Question
The first point of the parabola is at (-9, -1), the second point is at (2, 15). Compare the magnitude of the estimated average rates of change of the exponential function pictured above and the quadratic function f(x) = x2 - 20 over the interval [-9, 2] and identify which
function has a greater rate of change than the other. (1 point)
A. The estimated average rate of change of the exponential function over the interval [-9, 2] is negative, while the estimated average rate of change of f (x) over the interval [-9, 2] is positive.
B. The Magnitude of the estimated average rate of change of the exponential function over the interval [-9, 2] estimated is less than the magnitude of the estimated average rate of change of f (x) over the interval
[-9, 2]
C. The magnitude of the estimated average rate of change of the exponential function over the interval [-9,2] is greater than the magnitude of the estimated average rate of change of f (x) over the interval [-9, 2]
D. The estimated average rate of change of both the exponential function and f (x) over the interval [-9, 2] is approximately 1.5
function has a greater rate of change than the other. (1 point)
A. The estimated average rate of change of the exponential function over the interval [-9, 2] is negative, while the estimated average rate of change of f (x) over the interval [-9, 2] is positive.
B. The Magnitude of the estimated average rate of change of the exponential function over the interval [-9, 2] estimated is less than the magnitude of the estimated average rate of change of f (x) over the interval
[-9, 2]
C. The magnitude of the estimated average rate of change of the exponential function over the interval [-9,2] is greater than the magnitude of the estimated average rate of change of f (x) over the interval [-9, 2]
D. The estimated average rate of change of both the exponential function and f (x) over the interval [-9, 2] is approximately 1.5
Answers
GPT-4o mini
To compare the average rates of change of the exponential function and the quadratic function \( f(x) = x^2 - 20 \) over the interval \([-9, 2]\), we need to find the average rates of change for both functions.
1. **Average Rate of Change** of a function \( g(x) \) over the interval \([a, b]\) is calculated as:
\[
\text{Average Rate of Change} = \frac{g(b) - g(a)}{b - a}
\]
2. **Quadratic Function**:
We have \( f(x) = x^2 - 20 \).
- At \( x = -9 \):
\[
f(-9) = (-9)^2 - 20 = 81 - 20 = 61
\]
- At \( x = 2 \):
\[
f(2) = (2)^2 - 20 = 4 - 20 = -16
\]
- Now, we can calculate the average rate of change of \( f(x) \) over \([-9, 2]\):
\[
\text{Average Rate of Change of } f = \frac{-16 - 61}{2 - (-9)} = \frac{-77}{11} = -7
\]
3. **Exponential Function**:
From the points provided, we have two points for the exponential function: \( (-9, -1) \) and \( (2, 15) \).
- The average rate of change is calculated as follows:
\[
\text{Average Rate of Change of Exponential Function} = \frac{15 - (-1)}{2 - (-9)} = \frac{15 + 1}{2 + 9} = \frac{16}{11}
\]
4. **Comparing the Rates of Change**:
- Average rate of change of the exponential function is \( \frac{16}{11} \) (approximately 1.45).
- Average rate of change of the quadratic function \( f(x) = x^2 - 20 \) is \( -7 \).
5. **Magnitude Comparison**:
- The magnitude of the average rate of change of the exponential function is:
\[
\left| \frac{16}{11} \right| \approx 1.45
\]
- The magnitude of the average rate of change of the quadratic function is:
\[
|-7| = 7
\]
Now, based on our calculations:
- The exponential function has a positive average rate of change of approximately \( 1.45 \).
- The quadratic function has a negative average rate of change of \( -7 \).
Thus, we can determine:
- The magnitude of the average rate of change of the exponential function is less than the magnitude of the average rate of change of the quadratic function.
Thus, the correct answer is **B**: The magnitude of the estimated average rate of change of the exponential function over the interval \([-9, 2]\) is less than the magnitude of the estimated average rate of change of \( f(x) \) over the interval \([-9, 2]\).
1. **Average Rate of Change** of a function \( g(x) \) over the interval \([a, b]\) is calculated as:
\[
\text{Average Rate of Change} = \frac{g(b) - g(a)}{b - a}
\]
2. **Quadratic Function**:
We have \( f(x) = x^2 - 20 \).
- At \( x = -9 \):
\[
f(-9) = (-9)^2 - 20 = 81 - 20 = 61
\]
- At \( x = 2 \):
\[
f(2) = (2)^2 - 20 = 4 - 20 = -16
\]
- Now, we can calculate the average rate of change of \( f(x) \) over \([-9, 2]\):
\[
\text{Average Rate of Change of } f = \frac{-16 - 61}{2 - (-9)} = \frac{-77}{11} = -7
\]
3. **Exponential Function**:
From the points provided, we have two points for the exponential function: \( (-9, -1) \) and \( (2, 15) \).
- The average rate of change is calculated as follows:
\[
\text{Average Rate of Change of Exponential Function} = \frac{15 - (-1)}{2 - (-9)} = \frac{15 + 1}{2 + 9} = \frac{16}{11}
\]
4. **Comparing the Rates of Change**:
- Average rate of change of the exponential function is \( \frac{16}{11} \) (approximately 1.45).
- Average rate of change of the quadratic function \( f(x) = x^2 - 20 \) is \( -7 \).
5. **Magnitude Comparison**:
- The magnitude of the average rate of change of the exponential function is:
\[
\left| \frac{16}{11} \right| \approx 1.45
\]
- The magnitude of the average rate of change of the quadratic function is:
\[
|-7| = 7
\]
Now, based on our calculations:
- The exponential function has a positive average rate of change of approximately \( 1.45 \).
- The quadratic function has a negative average rate of change of \( -7 \).
Thus, we can determine:
- The magnitude of the average rate of change of the exponential function is less than the magnitude of the average rate of change of the quadratic function.
Thus, the correct answer is **B**: The magnitude of the estimated average rate of change of the exponential function over the interval \([-9, 2]\) is less than the magnitude of the estimated average rate of change of \( f(x) \) over the interval \([-9, 2]\).