Answer
$f(x)=\begin{cases}
0.505x^2-1.47x+6.3,1\leq x\leq 6\\
-1.97x+26.3,6\lt x\leq 12
\end{cases}$
Work Step by Step
We are given the function:
$f(x)=\begin{cases}
-1.97x+26.3\\
0.505x^2-1.47x+6.3
\end{cases}$
From the data we notice that the values of $y$ increase for $x\in\{1,2,3,4,5,6\}$ and decrease for $\{6,7,8,9,10,11,12\}$.
The function $y=0.505x^2-1.47x+6.3$ is increasing for $x\in[1,6]$, while the function $y=-1.97x+26.3$ is decreasing, having a negative slope for $x\in (6,12]$.
Therefore we can write the function:
$f(x)=\begin{cases}
0.505x^2-1.47x+6.3,1\leq x\leq 6\\
-1.97x+26.3,6\lt x\leq 12
\end{cases}$
Graph the model.
