Question
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 = 1/4(2)^x (4 points)
The average rate of change for the quadratic function for the interval [2,6] is _. The average rate of change for exponential function is _. The average rate of change of the quadratic function is _ _ than the average rate of change of the exponential function over the interval [2, 6]
The average rate of change for the quadratic function for the interval [2,6] is _. The average rate of change for exponential function is _. The average rate of change of the quadratic function is _ _ than the average rate of change of the exponential function over the interval [2, 6]
Answers
GPT-4o mini
To compare the average rates of change for both the quadratic function and the exponential function over the interval [2, 6], we first need to define both functions clearly, and then calculate their average rates of change.
### Step 1: Calculate Average Rate of Change for the Quadratic Function
Assume the quadratic function is represented as \(f(x) = ax^2 + bx + c\). However, since we don't have the exact quadratic function provided, we will use generic values.
The average rate of change is calculated using the formula:
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
For the interval \([2, 6]\):
- Let \(a = 2\)
- Let \(b = 6\)
So we will evaluate \(f(2)\) and \(f(6)\) based on the quadratic function values.
### Step 2: Calculate Average Rate of Change for the Exponential Function
Given the exponential function
\[
y = \frac{1}{4}(2)^x
\]
We will also calculate:
\[
\text{Average Rate of Change} = \frac{g(6) - g(2)}{6 - 2}
\]
where \(g(x) = \frac{1}{4}(2)^x\).
Calculating:
- \(g(2) = \frac{1}{4}(2^2) = \frac{1}{4} \cdot 4 = 1\)
- \(g(6) = \frac{1}{4}(2^6) = \frac{1}{4} \cdot 64 = 16\)
So the average rate of change for \(g\) is:
\[
\text{Average Rate of Change} = \frac{16 - 1}{6 - 2} = \frac{15}{4} = 3.75
\]
### Step 3: Fill in the Blanks
Assuming the average rate of change for the quadratic function over \([2, 6]\) you calculated (for example):
- The average rate of change for the quadratic function for the interval [2,6] is \(k\) (assuming \(k\) is a placeholder).
- The average rate of change for the exponential function is \(3.75\).
- The average rate of change of the quadratic function is \(k\) compared to \(3.75\).
### Final Result
You can now fill in the values based on your calculated quadratic function rate of change:
- average rate of change for the quadratic function: \(k\)
- average rate of change for the exponential function: \(3.75\)
- relation statement: (greater than, less than, or equal to)
Hence, combine to complete the final statement:
"The average rate of change of the quadratic function is \(k\) ___ than the average rate of change of the exponential function over the interval [2, 6]."
Make sure to verify or calculate the specific value of \(k\) based on your quadratic function.
### Step 1: Calculate Average Rate of Change for the Quadratic Function
Assume the quadratic function is represented as \(f(x) = ax^2 + bx + c\). However, since we don't have the exact quadratic function provided, we will use generic values.
The average rate of change is calculated using the formula:
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
For the interval \([2, 6]\):
- Let \(a = 2\)
- Let \(b = 6\)
So we will evaluate \(f(2)\) and \(f(6)\) based on the quadratic function values.
### Step 2: Calculate Average Rate of Change for the Exponential Function
Given the exponential function
\[
y = \frac{1}{4}(2)^x
\]
We will also calculate:
\[
\text{Average Rate of Change} = \frac{g(6) - g(2)}{6 - 2}
\]
where \(g(x) = \frac{1}{4}(2)^x\).
Calculating:
- \(g(2) = \frac{1}{4}(2^2) = \frac{1}{4} \cdot 4 = 1\)
- \(g(6) = \frac{1}{4}(2^6) = \frac{1}{4} \cdot 64 = 16\)
So the average rate of change for \(g\) is:
\[
\text{Average Rate of Change} = \frac{16 - 1}{6 - 2} = \frac{15}{4} = 3.75
\]
### Step 3: Fill in the Blanks
Assuming the average rate of change for the quadratic function over \([2, 6]\) you calculated (for example):
- The average rate of change for the quadratic function for the interval [2,6] is \(k\) (assuming \(k\) is a placeholder).
- The average rate of change for the exponential function is \(3.75\).
- The average rate of change of the quadratic function is \(k\) compared to \(3.75\).
### Final Result
You can now fill in the values based on your calculated quadratic function rate of change:
- average rate of change for the quadratic function: \(k\)
- average rate of change for the exponential function: \(3.75\)
- relation statement: (greater than, less than, or equal to)
Hence, combine to complete the final statement:
"The average rate of change of the quadratic function is \(k\) ___ than the average rate of change of the exponential function over the interval [2, 6]."
Make sure to verify or calculate the specific value of \(k\) based on your quadratic function.