To compare the estimated average rates of change for the quadratic function and the exponential function over the interval \([2,6]\), we need to calculate the average rate of change for both functions over this interval.
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Quadratic Function: Let's denote the quadratic function as \(f(x)\). The average rate of change of \(f(x)\) over the interval \([2, 6]\) is calculated using the formula:
\[ \text{Average Rate of Change} = \frac{f(6) - f(2)}{6 - 2} \]
You would need the specific quadratic function to get the values of \(f(6)\) and \(f(2)\).
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Exponential Function: For the exponential function \(y = 34(2^x)\), we calculate the average rate of change over the interval \([2, 6]\):
\[ \text{Average Rate of Change} = \frac{f(6) - f(2)}{6 - 2} \]
First, we find \(f(2)\) and \(f(6)\):
- \(f(2) = 34(2^2) = 34(4) = 136\)
- \(f(6) = 34(2^6) = 34(64) = 2176\)
Now we calculate the average rate of change for \(y = 34(2^x)\) over \([2, 6]\):
\[ \text{Average Rate of Change} = \frac{2176 - 136}{6 - 2} = \frac{2040}{4} = 510 \]
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Comparison: Assuming we've calculated the average rate of change for the quadratic function \(f(x)\) and found it to be some value, let’s call it \(A\).
Thus, the comparison becomes:
- Exponential Rate of Change = 510
- Quadratic Rate of Change = \(A\)
If the quadratic function is provided, we would calculate its average rate of change, say it turned out to be \(A = 502\), then:
\[ 510 - 502 = 8 \]
Thus the difference in average rates of change would be \(8\), and we would conclude that the exponential function has an average rate of change that is greater by that amount.
To finalize which response is correct, it's vital to have the estimated average rate of change of the quadratic function. However, based on the options you provided, if we end up with a numerical difference of, say, \(7.25\), and we find the quadratic rate's change \(A\) is less, we would choose the appropriate response accordingly.
Please provide the specific quadratic function if you'd like to see the numerical comparison finalized!
1 answer