1. **Problem Statement:** Convert the quadratic equation $y = 2x^2 - 8x + 7$ into graphing form (vertex form) and find the vertex and line of symmetry. 2. **Formula and Rules:** The vertex form of a quadratic is $$y = a(x-h)^2 + k$$ where $(h,k)$ is the vertex. To convert from standard form $ax^2 + bx + c$ to vertex form, complete the square. 3. **Step-by-step Solution:** - Start with the equation: $$y = 2x^2 - 8x + 7$$ - Factor out the coefficient of $x^2$ from the first two terms: $$y = 2(x^2 - 4x) + 7$$ - Complete the square inside the parentheses: Take half of $-4$, which is $-2$, and square it: $(-2)^2 = 4$. - Add and subtract 4 inside the parentheses to keep the equation balanced: $$y = 2(x^2 - 4x + 4 - 4) + 7$$ - Group the perfect square trinomial and simplify: $$y = 2((x - 2)^2 - 4) + 7$$ - Distribute the 2: $$y = 2(x - 2)^2 - 8 + 7$$ - Simplify constants: $$y = 2(x - 2)^2 - 1$$ 4. **Vertex and Line of Symmetry:** - Vertex is at $(2, -1)$. - Line of symmetry is the vertical line $x = 2$. **Final answer:** $$y = 2(x - 2)^2 - 1$$ with vertex $(2, -1)$ and line of symmetry $x = 2$.