1. **State the problem:** We are given the quadratic function $$y = 2x^2 - 8x + 3$$ and want to analyze its properties, including vertex and points on the curve. 2. **Formula and rules:** The vertex of a parabola given by $$y = ax^2 + bx + c$$ is at $$x = -\frac{b}{2a}$$. The parabola opens upwards if $$a > 0$$ and downwards if $$a < 0$$. 3. **Calculate the vertex:** Here, $$a = 2$$ and $$b = -8$$. $$x_{vertex} = -\frac{-8}{2 \times 2} = \frac{8}{4} = 2$$ Substitute $$x=2$$ into the function to find $$y_{vertex}$$: $$y = 2(2)^2 - 8(2) + 3 = 2 \times 4 - 16 + 3 = 8 - 16 + 3 = -5$$ So, the vertex is at $$(2, -5)$$. 4. **Check the point near (5,5):** Substitute $$x=5$$: $$y = 2(5)^2 - 8(5) + 3 = 2 \times 25 - 40 + 3 = 50 - 40 + 3 = 13$$ The point (5,5) is not exactly on the curve; the actual point is (5,13). 5. **Summary:** The parabola opens upwards (since $$a=2>0$$), has vertex at $$(2,-5)$$, and passes through points like $$(0,3)$$ and $$(5,13)$$. **Final answer:** Vertex at $$(2,-5)$$, parabola opens upwards, equation $$y=2x^2 - 8x + 3$$.