Question
Calculate the average rate of change over the interval [2,4] given the function table for f(x)=-x^2+12. The table is as follows: x: 1, 2, 3, 4/ f(x): 11, 8, 3, -4
Answers
GPT-4o mini
To calculate the average rate of change of the function \( f(x) = -x^2 + 12 \) over the interval \([2, 4]\), we can use the average rate of change formula:
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
where \( a = 2 \) and \( b = 4 \).
From the provided function table, we have:
- \( f(2) = 8 \)
- \( f(4) = -4 \)
Now, we can substitute these values into the formula:
\[
\text{Average Rate of Change} = \frac{f(4) - f(2)}{4 - 2} = \frac{-4 - 8}{4 - 2}
\]
\[
= \frac{-12}{2} = -6
\]
Therefore, the average rate of change of the function \( f(x) = -x^2 + 12 \) over the interval \([2, 4]\) is \(-6\).
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
where \( a = 2 \) and \( b = 4 \).
From the provided function table, we have:
- \( f(2) = 8 \)
- \( f(4) = -4 \)
Now, we can substitute these values into the formula:
\[
\text{Average Rate of Change} = \frac{f(4) - f(2)}{4 - 2} = \frac{-4 - 8}{4 - 2}
\]
\[
= \frac{-12}{2} = -6
\]
Therefore, the average rate of change of the function \( f(x) = -x^2 + 12 \) over the interval \([2, 4]\) is \(-6\).