A function is given.
f(z) = 3 − 4z square; 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.
To find the net change between the given values of the variable, we need to evaluate the function at those values and subtract the smaller value from the larger value.
For z = -2, we have:
f(-2) = 3 - 4*(-2)^2
f(-2) = 3 - 4*(4)
f(-2) = 3 - 16
f(-2) = -13
For z = 0, we have:
f(0) = 3 - 4*0^2
f(0) = 3 - 4*0
f(0) = 3 - 0
f(0) = 3
Therefore, the net change between the given values of the variable is:
-13 - 3 = -16
To find the average rate of change between the given values of the variable, we need to calculate the difference in the function values and divide by the difference in the variable values.
The difference in the function values is:
f(0) - f(-2) = 3 - (-13) = 3 + 13 = 16
The difference in the variable values is:
0 - (-2) = 0 + 2 = 2
Therefore, the average rate of change between the given values of the variable is:
16/2 = 8
To determine the net change between the given values of the variable, we need to find the difference in the function values at those points.
(a) To find the net change, substitute the given values of z into the function f(z):
For z = -2:
f(-2) = 3 - 4(-2)^2
= 3 - 4(4)
= 3 - 16
= -13
For z = 0:
f(0) = 3 - 4(0)^2
= 3 - 4(0)
= 3
The net change between the given values of the variable is the difference between these two values:
Net change = f(0) - f(-2)
= 3 - (-13)
= 3 + 13
= 16
Therefore, the net change between the given values of the variable is 16.
(b) To determine the average rate of change, we need to calculate the slope of the line connecting the two points (-2, f(-2)) and (0, f(0)).
Average rate of change = (f(0) - f(-2)) / (0 - (-2))
= (3 - (-13)) / 2
= (3 + 13) / 2
= 16 / 2
= 8
Therefore, the average rate of change between the given values of the variable is 8.