To graph the function \( y = \sqrt{-2x + 1} - 3 \), we need to calculate the corresponding \( y \)-values for the given \( x \)-values: -1, 0, and 3.
1. For \( x = -1 \):
\[ y = \sqrt{-2(-1) + 1} - 3 \] \[ = \sqrt{2 + 1} - 3 \] \[ = \sqrt{3} - 3 \approx 1.732 - 3 \approx -1.268 \]
2. For \( x = 0 \):
\[ y = \sqrt{-2(0) + 1} - 3 \] \[ = \sqrt{0 + 1} - 3 \] \[ = \sqrt{1} - 3 \] \[ = 1 - 3 = -2 \]
3. For \( x = 3 \):
\[ y = \sqrt{-2(3) + 1} - 3 \] \[ = \sqrt{-6 + 1} - 3 \] \[ = \sqrt{-5} - 3 \]
Since the square root of a negative number is not defined in the set of real numbers, there is no real value for \( y \) when \( x = 3 \).
Summary of the Table:
| \( x \) | \( y \) | |------------|--------------------------| | -1 | \( \sqrt{3} - 3 \approx -1.268 \) | | 0 | -2 | | 3 | Not defined (Complex) |
Thus, the completed table is:
| \( x \) | \( y \) | |------------|--------------------------| | -1 | \(-1.268\) | | 0 | \(-2\) | | 3 | Not defined |