To determine the intervals where the average rate of change of a function is negative, you will need to look at the behavior of the function over different intervals and assess how the function values (y-values) are changing as you move from the left endpoint to the right endpoint of each interval.
The average rate of change of a function \( f \) over the interval \([a, b]\) can be calculated using the formula:
\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
If \( f(b) < f(a) \), then the average rate of change is negative, indicating that the function is decreasing over that interval.
To identify specific intervals, follow these steps:
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Identify intervals based on the endpoints shown on the graph. For example, if your graph shows intervals like [a,b], [b,c], [c,d], etc., record these intervals.
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Evaluate the function values at the endpoints:
- For each interval, determine \( f(a) \) and \( f(b) \).
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Calculate the average rate of change for each interval:
- Apply the formula mentioned above to determine if the average rate of change is negative.
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Determine the intervals with a negative average rate of change by checking the condition \( f(b) < f(a) \).
Once you have completed this analysis, you can fill in the statement accordingly. For example:
"The average rate of change is negative over the interval \([x_1, x_2]\), where \( f(x_1) > f(x_2) \)."
Make sure to specify the actual intervals based on your observations from the graph.