The function \( f(x) = 2x^2 - 4x + 1 \) is a quadratic function. Find the vertex using the formula \( x = - rac{b}{2a} \).
["# Finding the Vertex of the Quadratic Function ( f(x) = 2x^2 - 4x + 1 )", "Understanding the behavior of quadratic functions is essential in algebra and calculus, and one of the most important features is the vertex — the turning point of the parabola. In this article, we explore the quadratic function ( f(x) = 2x^2 - 4x + 1 ), and learn how to find its vertex using the standard formula ( x = -\frac{b}{2a} ).", "## What is a Quadratic Function?", "A quadratic function is a second-degree polynomial of the form:\n[\nf(x) = ax^2 + bx + c\n]\nwhere ( a ), ( b ), and ( c ) are constants, and ( a <br/>\ne 0 ). The graph of a quadratic function is a parabola, which opens upwards if ( a > 0 ) and downwards if ( a < 0 ). The vertex represents the minimum or maximum point of the parabola and determines whether the function has a maximum or minimum value.", "## The Standard Formula for the Vertex", "For any quadratic function ( f(x) = ax^2 + bx + c ), the x-coordinate of the vertex is given by:\n[\nx = -\frac{b}{2a}\n]\nOnce this ( x )-value is found, substituting it back into the function gives the y-coordinate of the vertex.", "## Step-by-Step: Finding the Vertex of ( f(x) = 2x^2 - 4x + 1 )", "### Step 1: Identify the coefficients\nFrom the given function:\n[\nf(x) = 2x^2 - 4x + 1\n]\nWe identify:\n- ( a = 2 )\n- ( b = -4 )\n- ( c = 1 )", "### Step 2: Apply the vertex formula\nUsing ( x = -\frac{b}{2a} ):\n[\nx = -\frac{-4}{2 \cdot 2} = \frac{4}{4} = 1\n]", "### Step 3: Find the y-coordinate\nSubstitute ( x = 1 ) back into the function:\n[\nf(1) = 2(1)^2 - 4(1) + 1 = 2 - 4 + 1 = -1\n]\nThus, the y-coordinate is ( -1 ).", "### Step 4: Write the vertex\nThe vertex of the quadratic function is located at:\n[\n\boxed{(1, -1)}\n]", "## Conclusion", "The vertex of the quadratic function ( f(x) = 2x^2 - 4x + 1 ) is at ( (1, -1) ). This point confirms that the parabola opens upwards (since ( a = 2 > 0 )) and the vertex represents the minimum value of the function. Using the formula ( x = -\frac{b}{2a} ) efficiently locates the x-coordinate of the vertex, and substituting this back yields the complete vertex coordinate. Mastering this technique is vital for solving optimization problems and analyzing quadratic behavior in diverse fields such as physics, economics, and engineering."]
