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Sagot :
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]
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]
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