To determine which type of function best models the data, we can analyze the rate of change in the values of \(y\) as \(x\) increases.
Given the pairs from the table:
- When \(x=2\), \(y=8.5\)
- When \(x=4\), \(y=46\)
- When \(x=6\), \(y=107.5\)
- When \(x=8\), \(y=193\)
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Linear Function: A linear function would show a constant rate of change. However, we can calculate the differences:
- From \(y=8.5\) to \(y=46\): \(46 - 8.5 = 37.5\)
- From \(y=46\) to \(y=107.5\): \(107.5 - 46 = 61.5\)
- From \(y=107.5\) to \(y=193\): \(193 - 107.5 = 85.5\)
The differences are not constant, ruling out a linear function.
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Quadratic Function: A quadratic function would have a second difference that is constant. Here are the first differences again:
- First differences: \(37.5\), \(61.5\), \(85.5\)
- Second differences:
- \(61.5 - 37.5 = 24\)
- \(85.5 - 61.5 = 24\)
The second difference is consistent, suggesting that a quadratic function might be appropriate.
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Cubic Function and Exponential Function:
- A cubic function would generally show a changing second difference. The second difference is consistent, so a cubic model is not suggested here.
- An exponential function tends to grow at an increasing rate, but the values do not appear to follow an exponential growth pattern since the differences grow but in a way that aligns more with a quadratic trend.
Based on this analysis, quadratic is the most suitable type of function to model this data. Therefore, the answer is:
quadratic