Question

given the function ( f(x)=x^{2}-2x - 6 ), determine the average rate of change of the function over the interval ( -1leq xleq6 ).

Explanation:

Step1: Recall the formula for average rate of change

The formula for the average rate of change of a function \(y = f(x)\) over the interval \([a,b]\) is \(\frac{f(b)-f(a)}{b - a}\). Here, \(a=-1\) and \(b = 6\).

Step2: Calculate \(f(-1)\)

Substitute \(x=-1\) into \(f(x)=x^{2}-2x - 6\).
\(f(-1)=(-1)^{2}-2\times(-1)-6=1 + 2-6=-3\)

Step3: Calculate \(f(6)\)

Substitute \(x = 6\) into \(f(x)=x^{2}-2x - 6\).
\(f(6)=6^{2}-2\times6-6=36-12 - 6=18\)

Step4: Calculate the average rate of change

Substitute \(f(-1)=-3\), \(f(6)=18\), \(a=-1\), and \(b = 6\) into the formula \(\frac{f(b)-f(a)}{b - a}\).
\(\frac{f(6)-f(-1)}{6-(-1)}=\frac{18-(-3)}{6 + 1}=\frac{18 + 3}{7}=\frac{21}{7}=3\)

Answer:

\(3\)