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What is the equation of the quadratic function represented by this table?

[tex]\[
\begin{tabular}{|c|c|}
\hline
x & y \\
\hline
-3 & 3.75 \\
\hline
-2 & 4 \\
\hline
-1 & 3.75 \\
\hline
0 & 3 \\
\hline
1 & 1.75 \\
\hline
\end{tabular}
\][/tex]

[tex]\[
y = \square(x - \square)^2 + \square
\][/tex]

Sagot :

GinnyAnsweridn
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]
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