Question
A U-shaped upward opening curve with 2 plotted points is on a coordinate plane. The x-axis ranges from negative 4 to 7 in increments of 1. The y-axis ranges from negative 3 to 15 in increments of 1.
Compare the estimated 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)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 over 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 over the interval left bracket 2 comma 6 right bracket.
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 over 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 over the interval left bracket 2 comma 6 right bracket.
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 over 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 over the interval left bracket 2 comma 6 right bracket.
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 over the interval [2,6]
.
Compare the estimated 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)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 over 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 over the interval left bracket 2 comma 6 right bracket.
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 over 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 over the interval left bracket 2 comma 6 right bracket.
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 over 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 over the interval left bracket 2 comma 6 right bracket.
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 over the interval [2,6]
.
Answers
GPT-4o mini
To compare the average rates of change of the two functions over the interval \([2, 6]\), we can calculate the average rate of change for each function.
1. **Quadratic Function**:
The average rate of change of a function \( f \) over an interval \([a, b]\) is given by the formula:
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
Assuming that we need to evaluate the quadratic function at \( x = 2 \) and \( x = 6 \), we denote the values at these points as \( f(2) \) and \( f(6) \).
2. **Exponential Function** \( y = 34(2^x) \):
We can also apply the same formula to find the average rate of change over the interval \([2, 6]\):
\[
\text{Average Rate of Change} = \frac{f(6) - f(2)}{6 - 2}
\]
where \( f(x) = 34 \cdot 2^x \).
Let's compute the average rates of change using specific values for \( f(2) \) and \( f(6) \):
- **For the exponential function**:
- \( f(2) = 34 \cdot 2^2 = 34 \cdot 4 = 136 \)
- \( f(6) = 34 \cdot 2^6 = 34 \cdot 64 = 2176 \)
Now calculate the average rate of change:
\[
\text{Average Rate of Change} = \frac{2176 - 136}{6 - 2} = \frac{2040}{4} = 510
\]
3. Now we need to compare this with the estimated average rate of change of the quadratic function, which we can assume has been given or calculated as needed.
After calculating both of them, let’s say if the average rate of change for the quadratic was found to be \( x \) (you can substitute this with the actual value based on the graph data you have).
4. Finally, to determine the difference:
If \( x \) is the average rate of change for the quadratic function, then the comparison can be phrased as follows:
- If \( x - 510 = 0.16 \), then the quadratic function's average rate is greater.
- If \( 510 - x = 0.16 \), then the exponential function's average rate is greater.
- If the difference is found to be \( 7.25 \) either way, you make the corresponding claim.
Given that without specific values for the quadratic function's average rate of change it's hard to choose a final response from the provided options, simply calculate and compare as shown.
Make sure you substitute actual results where required! My computations above follow the expected logic; just let me know if further clarification or a different approach is needed.
1. **Quadratic Function**:
The average rate of change of a function \( f \) over an interval \([a, b]\) is given by the formula:
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
Assuming that we need to evaluate the quadratic function at \( x = 2 \) and \( x = 6 \), we denote the values at these points as \( f(2) \) and \( f(6) \).
2. **Exponential Function** \( y = 34(2^x) \):
We can also apply the same formula to find the average rate of change over the interval \([2, 6]\):
\[
\text{Average Rate of Change} = \frac{f(6) - f(2)}{6 - 2}
\]
where \( f(x) = 34 \cdot 2^x \).
Let's compute the average rates of change using specific values for \( f(2) \) and \( f(6) \):
- **For the exponential function**:
- \( f(2) = 34 \cdot 2^2 = 34 \cdot 4 = 136 \)
- \( f(6) = 34 \cdot 2^6 = 34 \cdot 64 = 2176 \)
Now calculate the average rate of change:
\[
\text{Average Rate of Change} = \frac{2176 - 136}{6 - 2} = \frac{2040}{4} = 510
\]
3. Now we need to compare this with the estimated average rate of change of the quadratic function, which we can assume has been given or calculated as needed.
After calculating both of them, let’s say if the average rate of change for the quadratic was found to be \( x \) (you can substitute this with the actual value based on the graph data you have).
4. Finally, to determine the difference:
If \( x \) is the average rate of change for the quadratic function, then the comparison can be phrased as follows:
- If \( x - 510 = 0.16 \), then the quadratic function's average rate is greater.
- If \( 510 - x = 0.16 \), then the exponential function's average rate is greater.
- If the difference is found to be \( 7.25 \) either way, you make the corresponding claim.
Given that without specific values for the quadratic function's average rate of change it's hard to choose a final response from the provided options, simply calculate and compare as shown.
Make sure you substitute actual results where required! My computations above follow the expected logic; just let me know if further clarification or a different approach is needed.