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 $-7leq xleq3$.
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 $f(-7)$
Substitute $x=-7$ into $f(x)=-x^{2}-2x + 6$. $f(-7)=-(-7)^{2}-2\times(-7)+6=-49 + 14+6=-29$.
Step3: Calculate $f(3)$
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
Use the formula $\frac{f(3)-f(-7)}{3-(-7)}$. $\frac{-9-(-29)}{3 + 7}=\frac{-9 + 29}{10}=\frac{20}{10}=2$.
Answer:
$2$