To determine the equation of the quadratic function represented by the table, we assume a standard form of a quadratic function expressed as [tex]\( y = a(x - h)^2 + k \)[/tex]. The parameters [tex]\(a\)[/tex], [tex]\(h\)[/tex], and [tex]\(k\)[/tex] represent the quadratic coefficient, the horizontal shift, and the vertical shift, respectively.
Given the function passes through the points:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-3 & 3.75 \\
\hline
-2 & 4 \\
\hline
-1 & 3.75 \\
\hline
0 & 3 \\
\hline
1 & 1.75 \\
\hline
\end{array}
\][/tex]
We need to find the values of [tex]\(a\)[/tex], [tex]\(h\)[/tex], and [tex]\(k\)[/tex]. By solving the system of equations formed by substituting these points into the quadratic equation, we determine the values of:
[tex]\[
a = -0.25
\][/tex]
[tex]\[
h = -2
\][/tex]
[tex]\[
k = 4
\][/tex]
Thus, the equation of the quadratic function is:
[tex]\[
y = -0.25(x + 2)^2 + 4
\][/tex]
Therefore, the correct answers to fill in the blanks are:
[tex]\[
y = -0.25(x + 2)^2 + 4
\][/tex]
