A function \( f(x) = 3x^2 - 12x + 7 \) is given. Find the vertex of this quadratic function.

["Understanding the Vertex of Quadratic Functions: Analyzing ( f(x) = 3x^2 - 12x + 7 )", "When studying quadratic functions, one of the most fundamental concepts is identifying the vertex—the key point that reveals the function’s maximum or minimum value. For the quadratic function ( f(x) = 3x^2 - 12x + 7 ), finding the vertex provides essential insight into its graph’s shape and behavior.", "### What Is a Vertex?", "In a quadratic function expressed in standard form ( f(x) = ax^2 + bx + c ), the vertex is the point ((h, k)) where the parabola reaches its peak (maximum) or trough (minimum). Since the coefficient of ( x^2 ) determines the direction of the parabola, ( f(x) = 3x^2 - 12x + 7 ) opens upwards (because ( a = 3 > 0 )) and thus has a minimum vertex point.", "### How to Find the Vertex", "There are multiple methods to determine the vertex: using the vertex formula, completing the square, or using calculus. Here, we’ll explore the vertex formula, which is both efficient and widely used.", "#### Step 1: Identify coefficients\nFrom ( f(x) = 3x^2 - 12x + 7 ), identify ( a = 3 ), ( b = -12 ), and ( c = 7 ).", "#### Step 2: Use the vertex formula\nThe x-coordinate of the vertex for a quadratic ( ax^2 + bx + c ) is given by:\n[\nh = -\frac{b}{2a}\n]\nSubstituting ( a = 3 ) and ( b = -12 ):\n[\nh = -\frac{-12}{2 \ imes 3} = \frac{12}{6} = 2\n]", "#### Step 3: Find the y-coordinate\nSubstitute ( x = 2 ) into the original function to find ( k ):\n[\nf(2) = 3(2)^2 - 12(2) + 7 = 3(4) - 24 + 7 = 12 - 24 + 7 = -5\n]", "Thus, the vertex is at the point ( (2, -5) ).", "### Why the Vertex Matters", "- Determines the minimum value of the function (since the parabola opens upward).\n- Indicates the axis of symmetry: ( x = 2 ).\n- Essential for graph sketching, optimization problems, and understanding function transformations.", "### Conclusion", "The vertex of the quadratic function ( f(x) = 3x^2 - 12x + 7 ) is ( (2, -5) ). This point not only reveals the function’s lowest value but also serves as a turning point and key reference for analysis. Whether you're solving equations, maximizing profit scenarios, or graphing parabolas, identifying the vertex is foundational. Use the vertex formula or algebraic substitution to confidently locate this pivotal point in any quadratic function.", "---", "Keywords: quadratic function vertex, find vertex of ( ax^2 + bx + c ), function vertex formula, parabola vertex calculation, vertex of ( f(x) = 3x^2 - 12x + 7 ), computational algebra, coordinate geometry."]