Quadratic Graph 6A274E

1. **State the problem:** We are given the quadratic equation $$y = 2x^2 - 8x + 3$$ and asked to describe its graph. 2. **Identify the type of graph:** This is a quadratic function, so its graph is a parabola. 3. **Determine the direction of the parabola:** The coefficient of $x^2$ is $2$, which is positive, so the parabola opens upward. 4. **Find the vertex:** The vertex of a parabola given by $$y = ax^2 + bx + c$$ is at $$x = -\frac{b}{2a}$$. Here, $a = 2$ and $b = -8$, so $$x = -\frac{-8}{2 \times 2} = \frac{8}{4} = 2.$$ 5. **Calculate the $y$-coordinate of the vertex:** Substitute $x = 2$ into the equation: $$y = 2(2)^2 - 8(2) + 3 = 2 \times 4 - 16 + 3 = 8 - 16 + 3 = -5.$$ 6. **Vertex coordinates:** The vertex is at $(2, -5)$. 7. **Summary:** The graph is an upward-opening parabola with vertex at $(2, -5)$. This matches the description given: the parabola is centered near the middle of the image with the vertex at $x=2$, $y=-5$.