To determine which of the given equations matches the data points from the table, we will evaluate each equation at the provided [tex]\( x \)[/tex]-values and compare the results to the corresponding [tex]\( y \)[/tex]-values.
Given data points:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-5 & -23 \\
\hline
-4 & -13 \\
\hline
-3 & -7 \\
\hline
-2 & -5 \\
\hline
-1 & -7 \\
\hline
0 & -13 \\
\hline
1 & -23 \\
\hline
\end{array}
\][/tex]
Let's evaluate each equation—one by one—with these [tex]\( x \)[/tex]-values:
1. Equation 1: [tex]\( y = -2(x + 2)^2 - 5 \)[/tex]
[tex]\[
\begin{align*}
y(-5) &= -2(-5 + 2)^2 - 5 = -2(-3)^2 - 5 = -2(9) - 5 = -18 - 5 = -23 \\
y(-4) &= -2(-4 + 2)^2 - 5 = -2(-2)^2 - 5 = -2(4) - 5 = -8 - 5 = -13 \\
y(-3) &= -2(-3 + 2)^2 - 5 = -2(-1)^2 - 5 = -2(1) - 5 = -2 - 5 = -7 \\
y(-2) &= -2(-2 + 2)^2 - 5 = -2(0)^2 - 5 = -2(0) - 5 = -5 \\
y(-1) &= -2(-1 + 2)^2 - 5 = -2(1)^2 - 5 = -2(1) - 5 = -2 - 5 = -7 \\
y(0) &= -2(0 + 2)^2 - 5 = -2(2)^2 - 5 = -2(4) - 5 = -8 - 5 = -13 \\
y(1) &= -2(1 + 2)^2 - 5 = -2(3)^2 - 5 = -2(9) - 5 = -18 - 5 = -23 \\
\end{align*}
\][/tex]
Results: [tex]\([-23, -13, -7, -5, -7, -13, -23]\)[/tex]
2. Equation 2: [tex]\( y = -2(x - 2)^2 + 5 \)[/tex]
[tex]\[
\begin{align*}
y(-5) &= -2(-5 - 2)^2 + 5 = -2(-7)^2 + 5 = -2(49) + 5 = -98 + 5 = -93 \\
y(-4) &= -2(-4 - 2)^2 + 5 = -2(-6)^2 + 5 = -2(36) + 5 = -72 + 5 = -67 \\
y(-3) &= -2(-3 - 2)^2 + 5 = -2(-5)^2 + 5 = -2(25) + 5 = -50 + 5 = -45 \\
y(-2) &= -2(-2 - 2)^2 + 5 = -2(-4)^2 + 5 = -2(16) + 5 = -32 + 5 = -27 \\
y(-1) &= -2(-1 - 2)^2 + 5 = -2(-3)^2 + 5 = -2(9) + 5 = -18 + 5 = -13 \\
y(0) &= -2(0 - 2)^2 + 5 = -2(2)^2 + 5 = -2(4) + 5 = -8 + 5 = -3 \\
y(1) &= -2(1 - 2)^2 + 5 = -2(-1)^2 + 5 = -2(1) + 5 = -2 + 5 = 3 \\
\end{align*}
\][/tex]
Results: [tex]\([-93, -67, -45, -27, -13, -3, 3]\)[/tex]
3. Equation 3: [tex]\( y = -(x + 2)^2 - 5 \)[/tex]
[tex]\[
\begin{align*}
y(-5) &= -(-5 + 2)^2 - 5 = -(-3)^2 - 5 = -(9) - 5 = -9 - 5 = -14 \\
y(-4) &= -(-4 + 2)^2 - 5 = -(-2)^2 - 5 = -(4) - 5 = -4 - 5 = -9 \\
y(-3) &= -(-3 + 2)^2 - 5 = -(-1)^2 - 5 = -(1) - 5 = -1 - 5 = -6 \\
y(-2) &= -(-2 + 2)^2 - 5 = -(0)^2 - 5 = -(0) - 5 = -5 \\
y(-1) &= -(-1 + 2)^2 - 5 = -(1)^2 - 5 = -(1) - 5 = -1 - 5 = -6 \\
y(0) &= -(0 + 2)^2 - 5 = -(2)^2 - 5 = -(4) - 5 = -4 - 5 = -9 \\
y(1) &= -(1 + 2)^2 - 5 = -(3)^2 - 5 = -(9) - 5 = -9 - 5 = -14 \\
\end{align*}
\][/tex]
Results: [tex]\([-14, -9, -6, -5, -6, -9, -14]\)[/tex]
4. Equation 4: [tex]\( y = -(x - 2)^2 + 5 \)[/tex]
[tex]\[
\begin{align*}
y(-5) &= -(-5 - 2)^2 + 5 = -(-7)^2 + 5 = -(49) + 5 = -49 + 5 = -44 \\
y(-4) &= -(-4 - 2)^2 + 5 = -(-6)^2 + 5 = -(36) + 5 = -36 + 5 = -31 \\
y(-3) &= -(-3 - 2)^2 + 5 = -(-5)^2 + 5 = -(25) + 5 = -25 + 5 = -20 \\
y(-2) &= -(-2 - 2)^2 + 5 = -(-4)^2 + 5 = -(16) + 5 = -16 + 5 = -11 \\
y(-1) &= -(-1 - 2)^2 + 5 = -(-3)^2 + 5 = -(9) + 5 = -9 + 5 = -4 \\
y(0) &= -(0 - 2)^2 + 5 = -(2)^2 + 5 = -(4) + 5 = -4 + 5 = 1 \\
y(1) &= -(1 - 2)^2 + 5 = -(-1)^2 + 5 = -(1) + 5 = -1 + 5 = 4 \\
\end{align*}
\][/tex]
Results: [tex]\([-44, -31, -20, -11, -4, 1, 4]\)[/tex]
Comparing the calculated [tex]\( y \)[/tex]-values to the given [tex]\( y \)[/tex]-values [tex]\([-23, -13, -7, -5, -7, -13, -23]\)[/tex], we find that only the equation [tex]\( y = -2(x + 2)^2 - 5 \)[/tex] produces the correct results for all [tex]\( x \)[/tex]-values.
Hence, the equation of the quadratic function is:
[tex]\[ y = -2(x + 2)^2 - 5 \][/tex]
