given the function f(x)=-x^2 - 2x + 6, determine the average rate of change of the function over the…
given the function f(x)=-x^2 - 2x + 6, determine the average rate of change of the function over the interval -7 ≤ x ≤ 3.
Answer
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, $a=-7$ and $b = 3$.
Step2: Calculate $f(a)$
Substitute $x=-7$ into $f(x)=-x^{2}-2x + 6$. $f(-7)=-(-7)^{2}-2\times(-7)+6=-49 + 14+6=-29$.
Step3: Calculate $f(b)$
Substitute $x = 3$ into $f(x)=-x^{2}-2x + 6$. $f(3)=-3^{2}-2\times3 + 6=-9-6 + 6=-9$.
Step4: Calculate the average rate of change
$\frac{f(3)-f(-7)}{3-(-7)}=\frac{-9-(-29)}{3 + 7}=\frac{-9 + 29}{10}=\frac{20}{10}=2$.
Answer:
$2$