To solve the problem, let's analyze the graph and check the intervals for the average rate of change:

Step 1: Interpret the intervals and their slopes

• The average rate of change is determined by the slope of the line segments between two points. A negative slope indicates that the function is decreasing over the interval.

Step 2: Analyze each interval on the graph

1. Interval [0, 1]: The function decreases from $f(0) = -1$ to $f(1) = -2$. The slope is negative.
2. Interval [1, 3]: The function increases from $f(1) = -2$ to $f(3) = 0$. The slope is positive.
3. Interval [3, 4]: The function decreases from $f(3) = 0$ to $f(4) = -1$. The slope is negative.
4. Interval [4, 6]: The function increases from $f(4) = -1$ to $f(6) = 3$. The slope is positive.
5. Interval [6, 7]: The function decreases from $f(6) = 3$ to $f(7) = 1$. The slope is negative.

Step 3: Match the intervals with the options

From the analysis:

• The average rate of change is negative on [0, 1], [3, 4], and [6, 7].
• The correct statement is:

  "The average rate of change is negative on the interval [0, 1] and on the interval [3, 4]."

Final Answer:

The last option is correct.

Would you like further details or a breakdown of the calculations?

Here are 5 related questions to expand your understanding:

1. How is the average rate of change calculated for a specific interval?
2. What does a positive slope indicate about the behavior of a function?
3. How do you identify intervals of increasing or decreasing functions from a graph?
4. Can a function have multiple intervals with the same type of slope?
5. How would the analysis change if this were a continuous curve instead of piecewise linear?

Tip: To find the average rate of change on any interval, use the formula:
$\text{Slope} = \frac{f(b) - f(a)}{b - a}$
where $a$ and $b$ are the endpoints of the interval.