
• $[5, 6]$

These are the intervals where the function has a negative average rate of change.

Would you like more details or have any questions?

Here are some related questions for further exploration:

1. How do you calculate the average rate of change over an interval?
2. Why is the rate of change zero in intervals where the function is constant?
3. Can the function have a positive average rate of change over other intervals? If so, where?
4. How would you define the rate of change for a non-linear function?
5. What is the significance of a negative rate of change in real-world applications?

Tip: The average rate of change between two points $(x_1, f(x_1))$ and $(x_2, f(x_2))$ is calculated as $\frac{f(x_2) - f(x_1)}{x_2 - x_1}$. This formula helps identify increasing or decreasing behavior.