Question

h(x)=x^{2}-1
what is the average rate of change of h over the interval -3≤x≤-1?

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, \(a=-3\) and \(b = - 1\).

Step2: Calculate \(h(-3)\) and \(h(-1)\)

Substitute \(x=-3\) into \(h(x)=x^{2}-1\): \(h(-3)=(-3)^{2}-1=9 - 1=8\).
Substitute \(x=-1\) into \(h(x)=x^{2}-1\): \(h(-1)=(-1)^{2}-1=1 - 1=0\).

Step3: Compute the average rate of change

Using the formula \(\frac{h(b)-h(a)}{b - a}\), with \(h(-1) = 0\), \(h(-3)=8\), \(a=-3\), and \(b=-1\), we have \(\frac{h(-1)-h(-3)}{-1-(-3)}=\frac{0 - 8}{-1 + 3}=\frac{-8}{2}=-4\).

Answer:

\(-4\)