To determine which quadratic function best fits the given data points, we can calculate the sum of squared errors (SSE) for each candidate function. The function with the smallest SSE will be the best fit.
Here are the data points provided:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 32 \\
\hline
2 & 78 \\
\hline
3 & 178 \\
\hline
4 & 326 \\
\hline
5 & 390 \\
\hline
6 & 337 \\
\hline
\end{array}
\][/tex]
The candidate quadratic functions are:
1. [tex]\( y = 11.41 x^2 + 154.42 x - 143.9 \)[/tex]
2. [tex]\( y = 11.41 x^2 + 154.42 x + 143.9 \)[/tex]
3. [tex]\( y = -11.41 x^2 + 154.42 x - 143.9 \)[/tex]
4. [tex]\( y = -11.41 x^2 + 154.42 x + 143.9 \)[/tex]
We have the sum of squared errors (SSE) for each function as follows:
1. [tex]\( \text{SSE}_1 = 1,193,750.3863 \)[/tex]
2. [tex]\( \text{SSE}_2 = 2,886,089.2143 \)[/tex]
3. [tex]\( \text{SSE}_3 = 8,921.3815 \)[/tex]
4. [tex]\( \text{SSE}_4 = 505,957.7375 \)[/tex]
The smallest SSE value is [tex]\( \text{SSE}_3 = 8,921.3815 \)[/tex].
Thus, the quadratic function that best fits the given data is:
[tex]\[
y = -11.41 x^2 + 154.42 x - 143.9
\][/tex]
