Answer
Model: $ f(x) = -(x-6)^2+14$
Domain: All real numbers
Range: $(-\infty, 14]$
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 $(h,k)=(6,14)$.
Step 2: Set these into the standard vertex form of a parabola. This gives: $$\begin{aligned}
f(x) &= a(x-6)^2+14.
\end{aligned}$$ Step 3:
Choose any point from the scatter plot to find the value of the constant, $a$. Let's take the point $(x,y)=(4,10)$ and insert these into the above equation to find $a$. $$\begin{aligned}
10 &= a(4-6)^2+14\\
10-14& = a(-2)^2\\
4a & = -4\\
a& = -1.
\end{aligned}$$ Hence, the parabola that best fit the data is $$f(x) =-(x-6)^2+14.$$ The domain and range of this function are:
Domain: All real numbers,
Range: $(-\infty, 14]$.
Plot the function and see if it fits the data.
