Answer
Model: $ f(x) = 4x^2-6$
Domain: All real numbers
Range: $[-6,\infty )$
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 $(h,k)=(0,-6)$.
Step 2: Set these into the standard vertex form of a parabola. This gives: $$\begin{aligned}
f(x) &= ax^2-6.
\end{aligned}$$ Step 3:
Choose any point from the scatter plot to find the value of the constant $a$. Let's take the point $(x,y)=(3,30)$ and insert this into the above equation to find $a$. $$\begin{aligned}
30 &= a3^2-6\\
30+6.5& = 9a\\
9a & = 36\\
a& = 4 .
\end{aligned}$$ Hence, the parabola that best fit the data is $$f(x) =4x^2-6.$$ The domain and range of this function are:
Domain: All real numbers,
Range: $[-6,\infty )$.
Plot the function and see if it fits the data.
