To find the correct response, we need to determine the average rate of change of both the quadratic function graphed in the image over the interval \([2, 6]\) and the exponential function \(y = 34(2)^x\) over the same interval.
Step 1: Average Rate of Change for the Quadratic Function
Let’s denote the quadratic function as \(f(x)\). The average rate of change of a function \(f\) over an interval \([a, b]\) is given by:
\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
Assuming from the information provided in the question that at \(x = 2\) and \(x = 6\), we can denote the function values as \(f(2)\) and \(f(6)\) (specific values should derive from the image, which we cannot see directly).
Calculate:
- \(f(2)\) (the y-value on the U-shaped curve at \(x = 2\))
- \(f(6)\) (the y-value on the U-shaped curve at \(x = 6\))
Let’s estimate those values from what you described:
- Suppose \(f(2) = y_2\)
- Suppose \(f(6) = y_6\)
Then: \[ \text{Average Rate of Change for } f(x) = \frac{y_6 - y_2}{6 - 2} = \frac{y_6 - y_2}{4} \]
Step 2: Average Rate of Change for the Exponential Function
For the exponential function \(y = 34(2^x)\):
\[ f(2) = 34(2^2) = 34 \cdot 4 = 136 \] \[ f(6) = 34(2^6) = 34 \cdot 64 = 2176 \]
So the average rate of change over the interval \([2, 6]\) is:
\[ \text{Average Rate of Change} = \frac{2176 - 136}{6 - 2} = \frac{2040}{4} = 510 \]
Step 3: Compare the Rates
Now compare this to the average rate of change of the quadratic function:
- If we calculated, say for example, \(y_6 = 8\) and \(y_2 = 0\) (these need to be estimated correctly from the image, leading to a hypothetical case of): \[ \text{Average Rate of Change for the quadratic} = \frac{8 - 0}{4} = 2 \]
Conclusion
At this point, it looks like the average rate of change for the quadratic might be much lower than the exponential. Let's say:
- Average rate for the quadratic function = \(R_q\)
- Average rate for the exponential function = \(R_e = 510\)
To summarize this:
- Calculate \(R_e - R_q\).
The answers you are considering have the estimated values differing by numeric factors. However, without explicit values from the image, concluding outright is impossible.
Given the results you presented, it sounds like either the quadratic function's average change is either close to or significantly less than the exponential function's average. Depending on hypothetical values you've collected or the direct observation from the image, your choice should reflect that comparison.
If you estimate \(R_q\) was, for instance, \(2\), then:
\( 510 - 2 = 508\)
The statements you presented more focused on whether the quadratic or the exponential rate was higher.
To select the most coherent statement based on that:
- If \(R_e\) is higher, it's likely that: "The estimated average rate of change of the exponential function is higher than that of the quadratic function."
Please replace the estimates from the quadratic function with the actual values derived from the image for a definitive answer!
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