Question

given the function ( h(x) = -x^{2} - 6x + 13 ), determine the average rate of change of the function over the interval ( -7 leq x leq 3 ).

Explanation:

Step1: Recall the formula for average rate of change

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 \(a=-7\) and \(b = 3\).

Step2: Calculate \(h(-7)\)

Substitute \(x=-7\) into \(h(x)=-x^{2}-6x + 13\):

$$
LATEXBLOCK0
$$

Step3: Calculate \(h(3)\)

Substitute \(x = 3\) into \(h(x)=-x^{2}-6x + 13\):

$$
LATEXBLOCK1
$$

Step4: Calculate the average rate of change

Using the formula \(\frac{h(b)-h(a)}{b - a}\), with \(a=-7\), \(b = 3\), \(h(-7)=6\) and \(h(3)=-14\):

$$
LATEXBLOCK2
$$

Answer:

\(-2\)