Answer
Model: $ f(x) = -0.0363(x+4)^2+150$
Domain: All real numbers
Range: $(-\infty , 150]$
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 $(h,k)=(-4,150)$.
Step 2: Set these into the standard vertex form of a parabola. This gives: $$\begin{aligned}
f(x) &= a(x+4)^2+150.
\end{aligned}$$ Step 3:
Choose any point from the scatter plot to find the value of the constant, $a$. Let's take the point $(x,y)=(50,44.2)$ and insert these into the above equation to find $a$. $$\begin{aligned}
44.2 &= a(50+4)^2+150\\
2916a &= 44.2-150 = -105.8\\
a& = -0.0363.
\end{aligned}$$ Hence, the parabola that best fit the data is $$f(x) = -0.04(x+4)^2+150.$$ The domain and range of this function are:
Domain: All real numbers,
Range: $(-\infty , 150]$.
Plot the function and see if it fits the data.
