To complete the table of data points for the function \( y = \sqrt{13x - 3} + 2 \), we will calculate the values of \( y \) for the given values of \( x \): 3, 4, and 7.
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For \( x = 3 \): \[ y = \sqrt{13(3) - 3} + 2 = \sqrt{39 - 3} + 2 = \sqrt{36} + 2 = 6 + 2 = 8 \] So, the point is \( (3, 8) \).
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For \( x = 4 \): \[ y = \sqrt{13(4) - 3} + 2 = \sqrt{52 - 3} + 2 = \sqrt{49} + 2 = 7 + 2 = 9 \] So, the point is \( (4, 9) \).
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For \( x = 7 \): \[ y = \sqrt{13(7) - 3} + 2 = \sqrt{91 - 3} + 2 = \sqrt{88} + 2 \approx 9.38 + 2 = 11.38 \] So, the point is \( (7, \sqrt{88} + 2) \approx (7, 11.38) \).
Now we can summarize the points:
\[ \begin{array}{|c|c|} \hline x & y \ \hline 3 & 8 \ 4 & 9 \ 7 & 11.38 \ \hline \end{array} \]
So, the completed table of data points is:
- \( (3, 8) \)
- \( (4, 9) \)
- \( (7, 11.38) \)
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