- Examine the values of \(y\) for \(x = -2, -1, 0, 1\) from the provided table: \((-2, -12), (-1, 0), (0, 6), (1, 7.5)\).
- Note that the values of \(y\) strictly increase as \(x\) increases across this interval.
- Conclude that the function is increasing on the interval \([-2, 1]\).
- The correct description is: \(\boxed{\text{The function is increasing on the interval } [-2, 1]}\).
Explanation
- Analyze the Data Points in the Interval [-2, 1]
To determine the behavior of the cubic polynomial function \(f(x)\) on the interval \([-2, 1]\), we analyze the values of \(y\) provided in the table for the corresponding values of \(x\). The table gives us the following points: \((-2, -12), (-1, 0), (0, 6), (1, 7.5), (2, 6), (3, 3)\). Specifically, for the interval \([-2, 1]\), we look at the values at \(x = -2, -1, 0,\) and \(1\).
- Determine Function Behavior on the Interval [-2, 1]
We observe the changes in \(y\) as \(x\) increases from \(-2\) to \(1\):
- At \(x = -2, y = -12\)
- At \(x = -1, y = 0\)
- At \(x = 0, y = 6\)
- At \(x = 1, y = 7.5\)
Comparing consecutive points, we see that:
- From \(x = -2\) to \(x = -1\), the value of \(y\) increases from \(-12\) to \(0\).
- From \(x = -1\) to \(x = 0\), the value of \(y\) increases from \(0\) to \(6\).
- From \(x = 0\) to \(x = 1\), the value of \(y\) increases from \(6\) to \(7.5\).
Since the values of \(y\) are consistently increasing as \(x\) increases across the entire interval \([-2, 1]\), the function is increasing on this interval.
- Final Conclusion
Based on the analysis, the function shows an increasing trend throughout the interval \([-2, 1]\). We have verified that for every increase in \(x\) within this range, there is a corresponding increase in the function value \(y\). Therefore, the sentence that best describes the behavior of the function on the interval \([-2, 1]\) is that the function is increasing. The final answer is "The function is increasing on the interval \([-2, 1]\)."
Examples
Understanding whether a function is increasing or decreasing is crucial in fields like economics, where you might model profit relative to production levels. For example, if a company's profit function \(P(x)\) is increasing on an interval of production levels \(x \in [a, b]\), it means that increasing production within this range leads to higher total profit. By analyzing points of data, managers can determine optimal ranges for production to maximize their growth.