Calculate the average rate of change over the interval [2,4]

Question

given the function table for f(x)=−x2+12
.

x f(x)
1 11
2 8
3 3
4 −4
(1 point)
Responses

−2
negative 2

2
2

6
6

−6
negative 6
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Answer 1

To calculate the average rate of change of the function \( f(x) = -x^2 + 12 \) over the interval \([2, 4]\), we 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 \( b = 4 \).

From the table you've provided, we can read the function values:

\[ f(2) = 8 \] \[ f(4) = -4 \]

Now, we substitute these values into the formula:

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

Thus, the average rate of change over the interval \([2, 4]\) is:

\(-6\)

So the correct response is:

−6