Question

find the average rate of change of $g(x)=-4x^{2}+5$ over the interval $-4,-2$. write your answer as an integer, fraction, or decimal rounded to the nearest tenth. simplify any fractions.

Explanation:

Step1: Recall the formula for average rate of change

The formula for the average rate of change of a function \(y = g(x)\) over the interval \([a,b]\) is \(\frac{g(b)-g(a)}{b - a}\). Here \(a=-4\), \(b = - 2\), and \(g(x)=-4x^{2}+5\).

Step2: Calculate \(g(-4)\) and \(g(-2)\)

• For \(x=-4\):

\(g(-4)=-4\times(-4)^{2}+5=-4\times16 + 5=-64 + 5=-59\)

• For \(x=-2\):

\(g(-2)=-4\times(-2)^{2}+5=-4\times4+5=-16 + 5=-11\)

Step3: Substitute into the average - rate - of - change formula

\(\frac{g(-2)-g(-4)}{-2-(-4)}=\frac{-11-(-59)}{-2 + 4}=\frac{-11 + 59}{2}=\frac{48}{2}\)

Answer:

\(24\)