A quadratic function is given by \( f(x) = 2x^2 - 4x + 1 \). Find the vertex of the parabola.
["# Find the Vertex of the Quadratic Function ( f(x) = 2x^2 - 4x + 1 )", "A quadratic function is a fundamental concept in algebra, often represented in the standard form:\n[ f(x) = ax^2 + bx + c ]\nIn this article, we will analyze the specific quadratic function ( f(x) = 2x^2 - 4x + 1 ) and determine its vertex — a key feature that reveals the peak (or trough) of the parabola.", "---", "## Understanding the Parabola and the Vertex", "The graph of a quadratic function is a parabola, which opens upward if ( a > 0 ) or downward if ( a < 0 ). Since the coefficient ( a = 2 ) is positive, the parabola opens upward, and the vertex represents the minimum point of the function.", "The x-coordinate of the vertex can be found using the formula:\n[ x = -\frac{b}{2a} ]\nwhere ( a ), ( b ), and ( c ) are the coefficients from ( f(x) = ax^2 + bx + c ).", "---", "## Step-by-Step: Finding the Vertex", "Given:\n[ f(x) = 2x^2 - 4x + 1 ]\nHere, ( a = 2 ), ( b = -4 ), and ( c = 1 ).", "1. Calculate the x-coordinate of the vertex\n[ x = -\frac{b}{2a} = -\frac{-4}{2 \cdot 2} = \frac{4}{4} = 1 ]", "2. Substitute ( x = 1 ) into the function to find the y-coordinate\n[ f(1) = 2(1)^2 - 4(1) + 1 = 2 - 4 + 1 = -1 ]", "---", "## Vertex Coordinates", "The vertex of the parabola is therefore the point:\n[ (1, -1) ]", "---", "## Visual and Practical Insight", "Plotting this parabola, the vertex at ( (1, -1) ) lies at the lowest point of the curve, confirming it is a minimum. Understanding this vertex helps solve optimization problems, model real-world phenomena like projectile motion, and analyze cost or revenue functions in economics.", "---", "## Summary", "- Function: ( f(x) = 2x^2 - 4x + 1 )\n- a = 2 (positive → upward opening parabola)\n- Vertex: ( \boxed{(1,\ -1)} )", "By using the vertex formula ( x = -\frac{b}{2a} ), we efficiently locate this critical point and deepen our insight into quadratic behavior.", "---", "Keywords for SEO: quadratic function vertex, find vertex of ( f(x) = 2x^2 - 4x + 1 ), vertex formula, parabola vertex, math tutorial quadratic, solve quadratic vertex."]
