To compare the estimated average rate of change of the quadratic function and the exponential function over the interval \([2, 6]\), we need to calculate the average rate of change for both functions in this range.
-
Quadratic Function: Let's denote the quadratic function as \(f(x)\). The average rate of change of a function over an interval \([a, b]\) can be calculated using the formula: \[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
Assuming we don't have the specific quadratic function, we will denote \(f(2)\) and \(f(6)\) as its values at those points.
-
Exponential Function: The given exponential function is: \[ y = \frac{3}{4} (2)^x \] We can calculate its values at \(x = 2\) and \(x = 6\): \[ y(2) = \frac{3}{4} (2)^2 = \frac{3}{4} \cdot 4 = 3 \] \[ y(6) = \frac{3}{4} (2)^6 = \frac{3}{4} \cdot 64 = 48 \] Now, let's find the average rate of change for this function over the interval \([2, 6]\): \[ \text{Average Rate of Change for the exponential function} = \frac{y(6) - y(2)}{6 - 2} = \frac{48 - 3}{4} = \frac{45}{4} = 11.25 \]
-
Comparing the Average Rates of Change: Now we need to calculate the average rate of change for the quadratic function \(f(x)\) over \([2, 6]\): \[ \text{Average Rate of Change for the quadratic function} = \frac{f(6) - f(2)}{6 - 2} \] This term is currently unspecified without knowing the specific quadratic function.
-
Conclusion: You need to compare the computed average rate of change of the quadratic function with \(11.25\) (the computed rate for the exponential function).
If you have values of \(f(2)\) and \(f(6)\) for the quadratic, calculate: \[ \text{Estimated Average Rate of Change} = \frac{f(6) - f(2)}{4} \] And compare it with \(11.25\).
Without specific values for the quadratic function, it is not possible to definitively state which of the provided multiple-choice responses is correct. Would you be able to provide the quadratic function so we can complete the calculation?