A parabola is given by the equation \( y = 2x^2 - 4x + 1 \). What is the vertex of the parabola?
["# Understanding the Vertex of the Parabola Defined by ( y = 2x^2 - 4x + 1 )", "A parabola is a fundamental shape in algebra and geometry, often represented by a quadratic equation in the standard form:\n[ y = ax^2 + bx + c ]\nIn this article, we examine the specific parabola defined by the equation:\n[ y = 2x^2 - 4x + 1 ]\nWe will explore how to determine the vertex — a critical point that reveals the parabola’s maximum or minimum value — and explain why this information is essential for understanding the curve’s behavior.", "---", "## How to Find the Vertex of a Parabola", "The vertex of a parabola in the form ( y = ax^2 + bx + c ) is the turning point that defines its peak or trough. It can be found using two primary methods:", "### 1. Using the Vertex Formula", "For a quadratic equation ( y = ax^2 + bx + c ), the x-coordinate of the vertex is given by:\n[\nx = -\frac{b}{2a}\n]", "From the equation ( y = 2x^2 - 4x + 1 ), we identify the coefficients:\n- ( a = 2 )\n- ( b = -4 )\n- ( c = 1 )", "Substitute into the formula:\n[\nx = -\frac{-4}{2 \cdot 2} = \frac{4}{4} = 1\n]", "Now, substitute ( x = 1 ) back into the original equation to find the y-coordinate:\n[\ny = 2(1)^2 - 4(1) + 1 = 2 - 4 + 1 = -1\n]", "Thus, the vertex is at:\n[\n(1,\ -1)\n]", "---", "### 2. Completing the Square (Alternative Method)", "We can also rewrite the equation in vertex form ( y = a(x - h)^2 + k ) by completing the square.", "Start with:\n[\ny = 2x^2 - 4x + 1\n]", "Factor out 2 from the first two terms:\n[\ny = 2(x^2 - 2x) + 1\n]", "To complete the square inside the parentheses:\n- Take half of (-2), which is (-1), then square it: ((−1)^2 = 1)\n- Add and subtract 1 inside the parentheses:\n[\ny = 2(x^2 - 2x + 1 - 1) + 1 = 2((x - 1)^2 - 1) + 1\n]", "Distribute the 2:\n[\ny = 2(x - 1)^2 - 2 + 1 = 2(x - 1)^2 - 1\n]", "Now the equation is in vertex form:\n[\ny = 2(x - 1)^2 - 1\n]", "From this form, it’s clear the vertex is at:\n[\n(h,\ k) = (1,\ -1)\n]", "---", "## Why the Vertex Matters", "The vertex is crucial for several reasons:", "- Directional Insight: Since the coefficient ( a = 2 ) is positive, the parabola opens upward, making the vertex the minimum point of the graph.\n- Graph Sketching: Knowing the vertex helps accurately sketch the curve, including its axis of symmetry (( x = 1 )) and curvature.\n- Real-World Applications: Parabolas model projectile motion, satellite dishes, and architectural arches — the vertex indicates peak height or optimal focus.", "---", "## Summary", "For the parabola given by ( y = 2x^2 - 4x + 1 ):", "- The vertex is at ( (1,\ -1) )\n- This point is found using either the vertex formula ( x = -\frac{b}{2a} ) or completing the square\n- The parabola opens upward due to ( a = 2 > 0 ), so the vertex represents the lowest point\n- Understanding the vertex is essential for graphing, analysis, and applying quadratic functions in physics, engineering, and economics", "Mastering vertex calculation empowers learners to interpret and utilize quadratic models with confidence. Whether solving equations or modeling real-world phenomena, the vertex remains the key reference point of any parabola."]
