question given the function f(x)=x² + 3x - 6, determine the average rate of change of the function over the…

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, $a=-6$ and $b = 1$.

Step2: Calculate $f(a)$

Substitute $x=-6$ into $f(x)=x^{2}+3x - 6$. $f(-6)=(-6)^{2}+3\times(-6)-6=36-18 - 6=12$.

Step3: Calculate $f(b)$

Substitute $x = 1$ into $f(x)=x^{2}+3x - 6$. $f(1)=1^{2}+3\times1-6=1 + 3-6=-2$.

Step4: Calculate average rate of change

$\frac{f(1)-f(-6)}{1-(-6)}=\frac{-2 - 12}{1 + 6}=\frac{-14}{7}=-2$.

Answer:

$-2$