To determine the average rate of change of the function \( f(x) = -x^2 - 2x + 3 \) over the interval \([-6, 2]\), we can use the formula for the average rate of change:
\[ \text{Average rate of change} = \frac{f(b) - f(a)}{b - a} \]
where \( a = -6 \) and \( b = 2 \).
First, we need to calculate \( f(-6) \) and \( f(2) \).
Calculating \( f(-6) \):
\[ f(-6) = -(-6)^2 - 2(-6) + 3 \] \[ = -36 + 12 + 3 \] \[ = -36 + 15 \] \[ = -21 \]
Calculating \( f(2) \):
\[ f(2) = -(2)^2 - 2(2) + 3 \] \[ = -4 - 4 + 3 \] \[ = -8 + 3 \] \[ = -5 \]
Now that we have both function values, we can calculate the average rate of change:
\[ \text{Average rate of change} = \frac{f(2) - f(-6)}{2 - (-6)} \] \[ = \frac{-5 - (-21)}{2 + 6} \] \[ = \frac{-5 + 21}{8} \] \[ = \frac{16}{8} \] \[ = 2 \]
Thus, the average rate of change of the function \( f(x) \) over the interval \([-6, 2]\) is \( \boxed{2} \).