To compare the average rates of change for both the quadratic and exponential functions over the interval \([2, 6]\), we first need to determine the average rate of change for each function over that interval.
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For the quadratic function: Let's denote it as \( f(x) \). The average rate of change over the interval \([a, b]\) is calculated using the formula: \[ \text{Average rate of change} = \frac{f(b) - f(a)}{b - a} \] For this interval, we need \(f(6)\) and \(f(2)\).
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For the exponential function \( y = 34(2^x) \): We will similarly calculate its average rate of change: \[ \text{Average rate of change} = \frac{y(6) - y(2)}{6 - 2} = \frac{34(2^6) - 34(2^2)}{4} \]
Calculating \(y(6)\): \[ y(6) = 34(2^6) = 34(64) = 2176 \] Calculating \(y(2)\): \[ y(2) = 34(2^2) = 34(4) = 136 \] Now calculating the average rate of change for the exponential function: \[ \text{Average rate of change} = \frac{2176 - 136}{4} = \frac{2040}{4} = 510 \]
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Now we assume an arbitrary quadratic function to calculate its average rate of change. Let's assume \( f(x) = ax^2 + bx + c \) (you'll need to provide the specific function if you want an exact calculation). We'll need its values at \( x=6 \) and \( x=2 \) to find: \[ f(6) \quad \text{and} \quad f(2) \] Then compute: \[ \text{Average rate of change for the quadratic function} = \frac{f(6) - f(2)}{6 - 2} \]
Finally, when you have both average rates of change, you can compare them by finding the difference.
Conclusion
Once you calculate both average rates of change, you can choose the correct response based on their comparison:
- If the average rate of change of the exponential function is greater, choose those options.
- If the average rate of change of the quadratic function is greater, choose the relevant options.
You can use the above steps to find the exact values you need. If you provide the quadratic function or its values, I can help you further!