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Sagot :
To determine which equation best fits the given set of data, we must compare the y-values generated by each equation with the provided y-values and assess which equation yields the smallest errors. This can be done by calculating the predicted y-values for each candidate equation and comparing them to the given y-values using the sum of squared errors (SSE).
Here are the candidate equations:
A. \( y = 11 \sqrt{x - 0.3} + 4.3 \)
B. \( y = 2x + 17 \)
C. \( y = 11 \sqrt{x + 0.3} - 4.3 \)
D. \( y = 2x - 17 \)
Let's calculate the predicted y-values for each equation:
### Equation A: \( y = 11 \sqrt{x - 0.3} + 4.3 \)
[tex]\[ \begin{align*} y(0) &= 11 \sqrt{0 - 0.3} + 4.3 &\approx \text{undefined (negative inside square root)} \\ y(2) &= 11 \sqrt{2 - 0.3} + 4.3 &= 11 \sqrt{1.7} + 4.3 &\approx 18.92 \\ y(4) &= 11 \sqrt{4 - 0.3} + 4.3 &= 11 \sqrt{3.7} + 4.3 &\approx 25.16 \\ y(6) &= 11 \sqrt{6 - 0.3} + 4.3 &= 11 \sqrt{5.7} + 4.3 &\approx 29.57 \\ y(8) &= 11 \sqrt{8 - 0.3} + 4.3 &= 11 \sqrt{7.7} + 4.3 &\approx 33.06 \\ y(10) &= 11 \sqrt{10 - 0.3} + 4.3 &= 11 \sqrt{9.7} + 4.3 &\approx 36.19 \\ y(12) &= 11 \sqrt{12 - 0.3} + 4.3 &= 11 \sqrt{11.7} + 4.3 &\approx 38.84 \\ y(14) &= 11 \sqrt{14 - 0.3} + 4.3 &= 11 \sqrt{13.7} + 4.3 &\approx 41.18 \\ y(16) &= 11 \sqrt{16 - 0.3} + 4.3 &= 11 \sqrt{15.7} + 4.3 &\approx 43.33 \\ y(18) &= 11 \sqrt{18 - 0.3} + 4.3 &= 11 \sqrt{17.7} + 4.3 &\approx 45.30 \\ \end{align*} \][/tex]
Already, we can notice a significant discrepancy: the predicted values do not match closely with the data points.
### Equation B: \( y = 2x + 17 \)
[tex]\[ \begin{align*} y(0) &= 2(0) + 17 &= 17 \\ y(2) &= 2(2) + 17 &= 21 \\ y(4) &= 2(4) + 17 &= 25 \\ y(6) &= 2(6) + 17 &= 29 \\ y(8) &= 2(8) + 17 &= 33 \\ y(10) &= 2(10) + 17 &= 37 \\ y(12) &= 2(12) + 17 &= 41 \\ y(14) &= 2(14) + 17 &= 45 \\ y(16) &= 2(16) + 17 &= 49 \\ y(18) &= 2(18) + 17 &= 53 \\ \end{align*} \][/tex]
This equation's predicted values align very closely with the provided y-values, suggesting a strong fit.
### Equation C: \( y = 11 \sqrt{x + 0.3} - 4.3 \)
[tex]\[ \begin{align*} y(0) &= 11 \sqrt{0 + 0.3} - 4.3 &= 11 \sqrt{0.3} - 4.3 &\approx 1.71 \\ y(2) &= 11 \sqrt{2 + 0.3} - 4.3 &= 11 \sqrt{2.3} - 4.3 &\approx 8.41 \\ y(4) &= 11 \sqrt{4 + 0.3} - 4.3 &= 11 \sqrt{4.3} - 4.3 &\approx 18.46 \\ y(6) &= 11 \sqrt{6 + 0.3} - 4.3 &= 11 \sqrt{6.3} - 4.3 &\approx 25.35 \\ y(8) &= 11 \sqrt{8 + 0.3} - 4.3 &= 11 \sqrt{8.3} - 4.3 &\approx 31.12 \\ y(10) &= 11 \sqrt{10 + 0.3} - 4.3 &= 11 \sqrt{10.3} - 4.3 &\approx 35.79 \\ y(12) &= 11 \sqrt{12 + 0.3} - 4.3 &= 11 \sqrt{12.3} - 4.3 &\approx 39.63 \\ y(14) &= 11 \sqrt{14 + 0.3} - 4.3 &= 11 \sqrt{14.3} - 4.3 &\approx 42.85 \\ y(16) &= 11 \sqrt{16 + 0.3} - 4.3 &= 11 \sqrt{16.3} - 4.3 &\approx 45.59 \\ y(18) &= 11 \sqrt{18 + 0.3} - 4.3 &= 11 \sqrt{18.3} - 4.3 &\approx 47.88 \\ \end{align*} \][/tex]
This equation's predicted values are somewhat close but less accurate compared to Equation B.
### Equation D: \( y = 2x - 17 \)
[tex]\[ \begin{align*} y(0) &= 2(0) - 17 &= -17 \\ y(2) &= 2(2) - 17 &= -13 \\ y(4) &= 2(4) - 17 &= -9 \\ y(6) &= 2(6) - 17 &= -5 \\ y(8) &= 2(8) - 17 &= -1 \\ y(10) &= 2(10) - 17 &= 3 \\ y(12) &= 2(12) - 17 &= 7 \\ y(14) &= 2(14) - 17 &= 11 \\ y(16) &= 2(16) - 17 &= 15 \\ y(18) &= 2(18) - 17 &= 19 \\ \end{align*} \][/tex]
Clearly, this equation does not fit the data at all.
### Conclusion
Based on the predicted values, Equation B, \( y = 2x + 17 \), is the best fit for the given data set. The predicted values very closely match the provided y-values.
Thus, the correct answer is:
B. [tex]\( y = 2x + 17 \)[/tex]
Here are the candidate equations:
A. \( y = 11 \sqrt{x - 0.3} + 4.3 \)
B. \( y = 2x + 17 \)
C. \( y = 11 \sqrt{x + 0.3} - 4.3 \)
D. \( y = 2x - 17 \)
Let's calculate the predicted y-values for each equation:
### Equation A: \( y = 11 \sqrt{x - 0.3} + 4.3 \)
[tex]\[ \begin{align*} y(0) &= 11 \sqrt{0 - 0.3} + 4.3 &\approx \text{undefined (negative inside square root)} \\ y(2) &= 11 \sqrt{2 - 0.3} + 4.3 &= 11 \sqrt{1.7} + 4.3 &\approx 18.92 \\ y(4) &= 11 \sqrt{4 - 0.3} + 4.3 &= 11 \sqrt{3.7} + 4.3 &\approx 25.16 \\ y(6) &= 11 \sqrt{6 - 0.3} + 4.3 &= 11 \sqrt{5.7} + 4.3 &\approx 29.57 \\ y(8) &= 11 \sqrt{8 - 0.3} + 4.3 &= 11 \sqrt{7.7} + 4.3 &\approx 33.06 \\ y(10) &= 11 \sqrt{10 - 0.3} + 4.3 &= 11 \sqrt{9.7} + 4.3 &\approx 36.19 \\ y(12) &= 11 \sqrt{12 - 0.3} + 4.3 &= 11 \sqrt{11.7} + 4.3 &\approx 38.84 \\ y(14) &= 11 \sqrt{14 - 0.3} + 4.3 &= 11 \sqrt{13.7} + 4.3 &\approx 41.18 \\ y(16) &= 11 \sqrt{16 - 0.3} + 4.3 &= 11 \sqrt{15.7} + 4.3 &\approx 43.33 \\ y(18) &= 11 \sqrt{18 - 0.3} + 4.3 &= 11 \sqrt{17.7} + 4.3 &\approx 45.30 \\ \end{align*} \][/tex]
Already, we can notice a significant discrepancy: the predicted values do not match closely with the data points.
### Equation B: \( y = 2x + 17 \)
[tex]\[ \begin{align*} y(0) &= 2(0) + 17 &= 17 \\ y(2) &= 2(2) + 17 &= 21 \\ y(4) &= 2(4) + 17 &= 25 \\ y(6) &= 2(6) + 17 &= 29 \\ y(8) &= 2(8) + 17 &= 33 \\ y(10) &= 2(10) + 17 &= 37 \\ y(12) &= 2(12) + 17 &= 41 \\ y(14) &= 2(14) + 17 &= 45 \\ y(16) &= 2(16) + 17 &= 49 \\ y(18) &= 2(18) + 17 &= 53 \\ \end{align*} \][/tex]
This equation's predicted values align very closely with the provided y-values, suggesting a strong fit.
### Equation C: \( y = 11 \sqrt{x + 0.3} - 4.3 \)
[tex]\[ \begin{align*} y(0) &= 11 \sqrt{0 + 0.3} - 4.3 &= 11 \sqrt{0.3} - 4.3 &\approx 1.71 \\ y(2) &= 11 \sqrt{2 + 0.3} - 4.3 &= 11 \sqrt{2.3} - 4.3 &\approx 8.41 \\ y(4) &= 11 \sqrt{4 + 0.3} - 4.3 &= 11 \sqrt{4.3} - 4.3 &\approx 18.46 \\ y(6) &= 11 \sqrt{6 + 0.3} - 4.3 &= 11 \sqrt{6.3} - 4.3 &\approx 25.35 \\ y(8) &= 11 \sqrt{8 + 0.3} - 4.3 &= 11 \sqrt{8.3} - 4.3 &\approx 31.12 \\ y(10) &= 11 \sqrt{10 + 0.3} - 4.3 &= 11 \sqrt{10.3} - 4.3 &\approx 35.79 \\ y(12) &= 11 \sqrt{12 + 0.3} - 4.3 &= 11 \sqrt{12.3} - 4.3 &\approx 39.63 \\ y(14) &= 11 \sqrt{14 + 0.3} - 4.3 &= 11 \sqrt{14.3} - 4.3 &\approx 42.85 \\ y(16) &= 11 \sqrt{16 + 0.3} - 4.3 &= 11 \sqrt{16.3} - 4.3 &\approx 45.59 \\ y(18) &= 11 \sqrt{18 + 0.3} - 4.3 &= 11 \sqrt{18.3} - 4.3 &\approx 47.88 \\ \end{align*} \][/tex]
This equation's predicted values are somewhat close but less accurate compared to Equation B.
### Equation D: \( y = 2x - 17 \)
[tex]\[ \begin{align*} y(0) &= 2(0) - 17 &= -17 \\ y(2) &= 2(2) - 17 &= -13 \\ y(4) &= 2(4) - 17 &= -9 \\ y(6) &= 2(6) - 17 &= -5 \\ y(8) &= 2(8) - 17 &= -1 \\ y(10) &= 2(10) - 17 &= 3 \\ y(12) &= 2(12) - 17 &= 7 \\ y(14) &= 2(14) - 17 &= 11 \\ y(16) &= 2(16) - 17 &= 15 \\ y(18) &= 2(18) - 17 &= 19 \\ \end{align*} \][/tex]
Clearly, this equation does not fit the data at all.
### Conclusion
Based on the predicted values, Equation B, \( y = 2x + 17 \), is the best fit for the given data set. The predicted values very closely match the provided y-values.
Thus, the correct answer is:
B. [tex]\( y = 2x + 17 \)[/tex]
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