1. **State the problem:** We need to graph the parabola given by the equation $$y = x^2 - 2x + 3$$ and plot five points: the vertex, two points to the left of the vertex, and two points to the right of the vertex. 2. **Find the vertex:** The vertex of a parabola given by $$y = ax^2 + bx + c$$ is at $$x = -\frac{b}{2a}$$. Here, $$a = 1$$ and $$b = -2$$, so $$x = -\frac{-2}{2 \times 1} = \frac{2}{2} = 1$$. 3. **Calculate the y-coordinate of the vertex:** Substitute $$x = 1$$ into the equation: $$y = (1)^2 - 2(1) + 3 = 1 - 2 + 3 = 2$$. So, the vertex is at $$(1, 2)$$. 4. **Find two points to the left of the vertex:** Choose $$x = 0$$ and $$x = -1$$. For $$x = 0$$: $$y = 0^2 - 2(0) + 3 = 3$$ For $$x = -1$$: $$y = (-1)^2 - 2(-1) + 3 = 1 + 2 + 3 = 6$$ Points: $$(0, 3)$$ and $$(-1, 6)$$. 5. **Find two points to the right of the vertex:** Choose $$x = 2$$ and $$x = 3$$. For $$x = 2$$: $$y = 2^2 - 2(2) + 3 = 4 - 4 + 3 = 3$$ For $$x = 3$$: $$y = 3^2 - 2(3) + 3 = 9 - 6 + 3 = 6$$ Points: $$(2, 3)$$ and $$(3, 6)$$. 6. **Summary of points:** - Vertex: $$(1, 2)$$ - Left points: $$(0, 3)$$, $$(-1, 6)$$ - Right points: $$(2, 3)$$, $$(3, 6)$$ These points can be plotted on the Cartesian coordinate system to graph the parabola. **Final answer:** The parabola $$y = x^2 - 2x + 3$$ has vertex at $$(1, 2)$$ and passes through points $$(0, 3)$$, $$(-1, 6)$$, $$(2, 3)$$, and $$(3, 6)$$.