To determine the value of [tex]\(a\)[/tex] in the quadratic function that fits the given data points, follow these steps:
1. Identify the data points:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
0 & -3 \\
\hline
1 & -3.75 \\
\hline
2 & -4 \\
\hline
3 & -3.75 \\
\hline
4 & -3 \\
\hline
5 & -1.75 \\
\hline
\end{array}
\][/tex]
2. Fit a quadratic function [tex]\( y = ax^2 + bx + c \)[/tex] to these data points.
3. Determine the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex].
From the problem's calculations, we have:
- The quadratic polynomial that fits the data points is represented with coefficients approximately found.
- The coefficient for [tex]\(x^2\)[/tex], denoted as [tex]\(a\)[/tex], is approximately [tex]\(0.25\)[/tex].
4. Match the calculated value of [tex]\(a\)[/tex] with the provided options:
[tex]\[ a \approx 0.25 \][/tex]
5. Identify the correct option:
[tex]\[
\begin{aligned}
\text{A. } & \frac{1}{4} \\
\text{B. } & -\frac{1}{2} \\
\text{C. } & -\frac{1}{4} \\
\text{D. } & \frac{1}{2}
\end{aligned}
\][/tex]
Since [tex]\(a \approx 0.25\)[/tex], it corresponds to [tex]\(\frac{1}{4}\)[/tex].
Therefore, the correct answer is [tex]\(\boxed{A}\)[/tex].
