To determine which function best models the data you've provided in the table (with the pairs of \(x\) and \(y\)), we need to analyze the growth rate of the \(y\) values as \(x\) increases. Here’s a quick rundown of how each type of function behaves:
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Linear: The relationship between \(x\) and \(y\) is constant; the graph is a straight line.
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Quadratic: The relationship is parabolic; as \(x\) increases, \(y\) increases (or decreases) at an accelerating rate.
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Cubic: The relationship displays more complex growth than quadratic; the graph resembles a curve with possible inflection points.
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Exponential: As \(x\) increases, \(y\) increases at an accelerating rate, typically faster than polynomial functions.
Given the values you've indicated:
- When \(x = 1\), \(y = 10\)
- When \(x = 2\), \(y = 100\)
- When \(x = 3\), \(y = 1000\)
If we calculate the ratios of consecutive \(y\) values:
- From \(y=10\) to \(y=100\) (increasing by a factor of 10)
- From \(y=100\) to \(y=1000\) (again, increasing by a factor of 10)
This indicates that \(y\) is growing exponentially as \(x\) increases. Therefore, the best model among cubic, exponential, quadratic, and linear for the given data is exponential.