To determine the best type of function modeling the data in the table, we can analyze the values of \( x \) and \( y \):
\[ \begin{array}{|c|c|} \hline x & y \ \hline 2 & 8.5 \ 4 & 46 \ 6 & 107.5 \ 8 & 193 \ \hline \end{array} \]
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Linear: If this were a linear function, the change in \( y \) would be constant for equal changes in \( x \).
- The differences in \( y \) values:
- From \( 8.5 \) to \( 46 \): \( 46 - 8.5 = 37.5 \)
- From \( 46 \) to \( 107.5 \): \( 107.5 - 46 = 61.5 \)
- From \( 107.5 \) to \( 193 \): \( 193 - 107.5 = 85.5 \)
The differences are not constant, so this is not a linear function.
- The differences in \( y \) values:
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Quadratic: Quadratics have a constant second difference.
- First differences: \( 37.5, 61.5, 85.5 \)
- Second differences:
- From \( 37.5 \) to \( 61.5 \): \( 61.5 - 37.5 = 24 \)
- From \( 61.5 \) to \( 85.5 \): \( 85.5 - 61.5 = 24 \)
The second differences are constant (\( 24 \)), suggesting a quadratic relationship.
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Exponential: For an exponential function, the ratios of consecutive \( y \) values should be constant.
- Ratios:
- \( \frac{46}{8.5} \approx 5.41 \)
- \( \frac{107.5}{46} \approx 2.34 \)
- \( \frac{193}{107.5} \approx 1.80 \)
The ratios are not constant, therefore it is not exponential.
- Ratios:
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Cubic: Cubic functions would typically yield constant third differences, but given we have already found constant second differences for a quadratic, we can conclude it’s not cubic either.
Given this analysis, the best model for the data in the table is:
Quadratic