To find a model for the given data using quadratic regression, we'll fit a quadratic equation of the form [tex]\( y = ax^2 + bx + c \)[/tex] to the data points. The given data points are:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline
x & 5 & 7 & 8 & 10 & 11 & 13 & 15 \\
\hline
y & 31 & 51 & 56 & 62 & 61 & 51 & 31 \\
\hline
\end{array}
\][/tex]
The goal is to determine the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]. By performing quadratic regression, we obtain the following coefficients:
[tex]\[ a = -1.2 \][/tex]
[tex]\[ b = 24.7 \][/tex]
[tex]\[ c = -61.9 \][/tex]
These coefficients have already been rounded to the nearest tenth. Therefore, the quadratic model that best fits the given data is:
[tex]\[ y = -1.2x^2 + 24.7x - 61.9 \][/tex]
Putting it all together, the final quadratic equation representing the model is:
[tex]\[
y = -1.2x^2 + 24.7x - 61.9
\][/tex]
