Question

find the average rate of change of ( h(x)=-sqrt{x + 20} ) over the interval ( -2,9 ). 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 \(y = h(x)\) over the interval \([a,b]\) is \(\frac{h(b)-h(a)}{b - a}\). Here \(a=-2\), \(b = 8\), and \(h(x)=-\sqrt{x + 20}\).

Step2: Calculate \(h(a)\) and \(h(b)\)

• When \(x=a=-2\), \(h(-2)=-\sqrt{-2 + 20}=-\sqrt{18}=-3\sqrt{2}\).
• When \(x = b = 8\), \(h(8)=-\sqrt{8+20}=-\sqrt{28}=-2\sqrt{7}\).

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

\(\frac{h(8)-h(-2)}{8-(-2)}=\frac{-2\sqrt{7}-(-3\sqrt{2})}{8 + 2}=\frac{-2\sqrt{7}+3\sqrt{2}}{10}\approx\frac{-2\times2.646+3\times1.414}{10}=\frac{-5.292 + 4.242}{10}=\frac{-1.05}{10}=-0.105\)

Answer:

\(-0.105\)