\[
f(z)=4-3 z^{2} ; \quad z=-2, z=0
\]
(a) Determine the net change between the given values of the variable.
(b) Determine the average rate of change between the given values of the variable.
Final Answer: (a) \(\boxed{-8}\), (b) \(\boxed{-4.0}\)
Step 1 :Given the function \(f(z) = 4 - 3z^2\) and the values \(z = -2\) and \(z = 0\).
Step 2 :For part (a), the net change between the given values of the variable is given by the formula: \[f(b) - f(a)\]
Step 3 :Substitute the given values into the formula: \[f(0) - f(-2)\]
Step 4 :Simplify the expression to get the net change.
Step 5 :The net change between the given values of the variable is -8.
Step 6 :For part (b), the average rate of change between the given values of the variable is given by the formula: \[\frac{f(b) - f(a)}{b - a}\]
Step 7 :Substitute the given values into the formula: \[\frac{f(0) - f(-2)}{0 - (-2)}\]
Step 8 :Simplify the expression to get the average rate of change.
Step 9 :The average rate of change between the given values of the variable is -4.0.
Step 10 :Final Answer: (a) \(\boxed{-8}\), (b) \(\boxed{-4.0}\)