To determine the value of [tex]\( a \)[/tex] in the quadratic function's equation [tex]\( y = ax^2 + bx + c \)[/tex], we follow these steps:
1. Identify and list the 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. Calculate the first differences:
[tex]\[
\begin{aligned}
\Delta_1 &= -3.75 - (-3) = -0.75 \\
\Delta_2 &= -4 - (-3.75) = -0.25 \\
\Delta_3 &= -3.75 - (-4) = 0.25 \\
\Delta_4 &= -3 - (-3.75) = 0.75 \\
\Delta_5 &= -1.75 - (-3) = 1.25 \\
\end{aligned}
\][/tex]
3. Calculate the second differences:
[tex]\[
\begin{aligned}
\Delta^2_1 &= -0.25 - (-0.75) = 0.5 \\
\Delta^2_2 &= 0.25 - (-0.25) = 0.5 \\
\Delta^2_3 &= 0.75 - 0.25 = 0.5 \\
\Delta^2_4 &= 1.25 - 0.75 = 0.5 \\
\end{aligned}
\][/tex]
You can observe that all the second differences are constant and equal to [tex]\( 0.5 \)[/tex].
4. Find the coefficient [tex]\( a \)[/tex] from the second differences:
- For a quadratic function [tex]\( y = ax^2 + bx + c \)[/tex], the second differences are equal to [tex]\( 2a \)[/tex].
- Given that the second differences are [tex]\( 0.5 \)[/tex], we can set up the equation:
[tex]\[
2a = 0.5
\][/tex]
- Solving for [tex]\( a \)[/tex]:
[tex]\[
a = \frac{0.5}{2} = 0.25
\][/tex]
Therefore, the value of [tex]\( a \)[/tex] in the function's equation is [tex]\( \boxed{\frac{1}{4}} \)[/tex]. Thus, the correct answer is [tex]\( \text{C.} \frac{1}{4} \)[/tex].
