Answer
Model: $ f(x) = -4(x+5)^2+8$
Domain: All real numbers
Range: $(-\infty , 8]$
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 $(h,k)=(-5,8)$.
Step 2: Set these into the standard vertex form of a parabola. This gives: $$\begin{aligned}
f(x) &= a(x+5)^2+8.
\end{aligned}$$ Step 3:
Choose any point from the scatter plot to find the value of the constant, $a$. Let's take the point $(x,y)=(-7,-8)$ and insert these into the above equation to find $a$. $$\begin{aligned}
-8 &= a(-7+5)^2+8\\
-8&= a(-2)^2+8\\
-8& = 4a+8\\
4a& = -8-8= -16\\
a&= -4.
\end{aligned}$$ Hence, the parabola that best fit the data is $$f(x) = -4(x+5)^2+8.$$ The domain and range of this function are:
Domain: All real numbers,
Range $(-\infty , 8]$.
Plot the function and see if it fits the data.
