To determine the equation of the quadratic function [tex]\( f(x) \)[/tex], we need to find the coefficients of the quadratic polynomial [tex]\( f(x) = ax^2 + bx + c \)[/tex].
Given the provided table of values:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-8 & 7 \\
-7 & 2 \\
-6 & -1 \\
-5 & -2 \\
-4 & -1 \\
-3 & 2 \\
-2 & 7 \\
-1 & 14 \\
0 & 23 \\
\hline
\end{array}
\][/tex]
Let's fit a quadratic function to these values. We have the general form of the quadratic function:
[tex]\[
f(x) = ax^2 + bx + c
\][/tex]
The process of fitting the quadratic function yields the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
- [tex]\( a \)[/tex] is the coefficient of [tex]\( x^2 \)[/tex]
- [tex]\( b \)[/tex] is the coefficient of [tex]\( x \)[/tex]
- [tex]\( c \)[/tex] is the constant term.
After determining the best fit for the coefficients, we find:
[tex]\[
a \approx 1.0, \quad b \approx 10.0, \quad c \approx 23.0
\][/tex]
Therefore, the quadratic function [tex]\( f(x) \)[/tex] that best fits the given data points is:
[tex]\[
f(x) = 1.0x^2 + 10.0x + 23.0
\][/tex]
Simplifying the coefficients to just integers for clarity, we can write:
[tex]\[
f(x) = x^2 + 10x + 23
\][/tex]
Thus, the equation of [tex]\( f(x) \)[/tex] is:
[tex]\[
f(x) = x^2 + 10x + 23
\][/tex]
