To complete the table of data points for the function \( y = \sqrt{13x - 3} + 2 \), we will substitute the values of \( x \) into the equation and compute the corresponding \( y \) values.
For \( x = 3 \):
\[ y = \sqrt{13(3) - 3} + 2 \] \[ y = \sqrt{39 - 3} + 2 \] \[ y = \sqrt{36} + 2 \] \[ y = 6 + 2 = 8 \]
For \( x = 4 \):
\[ y = \sqrt{13(4) - 3} + 2 \] \[ y = \sqrt{52 - 3} + 2 \] \[ y = \sqrt{49} + 2 \] \[ y = 7 + 2 = 9 \]
For \( x = 7 \):
\[ y = \sqrt{13(7) - 3} + 2 \] \[ y = \sqrt{91 - 3} + 2 \] \[ y = \sqrt{88} + 2 \] Calculating \( \sqrt{88} \): \[ \sqrt{88} \approx 9.38 , \text{(exactly } 2\sqrt{22} \text{)} \] \[ y \approx 9.38 + 2 \approx 11.38 \]
Completed Table of Data Points
Now we can fill in the values in the table:
\[ \begin{array}{|c|c|} \hline x & y \ \hline 3 & 8 \ 4 & 9 \ 7 & 11.38 \ \hline \end{array} \]
So the completed table is:
\[ \begin{array}{|c|c|} \hline x & y \ \hline 3 & 8 \ 4 & 9 \ 7 & 11.38 \ \hline \end{array} \]