Question

given the function f(x)=x² - 2x - 6, determine the average rate of change of the function over the interval -1 ≤ x ≤ 6.

Explanation:

Step1: Recall average - rate - of - change formula

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 $f(a)$

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

Step3: Calculate $f(b)$

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

Step4: Calculate average rate of change

Use the formula $\frac{f(b)-f(a)}{b - a}=\frac{18-(-3)}{6-(-1)}=\frac{18 + 3}{6 + 1}=\frac{21}{7}=3$.

Answer:

3