Answer
Model: $ f(x) = (x-5)^2-4$
Domain: All real numbers
Range: $[-4 , \infty)$
Work Step by Step
Step 1: Make a scatter plot of the data and choose a vertex point that may either be the lowest or highest point. The vertex of the data point can be seen to be at $(h,k)=(5,-4)$.
Step 2: Set these into the standard vertex form of a parabola. This gives: $$\begin{aligned}
f(x) &= a(x-5)^2-4\\
\end{aligned}$$ Step 3:
Choose any point from the scatter plot to find the value of the constant, $a$. Let's take the point $(x,y)=(8,5)$ and insert these into the above equation to find $a$. $$\begin{aligned}
5 &= a(8-5)^2-4\\
9a &= 5+4 = 9\\
a& = 1.
\end{aligned}$$ Hence, the parabola that best fit the data is $$f(x) = (x-5)^2-4.$$ The domain and range of this function are:
Domain: All real numbers,
Range: $[-4 , \infty)$.
Plot the function and see if it fits the data.
