Answer
$f\left( x \right)=-\frac{1}{3}{{x}^{2}}+5x-12$
Work Step by Step
$f\left( x \right)=a{{x}^{2}}+bx+c$
Substitute the data point$\left( -3,-30 \right)$ into the standard quadratic equation$f\left( x \right)=a{{x}^{2}}+bx+c$,
$-30=a{{\left( -3 \right)}^{2}}+b\left( -3 \right)+c$
Substitute the data point$\left( 3,0 \right)$ into the standard quadratic equation$f\left( x \right)=a{{x}^{2}}+bx+c$,
$0=a{{\left( 3 \right)}^{2}}+b\left( 3 \right)+c$
Substitute the data point$\left( 6,6 \right)$ into the standard quadratic equation$f\left( x \right)=a{{x}^{2}}+bx+c$,
$6=a{{\left( 6 \right)}^{2}}+b\left( 6 \right)+c$
Simplify the equations further as shown below,
$-30=9a-3b+c$ …… (1)
$0=9a+3b+c$ …… (2)
$6=36a+6b+c$ …… (3)
Add equation (2) and (1),
$\begin{align}
& \underline{\begin{align}
& -30=9a-3b+c \\
& 0=9a+3b+c
\end{align}} \\
& -30=18a+2b \\
\end{align}$
Further,
$-15=9a+c$ …...(4)
Multiply equation (2) by 2 and subtract it from equation (3),
$\begin{align}
& \underline{\begin{align}
& 6=36a+6b+c \\
& 0=18a-6b-2c
\end{align}} \\
& 6=18a-c \\
\end{align}$
Further,
$6=18a-c$ …..(5)
Add equation (4) and (5),
$\begin{align}
& \underline{\begin{align}
& -6=18a-c \\
& -15=9a+c
\end{align}} \\
& -9=27a \\
\end{align}$
Further,
$\begin{align}
& a=-\frac{9}{27} \\
& =-\frac{1}{3}
\end{align}$
Therefore,
$a=-\frac{1}{3}$
Substitute $a=-\frac{1}{3}$ into the equation $6=18a-c$,
$\begin{align}
& 6=18\left( -\frac{1}{3} \right)-c \\
& 6=-6-c \\
& 12=-c \\
& c=-12
\end{align}$
Therefore,
$c=-12$
Substitute $a=-\frac{1}{3}$ and $c=-12$ into the equation $0=9a+3b+c$,
$\begin{align}
& 0=9\left( -\frac{1}{3} \right)+3b+12 \\
& 0=-3+3b-12 \\
& 0=-15+3b \\
& 15=3b
\end{align}$
Therefore,
$b=5$
Substitute the values $a=-\frac{1}{3},b=5$ and $c=-12$ in the standard quadratic equation$f\left( x \right)=a{{x}^{2}}+bx+c$,
$f\left( x \right)=-\frac{1}{3}{{x}^{2}}+5x-12$.
Therefore, the quadratic function that fits the set of data points $\left( -3,-30 \right),\left( 3,0 \right),\left( 6,6 \right)$ is $f\left( x \right)=-\frac{1}{3}{{x}^{2}}+5x-12$
