Let's find the equation of the given data set in the form [tex]\( y = a(x - b)^2 + c \)[/tex].
Given 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 know that the form of the equation we are looking for is:
[tex]\[
y = a(x - b)^2 + c
\][/tex]
By examining the data and fitting the best curve, we find the parameters [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] that fit the data points. Through this process, we determine:
- [tex]\( a = -0.25 \)[/tex]
- [tex]\( b = -2.0 \)[/tex]
- [tex]\( c = 4.0 \)[/tex]
Thus, substituting these values into the equation, we get:
[tex]\[
y = -0.25(x + 2)^2 + 4
\][/tex]
This is the equation that represents the given data points.
