To determine the intervals where the average rate of change of the function is negative, we need to analyze the provided points.
The points given are:
- (1, 0)
- (2, 1)
- (3, -1)
- (5, -1)
The average rate of change of a function between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula:
\[ \text{Average Rate of Change} = \frac{y_2 - y_1}{x_2 - x_1} \]
We can calculate the average rate of change over the intervals defined by the provided points:
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Between (1, 0) and (2, 1): \[ \text{Average Rate of Change} = \frac{1 - 0}{2 - 1} = \frac{1}{1} = 1 \quad \text{(Positive)} \]
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Between (2, 1) and (3, -1): \[ \text{Average Rate of Change} = \frac{-1 - 1}{3 - 2} = \frac{-2}{1} = -2 \quad \text{(Negative)} \]
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Between (3, -1) and (5, -1): \[ \text{Average Rate of Change} = \frac{-1 - (-1)}{5 - 3} = \frac{0}{2} = 0 \quad \text{(Neutral)} \]
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Between (1, 0) and (3, -1): \[ \text{Average Rate of Change} = \frac{-1 - 0}{3 - 1} = \frac{-1}{2} = -0.5 \quad \text{(Negative)} \]
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Between (2, 1) and (5, -1): \[ \text{Average Rate of Change} = \frac{-1 - 1}{5 - 2} = \frac{-2}{3} \quad \text{(Negative)} \]
From this analysis, we find that the average rate of change is negative over the following intervals:
- From \(x = 2\) to \(x = 3\)
- From \(x = 1\) to \(x = 3\)
- From \(x = 2\) to \(x = 5\)
Therefore, we can summarize:
The average rate of change is negative over the interval \( (2, 3) \) and over the interval \( (1, 3) \), and also \( (2, 5) \). If you seek only explicit intervals as defined between two points, the most distinct interval is \( (2, 3) \).
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