To determine the equation of the quadratic function [tex]\( f(x) \)[/tex] based on the given data points, we will assume the general form of a quadratic equation:
[tex]\[ f(x) = ax^2 + bx + c \][/tex]
Given data points are:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-8 & 13 \\
-7 & 6 \\
-6 & 1 \\
-5 & -2 \\
-4 & -3 \\
-3 & -2 \\
-2 & 1 \\
-1 & 6 \\
0 & 13 \\
\hline
\end{array}
\][/tex]
By fitting these points to the quadratic equation, we want to determine the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex].
Through a series of calculations and solving a system of linear equations using methods such as least squares fitting, the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are found to be:
[tex]\[
a = 1.0000000000000002, \quad b = 8.0, \quad c = 13.000000000000002
\][/tex]
Thus, the quadratic function that fits the given data points is:
[tex]\[
f(x) = 1.0000000000000002 \cdot x^2 + 8.0 \cdot x + 13.000000000000002
\][/tex]
Simplifying the coefficients (since they are very close to whole numbers due to floating point precision), we can approximate the function as:
[tex]\[
f(x) = x^2 + 8x + 13
\][/tex]
This is the equation of the quadratic function [tex]\( f(x) \)[/tex] that fits the provided data points.
