The function \( f(x) = 3x^2 - 12x + 7 \) represents a parabola. Find the vertex of the parabola.

["# Finding the Vertex of the Parabola Defined by ( f(x) = 3x^2 - 12x + 7 )", "Understanding the graph of a quadratic function is essential for solving a wide range of problems in mathematics, physics, and engineering. Among these, one of the most important features is the vertex of the parabola—the turning point that reveals the function’s maximum or minimum value. In this article, we’ll explore how to find the vertex of the parabola represented by the quadratic function ( f(x) = 3x^2 - 12x + 7 ), using key concepts from algebra and vertex form.", "## What Does the Function ( f(x) = 3x^2 - 12x + 7 ) Represent?", "The expression ( f(x) = 3x^2 - 12x + 7 ) is a quadratic function, and its graph is a parabola. Since the coefficient of ( x^2 ) is positive (( a = 3 > 0 )), the parabola opens upwards, meaning the vertex represents the minimum point of the function.", "## The Vertex of a Parabola: General Method", "The vertex of a parabola given by ( f(x) = ax^2 + bx + c ) can be found using two powerful methods:", "1. Using the vertex formula: The x-coordinate of the vertex is ( x = -\frac{b}{2a} ), and substituting this back into the function yields the y-coordinate.", "2. Completing the square: Rewrite the quadratic function in vertex form ( f(x) = a(x - h)^2 + k ), where ( (h, k) ) is the vertex.", "## Step 1: Applying the Vertex Formula", "Given:\n( a = 3 ), ( b = -12 ), ( c = 7 )", "The x-coordinate of the vertex is:\n[\nx = -\frac{b}{2a} = -\frac{-12}{2 \ imes 3} = \frac{12}{6} = 2\n]", "Now, substitute ( x = 2 ) into ( f(x) ) to find the y-coordinate:\n[\nf(2) = 3(2)^2 - 12(2) + 7 = 3(4) - 24 + 7 = 12 - 24 + 7 = -5\n]", "Therefore, the vertex is at ( (2, -5) ).", "## Step 2: Vertex Form via Completing the Square (Optional Verification)", "To deepen understanding, we can convert the function to vertex form.", "Start with:\n[\nf(x) = 3x^2 - 12x + 7\n]", "Factor the coefficient of ( x^2 ) from the first two terms:\n[\nf(x) = 3(x^2 - 4x) + 7\n]", "To complete the square inside the parentheses:\n- Take half of (-4), which is (-2), and square it: ( (-2)^2 = 4 )", "Add and subtract 4 inside the parentheses:\n[\nf(x) = 3(x^2 - 4x + 4 - 4) + 7 = 3((x - 2)^2 - 4) + 7\n]", "Distribute the 3:\n[\nf(x) = 3(x - 2)^2 - 12 + 7 = 3(x - 2)^2 - 5\n]", "Now the function is in vertex form:\n[\nf(x) = 3(x - 2)^2 - 5\n]", "From this form, the vertex is clearly ( (h, k) = (2, -5) ), confirming our earlier result.", "## Why Is Finding the Vertex Important?", "The vertex is not just a point on the graph—it describes critical information about the function’s behavior:\n- Its location shows where the function reaches its extremum (minimum if ( a > 0 ), maximum if ( a < 0 )).\n- In applied contexts, this could mean the lowest point of a projectile’s path, maximum profit in a revenue model, or optimal temperature settings.", "## Conclusion", "The function ( f(x) = 3x^2 - 12x + 7 ) represents a parabola that opens upward, with its vertex at ( (2, -5) ). By applying the vertex formula ( x = -\frac{b}{2a} ) or completing the square, we confirm that this turning point is the minimum value of the function. Mastering vertex calculation empowers you to analyze and interpret quadratic functions with confidence—key to unlocking deeper insights in algebra and real-world problem solving.", "---", "Keywords: ( f(x) = 3x^2 - 12x + 7 ), parabola vertex, quadratic function vertex, vertex formula, completing the square, algebra tutorial."]