To determine which analytical model best represents the given data, we will compare the calculated values from each model to the given [tex]\(y\)[/tex] values in the table.
Given data:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-6 & 252 \\
-5 & 175 \\
-4 & 112 \\
-3 & 63 \\
-2 & 28 \\
-1 & 7 \\
\hline
\end{array}
\][/tex]
Let's consider each proposed model:
1. Model: [tex]\( y = 7x^2 \)[/tex]
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-6 & 7(-6)^2 = 252 \\
-5 & 7(-5)^2 = 175 \\
-4 & 7(-4)^2 = 112 \\
-3 & 7(-3)^2 = 63 \\
-2 & 7(-2)^2 = 28 \\
-1 & 7(-1)^2 = 7 \\
\hline
\end{array}
\][/tex]
The values calculated using [tex]\( y = 7x^2 \)[/tex] perfectly match the given values.
2. Model: [tex]\( y = 252 \times 7^x \)[/tex]
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-6 & 252 \times 7^{-6} \approx 0.0021 \\
-5 & 252 \times 7^{-5} \approx 0.015 \\
-4 & 252 \times 7^{-4} \approx 0.105 \\
-3 & 252 \times 7^{-3} \approx 0.735 \\
-2 & 252 \times 7^{-2} \approx 5.143 \\
-1 & 252 \times 7^{-1} \approx 36.0 \\
\hline
\end{array}
\][/tex]
The calculated values significantly deviate from the given values.
3. Model: [tex]\( y = 7x + 252 \)[/tex]
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-6 & 7(-6) + 252 = 210 \\
-5 & 7(-5) + 252 = 217 \\
-4 & 7(-4) + 252 = 224 \\
-3 & 7(-3) + 252 = 231 \\
-2 & 7(-2) + 252 = 238 \\
-1 & 7(-1) + 252 = 245 \\
\hline
\end{array}
\][/tex]
The calculated values deviate significantly from the given values.
4. Model: [tex]\( y = 7x^2 - 5x + 28 \)[/tex]
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-6 & 7(-6)^2 - 5(-6) + 28 = 310 \\
-5 & 7(-5)^2 - 5(-5) + 28 = 228 \\
-4 & 7(-4)^2 - 5(-4) + 28 = 160 \\
-3 & 7(-3)^2 - 5(-3) + 28 = 106 \\
-2 & 7(-2)^2 - 5(-2) + 28 = 66 \\
-1 & 7(-1)^2 - 5(-1) + 28 = 40 \\
\hline
\end{array}
\][/tex]
The calculated values deviate from the given values.
From this analysis, we can see that the model [tex]\( y = 7x^2 \)[/tex] provides values that exactly match the given data. Therefore, the best analytical model that represents the data in the table is:
[tex]\[ y = 7x^2 \][/tex]
