Question

alculate the average rate of change over the interval [2,4] given the function table for f(x)=−x2+12

Answer 1

To calculate the average rate of change of the function \( f(x) = -x^2 + 12 \) over the interval \([2, 4]\), we can use the formula for the average rate of change:

\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]

where \( a = 2 \) and \( b = 4 \).

First, we need to find \( f(2) \) and \( f(4) \):

1. Calculate \( f(2) \):
\[
f(2) = -2^2 + 12 = -4 + 12 = 8
\]

2. Calculate \( f(4) \):
\[
f(4) = -4^2 + 12 = -16 + 12 = -4
\]

Now, we can use these values to find the average rate of change:

\[
\text{Average Rate of Change} = \frac{f(4) - f(2)}{4 - 2} = \frac{-4 - 8}{4 - 2} = \frac{-12}{2} = -6
\]

Therefore, the average rate of change of the function \( f(x) = -x^2 + 12 \) over the interval \([2, 4]\) is \(-6\).