Answer
Model: $ f(x) = 0.251(x-14)^2+30$
Domain: All real numbers
Range: $[30 , \infty)$
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 found to be at $(h,k)=(14,30)$.
Step 2: Set these into the standard vertex form of a parabola. This gives: $$\begin{aligned}
f(x) &= a(x-14)^2+30.
\end{aligned}$$ Step 3:
Choose any point from the scatter plot to find the value of the constant, $a$. Let's take the point $(x,y)=(21,42.3)$ and insert these into the above equation to find $a$. $$\begin{aligned}
42.3 &= a(21-14)^2+30\\
42.3-30& = a7^2\\
49a & = 12.3\\
a& = 0.251.
\end{aligned}$$ Hence, the parabola that best fit the data is $$f(x) =0.251(x-14)^2+30.$$ The domain and range of this function are:
Domain: All real numbers,
Range: $[30 , \infty)$.
Plot the function and see if it fits the data.
