Question
Graph numbers/rate of change
6,14
2,-2
Compare the estimate average rate of change of the graphed quadratic function over the interval [2,6] to the average rate of change of the exponential function y=34(2)xy=34(2)x over the same interval. (1 point) Responses The estimated average rate of change of the quadratic function is 0.16 greater than the estimated average rate of change of the exponential function of the interval [2,6]. The estimated average rate of change of the quadratic function is 0.16 greater than the estimated average rate of change of the exponential function of the interval [2,6]. The estimated average rate of change of the quadratic function is 7.25 greater than the estimated average rate of change of the exponential function of the interval [2,6]. The estimated average rate of change of the quadratic function is 7.25 greater than the estimated average rate of change of the exponential function of the interval [2,6]. The estimated average rate of change of the exponential function is 7.25 greater than the estimated average rate of change of the Quadratic function of the interval [2,6]. The estimated average rate of change of the exponential function is 7.25 greater than the estimated average rate of change of the Quadratic function of the interval [2,6]. The estimated average rate of change of the exponential function is 0.16 greater than the estimated average rate of change of the Quadratic function of the interval [2,6].
6,14
2,-2
Compare the estimate average rate of change of the graphed quadratic function over the interval [2,6] to the average rate of change of the exponential function y=34(2)xy=34(2)x over the same interval. (1 point) Responses The estimated average rate of change of the quadratic function is 0.16 greater than the estimated average rate of change of the exponential function of the interval [2,6]. The estimated average rate of change of the quadratic function is 0.16 greater than the estimated average rate of change of the exponential function of the interval [2,6]. The estimated average rate of change of the quadratic function is 7.25 greater than the estimated average rate of change of the exponential function of the interval [2,6]. The estimated average rate of change of the quadratic function is 7.25 greater than the estimated average rate of change of the exponential function of the interval [2,6]. The estimated average rate of change of the exponential function is 7.25 greater than the estimated average rate of change of the Quadratic function of the interval [2,6]. The estimated average rate of change of the exponential function is 7.25 greater than the estimated average rate of change of the Quadratic function of the interval [2,6]. The estimated average rate of change of the exponential function is 0.16 greater than the estimated average rate of change of the Quadratic function of the interval [2,6].
Answers
Answered by
GPT-4o mini
To compare the average rates of change of the quadratic function and the exponential function over the interval \([2, 6]\), we begin by defining what we need to calculate.
1. **Average Rate of Change of the Quadratic Function**:
The points given for the quadratic function are \((2, -2)\) and \((6, 14)\). The average rate of change is given by:
\[
\text{Average Rate of Change (Quadratic)} = \frac{f(6) - f(2)}{6 - 2} = \frac{14 - (-2)}{6 - 2} = \frac{14 + 2}{4} = \frac{16}{4} = 4
\]
2. **Average Rate of Change of the Exponential Function**:
The exponential function is given as \(y = 34 \cdot 2^x\). We need to find \(f(2)\) and \(f(6)\):
- \(f(2) = 34 \cdot 2^2 = 34 \cdot 4 = 136\)
- \(f(6) = 34 \cdot 2^6 = 34 \cdot 64 = 2176\)
Now, we can calculate the average rate of change:
\[
\text{Average Rate of Change (Exponential)} = \frac{f(6) - f(2)}{6 - 2} = \frac{2176 - 136}{6 - 2} = \frac{2040}{4} = 510
\]
3. **Compare the Average Rates of Change**:
- Average Rate of Change of the Quadratic: \(4\)
- Average Rate of Change of the Exponential: \(510\)
Now we can find the difference between the two average rates of change:
\[
\text{Difference} = \text{Average Rate of Change (Quadratic)} - \text{Average Rate of Change (Exponential)} = 4 - 510 = -506
\]
Thus, we can see that the average rate of change of the exponential function is significantly greater than that of the quadratic function by \(506\).
### Conclusion:
The estimated average rate of change of the exponential function is greater than the estimated average rate of change of the quadratic function of the interval \([2,6]\) by \(506\).
Based on your provided options, the closest correct interpretation is:
**The estimated average rate of change of the exponential function is 506 greater than the estimated average rate of change of the Quadratic function in the interval [2,6].** However, please ensure no option contains this, as it would suggest that your data or options might be miscommunicated or incorrect.
1. **Average Rate of Change of the Quadratic Function**:
The points given for the quadratic function are \((2, -2)\) and \((6, 14)\). The average rate of change is given by:
\[
\text{Average Rate of Change (Quadratic)} = \frac{f(6) - f(2)}{6 - 2} = \frac{14 - (-2)}{6 - 2} = \frac{14 + 2}{4} = \frac{16}{4} = 4
\]
2. **Average Rate of Change of the Exponential Function**:
The exponential function is given as \(y = 34 \cdot 2^x\). We need to find \(f(2)\) and \(f(6)\):
- \(f(2) = 34 \cdot 2^2 = 34 \cdot 4 = 136\)
- \(f(6) = 34 \cdot 2^6 = 34 \cdot 64 = 2176\)
Now, we can calculate the average rate of change:
\[
\text{Average Rate of Change (Exponential)} = \frac{f(6) - f(2)}{6 - 2} = \frac{2176 - 136}{6 - 2} = \frac{2040}{4} = 510
\]
3. **Compare the Average Rates of Change**:
- Average Rate of Change of the Quadratic: \(4\)
- Average Rate of Change of the Exponential: \(510\)
Now we can find the difference between the two average rates of change:
\[
\text{Difference} = \text{Average Rate of Change (Quadratic)} - \text{Average Rate of Change (Exponential)} = 4 - 510 = -506
\]
Thus, we can see that the average rate of change of the exponential function is significantly greater than that of the quadratic function by \(506\).
### Conclusion:
The estimated average rate of change of the exponential function is greater than the estimated average rate of change of the quadratic function of the interval \([2,6]\) by \(506\).
Based on your provided options, the closest correct interpretation is:
**The estimated average rate of change of the exponential function is 506 greater than the estimated average rate of change of the Quadratic function in the interval [2,6].** However, please ensure no option contains this, as it would suggest that your data or options might be miscommunicated or incorrect.