how do the average rates of change for the pair of functions compare over the given interval? f(x)= -3x²…

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = h(x)$ over the interval $[a,b]$ is $\frac{h(b)-h(a)}{b - a}$.

Step2: Calculate average rate of change of $f(x)$

For $f(x)=-3x^{2}$, $a=-6$, $b = - 4$. $f(-4)=-3\times(-4)^{2}=-3\times16=-48$. $f(-6)=-3\times(-6)^{2}=-3\times36=-108$. The average rate of change of $f(x)$ is $\frac{f(-4)-f(-6)}{-4-(-6)}=\frac{-48 - (-108)}{-4 + 6}=\frac{-48 + 108}{2}=\frac{60}{2}=30$.

Step3: Calculate average rate of change of $g(x)$

For $g(x)=-6x^{2}$, $a=-6$, $b=-4$. $g(-4)=-6\times(-4)^{2}=-6\times16=-96$. $g(-6)=-6\times(-6)^{2}=-6\times36=-216$. The average rate of change of $g(x)$ is $\frac{g(-4)-g(-6)}{-4-(-6)}=\frac{-96-(-216)}{-4 + 6}=\frac{-96 + 216}{2}=\frac{120}{2}=60$.

Step4: Find the ratio

To find how many times the average rate of change of $g(x)$ is that of $f(x)$, we calculate $\frac{\text{Average rate of change of }g(x)}{\text{Average rate of change of }f(x)}=\frac{60}{30}=2$.

Answer:

The average rate of change of $f(x)$ over $-6\leq x\leq - 4$ is $30$. The average rate of change of $g(x)$ over $-6\leq x\leq - 4$ is $60$. The average rate of change of $g(x)$ is $2$ times that of $f(x)$.