To compare the estimated average rate of change of the exponential function \( y = 9(13)^x \) and the quadratic function \( y = 9x^2 + 13x \) over the interval \([0.1, 0.6]\), we need to calculate the average rate of change for each function over that interval.
The average rate of change of a function \( f \) over the interval \([a, b]\) is given by the formula:
\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
For the exponential function \(y = 9(13)^x\):
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Evaluate at \(x = 0.1\): \[ y(0.1) = 9(13)^{0.1} \approx 9(1.34986) \approx 12.14874 \]
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Evaluate at \(x = 0.6\): \[ y(0.6) = 9(13)^{0.6} \approx 9(3.422424) \approx 30.801816 \]
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Calculate average rate of change: \[ \text{Average Rate of Change} = \frac{30.801816 - 12.14874}{0.6 - 0.1} = \frac{18.653076}{0.5} \approx 37.306152 \]
The average rate of change for the exponential function over \([0.1, 0.6]\) is approximately \(37.31\) (positive).
For the quadratic function \(y = 9x^2 + 13x\):
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Evaluate at \(x = 0.1\): \[ y(0.1) = 9(0.1)^2 + 13(0.1) = 9(0.01) + 1.3 = 0.09 + 1.3 = 1.39 \]
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Evaluate at \(x = 0.6\): \[ y(0.6) = 9(0.6)^2 + 13(0.6) = 9(0.36) + 7.8 = 3.24 + 7.8 = 11.04 \]
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Calculate average rate of change: \[ \text{Average Rate of Change} = \frac{11.04 - 1.39}{0.6 - 0.1} = \frac{9.65}{0.5} = 19.3 \]
The average rate of change for the quadratic function over \([0.1, 0.6]\) is \(19.3\) (also positive).
Summary of Results:
- Average rate of change of the exponential function over \([0.1, 0.6]\): approximately \(37.31\) (positive).
- Average rate of change of the quadratic function over \([0.1, 0.6]\): \(19.3\) (positive).
Conclusion:
Both functions have a positive estimated average rate of change over the interval \([0.1, 0.6]\). Therefore, the correct response is:
neither function.