Answer
Model: $ f(x) = 11.5x^2$
Domain: All real numbers
Range: $[0,\infty )$
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 $(h,k)=(0,0)$.
Step 2: Set these into the standard vertex form of a parabola. This gives: $$\begin{aligned}
f(x) &= ax^2.
\end{aligned}$$ Step 3:
Choose any point from the scatter plot to find the value of the constant $a$. Let take the point $(x,y)=(2,46)$ and insert this into the above equation to find $a$. $$\begin{aligned}
46 &= a2^2\\
46& = 4a\\
a& = 11.5
\end{aligned}$$ Hence, the parabola that best fit the data is $$f(x) = 11.5x^2.$$ The domain and range of this function are:
Domain: All real numbers,
Range: $(-\infty,0 ]$.
Plot the function and see if it fits the data.
