Answer
Model: $ f(x) = 12(x+55)^2-9000$
Domain: All real numbers
Range: $[-9000 , \infty)$
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 $(h,k)=(-55,-9000)$.
Step 2: Set these into the standard vertex form of a parabola. This gives: $$\begin{aligned}
f(x) &= a(x+55)^2-9000.
\end{aligned}$$ Step 3:
Choose any point from the scatter plot to find the value of the constant, $a$. Let's take the point $(x,y)=(-30,-1500)$ and insert these into the above equation to find $a$. $$\begin{aligned}
-1500 &= a(-30+55)^2-9000\\
9000-1500& = a25^2\\
625a & = 7500\\
a& = 12.
\end{aligned}$$ Hence, the parabola that best fit the data is $$f(x) = 12(x+55)^2-9000.$$ The domain and range of this function are:
Domain: All real numbers,
Range: $[-9000 , \infty)$.
Plot the function and see if it fits the data.
