To determine where the function has a negative average rate of change over specified intervals, you need to look at the endpoints of each interval on the graph. The average rate of change of a function \( f \) over an interval \([a, b]\) is given by the formula:
\[ \frac{f(b) - f(a)}{b - a} \]
This average rate of change is negative if \( f(b) < f(a) \), which means that the function is decreasing over that interval.
You can identify the intervals by examining the graph for segments where the function is clearly going downward as you move from left to right.
Here’s what you should do:
- Identify the endpoints of the intervals from the graph.
- Check the \( y \)-values of the function at those endpoints.
- Use the rule mentioned above to determine if the average rate of change is negative for each interval:
- If \( f(b) < f(a) \), then \(\frac{f(b) - f(a)}{b - a} < 0\).
Fill in the interval where the average rate of change is negative accordingly.
For example, if you observe from the graph that between two specific points, say \(x = 1\) and \(x = 3\), the function decreases from \(f(1) = 5\) to \(f(3) = 2\), then you would conclude that:
The average rate of change is negative at the interval
\[ [1, 3] \]
Replace the numbers accordingly based on your graph observations.
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