Calculate the rate of change for the quadratic function over the given interval: f(x)=x^2 + 4x +5 ; -4 =< x =< -2

The average rate of change for this quadratic function is -2.

Step-by-step explanation:

In this question, we have f(x) = y = x^{2} + 4x + 5.

Given a function y, the average rate of change S of y=f(x) in an interval  [x_s, x_f] will be given by the following equation:

S = \frac{f(x_{f}) - f(x_s)}{x_f - x_s}

So, in  your problem, f(x) = x^{2} + 4x + 5, x_{f} = -2 and x_{s} = -4.

Applying these informations to the equation S above, we have:

S = \frac{f(-2) - f(-4)}{-2-(-4)}

Where

f(-2) = (-2)^{2} + 4(-2) +5 = 4-8+5 = 9-8 = 1

f(-4) = (-4)^{2} + 4(-4) +5 = 16-16+5 = 5

So, the average rate of change S will be

S = \frac{1-5}{2} = -4