To find the quadratic function [tex]\( f(x) \)[/tex] that best fits the given data points:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-2 & 48 \\
\hline
-1 & 50 \\
\hline
0 & 48 \\
\hline
1 & 42 \\
\hline
2 & 32 \\
\hline
3 & 18 \\
\hline
4 & 0 \\
\hline
\end{array}
\][/tex]
we can model the function in the quadratic form:
[tex]\[
f(x) = ax^2 + bx + c
\][/tex]
Here are the determined coefficients for this quadratic equation:
- Coefficient [tex]\( a \)[/tex]: [tex]\(-2.000000000000003\)[/tex]
- Coefficient [tex]\( b \)[/tex]: [tex]\(-4.000000000000001\)[/tex]
- Constant term [tex]\( c \)[/tex]: [tex]\(48.000000000000036\)[/tex]
So, substituting these coefficients into the general quadratic form, we get the equation:
[tex]\[
f(x) = -2.000000000000003x^2 + -4.000000000000001x + 48.000000000000036
\][/tex]
For simplicity, we can round the coefficients to make the equation more readable while preserving its accuracy:
[tex]\[
f(x) = -2x^2 - 4x + 48
\][/tex]
Therefore, the equation that represents the function [tex]\( f(x) \)[/tex] fitting the given data points is:
[tex]\[
f(x) = -2x^2 - 4x + 48
\][/tex]
