1. **State the problem:** We need to complete the table for the function $$y = x^2 - 2x - 3$$ by calculating the value of $$y$$ for each given $$x$$ value: $$-4, -3, -2, -1, 0, 1, 2$$. 2. **Formula and explanation:** The function is a quadratic equation in the form $$y = ax^2 + bx + c$$ where $$a=1$$, $$b=-2$$, and $$c=-3$$. To find $$y$$ for each $$x$$, substitute $$x$$ into the equation and simplify. 3. **Calculate each value:** - For $$x = -4$$: $$y = (-4)^2 - 2(-4) - 3 = 16 + 8 - 3 = 21$$ - For $$x = -3$$: $$y = (-3)^2 - 2(-3) - 3 = 9 + 6 - 3 = 12$$ - For $$x = -2$$: $$y = (-2)^2 - 2(-2) - 3 = 4 + 4 - 3 = 5$$ - For $$x = -1$$: $$y = (-1)^2 - 2(-1) - 3 = 1 + 2 - 3 = 0$$ - For $$x = 0$$: $$y = 0^2 - 2(0) - 3 = 0 - 0 - 3 = -3$$ - For $$x = 1$$: $$y = 1^2 - 2(1) - 3 = 1 - 2 - 3 = -4$$ - For $$x = 2$$: $$y = 2^2 - 2(2) - 3 = 4 - 4 - 3 = -3$$ 4. **Completed table:** | x | -4 | -3 | -2 | -1 | 0 | 1 | 2 | |----|----|----|----|----|----|----|----| | y | 21 | 12 | 5 | 0 | -3 | -4 | -3 | 5. **Graph description:** The graph is a parabola opening upwards with vertex at $$x = \frac{-b}{2a} = \frac{2}{2} = 1$$. The vertex point is at $$y = 1^2 - 2(1) - 3 = -4$$. The parabola crosses the y-axis at $$y = -3$$ when $$x=0$$. This completes the table and describes the graph of the relation.