Answer
The average rate of change of $f\left( x \right)={{x}^{2}}-2x$ from ${{x}_{1}}=3\text{ to }{{x}_{2}}=6$ is $7$.
Work Step by Step
Consider the provided function $f\left( x \right)={{x}^{2}}-2x$
The value of the function $f\left( x \right)={{x}^{2}}-2x$ at ${{x}_{1}}=3$ is
$\begin{align}
& f\left( 3 \right)={{\left( 3 \right)}^{2}}-2\left( 3 \right) \\
& =9-6 \\
& =3
\end{align}$
The value of the function $f\left( x \right)={{x}^{2}}-2x$ a ${{x}_{2}}=6$ is
$\begin{align}
& f\left( 6 \right)={{\left( 6 \right)}^{2}}-2\left( 6 \right) \\
& =36-12 \\
& =24
\end{align}$
The average rate of change of $f$ from ${{x}_{1}}=3\text{ to }{{x}_{2}}=6$ is
$\begin{align}
& \frac{\Delta y}{\Delta x}=\frac{f\left( 6 \right)-f\left( 3 \right)}{6-3} \\
& =\frac{24-3}{6-3} \\
& =\frac{21}{3} \\
& =7
\end{align}$
The average rate of change of $f\left( x \right)={{x}^{2}}-2x$ from ${{x}_{1}}=3\text{ to }{{x}_{2}}=6$ is $7$.
