To compare the estimated average rates of change for the quadratic and exponential functions over the interval \([2, 6]\), we need to calculate the average rate of change for each function.
- Quadratic Function: The average rate of change of a function over an interval \([a, b]\) is calculated as: \[ \frac{f(b) - f(a)}{b - a} \] From the information given, we would need the function values \(f(2)\) and \(f(6)\).
However, since we do not have the specific values (due to the lack of the actual image with points on the curve), let's denote:
- \(f(2) = y_1\) (the function value at \(x = 2\))
- \(f(6) = y_2\) (the function value at \(x = 6\))
Plugging these into the formula gives: \[ \text{Average Rate of Change of the Quadratic} = \frac{y_2 - y_1}{6 - 2} = \frac{y_2 - y_1}{4} \]
- Exponential Function \(y = 34(2^x)\): Calculating this for \(x = 2\) and \(x = 6\):
- \(g(2) = 34(2^2) = 34 \times 4 = 136\)
- \(g(6) = 34(2^6) = 34 \times 64 = 2176\)
Now calculate the average rate of change: \[ \text{Average Rate of Change of the Exponential} = \frac{g(6) - g(2)}{6 - 2} = \frac{2176 - 136}{4} = \frac{2040}{4} = 510 \]
Finally, we compare the two average rates of change:
- If we hypothetically calculate the quadratic function's average rate of change and find it yields a certain value (let's say it found to be \(x\)), we can then formulate the comparison accordingly based on the information provided.
However, in this case, without a specific numerical output for the quadratic function (from the image), we can't definitively state the numerical relationship between the two average rates of change—you would need specific output data from the curve in the image.
Without that, the best way forward is to look at the difference estimations:
- If the quadratic's average rate was 502, then \(510 - 502 = 8\) (the exponential function would be greater).
- If the quadratic's average rate was 500, then the difference is 10.
Reference back to the details of your responses:
The expected outputs can derive from such estimations, finally arriving at a conclusion.
Since options state various differences, choose based on what figure might be more plausible. Given it states "exponential function is greater," indicating a significant difference likely closer to 7 or 8 might apply.
Select the option that closely represents the average you've found for the quadratic vs exponential, noting that it leans likely toward "exponential greater" in every option format.
Please choose based on calculated values or correctly input from visual data you reference if you're given further by that U-shaped curve.