To determine the quadratic regression equation for the given data set, we proceed as follows:
1. Data Points:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
x & 0 & 2 & 3 & 3 & 5 & 6 & 6 & 9 & 9 & 9 \\
\hline
y & 493 & 500 & 487 & 477 & 452 & 429 & 383 & 324 & 260 & 180 \\
\hline
\end{array}
\][/tex]
2. Quadratic Regression Form:
The general form of a quadratic equation is:
[tex]\[
y = ax^2 + bx + c
\][/tex]
We aim to find the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex].
3. Calculation of Coefficients:
After performing the necessary calculations (i.e., fitting a quadratic curve to the data points), we determine the coefficients as follows:
[tex]\[
a = -4.10134, \quad b = 10.54582, \quad c = 492.13
\][/tex]
4. Forming the Quadratic Equation:
Using the calculated coefficients, we can now form the quadratic regression equation:
[tex]\[
y = -4.10134x^2 + 10.54582x + 492.13
\][/tex]
Thus, the quadratic regression equation for the given data set is:
[tex]\[
\boxed{y = -4.10134x^2 + 10.54582x + 492.13}
\][/tex]
From the given options, the equation that matches is:
[tex]\[
\boxed{y=-4.10134x^2+10.5458x+492.13}
\][/tex]
