To find the quadratic regression equation that fits the given data points, we need to determine the quadratic polynomial [tex]\( y = ax^2 + bx + c \)[/tex] that best represents the relationship between the independent variable [tex]\( x \)[/tex] and the dependent variable [tex]\( y \)[/tex].
Here are the steps to derive the quadratic regression equation:
1. Gather Data: Collect the data points:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-4 & 40 \\
\hline
-3 & 28 \\
\hline
-2 & 10 \\
\hline
-1 & 8 \\
\hline
0 & 7 \\
\hline
1 & 10 \\
\hline
2 & 16 \\
\hline
3 & 26 \\
\hline
4 & 40 \\
\hline
\end{array}
\][/tex]
2. Set Up the Quadratic Regression: The quadratic regression model is [tex]\( y = ax^2 + bx + c \)[/tex].
3. Calculate the Coefficients: Based on statistical computations (using methods such as least squares), we determine the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex].
The computed coefficients for this problem are:
[tex]\[ a = 2.13 \][/tex]
[tex]\[ b = 0.13 \][/tex]
[tex]\[ c = 6.39 \][/tex]
4. Construct the Equation: Plug the coefficients into the quadratic equation:
[tex]\[ y = 2.13x^2 + 0.13x + 6.39 \][/tex]
Therefore, the quadratic regression equation that fits the given data points is:
[tex]\[ y = 2.13x^2 + 0.13x + 6.39 \][/tex]
Looking at the answer choices provided:
A. [tex]\( y = 0.82x^2 + 0.78x + 7.23 \)[/tex]
B. [tex]\( y = 1.64x^2 + 1.24x + 8.08 \)[/tex]
C. [tex]\( y = 2.13x^2 + 0.13x + 6.39 \)[/tex]
D. [tex]\( y = 16.76 \cdot 1.02^x \)[/tex]
The correct answer is:
[tex]\[ \boxed{C. \, y = 2.13x^2 + 0.13x + 6.39} \][/tex]
