Answer
Model: $ f(x) = 2.4(x+9)^2+3.5$
Domain: All real numbers
Range: $[3.5,\infty )$
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 $(h,k)=(-9,3.5)$.
Step 2: Set these into the standard vertex form of a parabola. This gives: $$\begin{aligned}
f(x) &= a(x+9)^2+3.5.
\end{aligned}$$ Step 3:
Choose any point from the scatter plot to find the value of the constant, $a$. Let's take the point, $(x,y)=(-3,89.9)$ and insert these into the above equation to find $a$. $$\begin{aligned}
89.9 &= a(-3+9)^2+3.5\\
89.9-3.5& = a(6)^2\\
36a & = 86.4\\
a& = 2.4
\end{aligned}$$ Hence, the parabola that best fit the data is $$f(x) =2.4(x+9)^2+3.5.$$ The domain and range of this function are:
Domain: All real numbers,
Range: $[3.5,\infty )$.
