The x-coordinate of the vertex is given by \( x = -\frac{b}{2a} \), where \( a = 3 \) and \( b = -12 \).

["# Understanding the x-Coordinate of a Parabola’s Vertex: A Guide for Students", "Learning how to find the vertex of a quadratic function is a fundamental skill in algebra and calculus. The x-coordinate of the vertex plays a crucial role in graphing parabolas and solving real-world problems involving projectile motion, optimization, and more.", "## The Vertex Formula: What is ( x = -\frac{b}{2a} )?", "For any quadratic function in standard form:", "[\nf(x) = ax^2 + bx + c\n]", "the x-coordinate of the vertex is given by the formula:", "[\nx = -\frac{b}{2a}\n]", "This formula derives from completing the square and provides the horizontal axis of symmetry for the parabola. Knowing this value allows you to pinpoint the peak (or trough) of the quadratic graph.", "## Step-by-Step: Calculating the x-Coordinate of the Vertex", "Given the values ( a = 3 ) and ( b = -12 ), we apply the vertex formula:", "[\nx = -\frac{b}{2a} = -\frac{-12}{2 \ imes 3} = \frac{12}{6} = 2\n]", "So, the x-coordinate of the vertex is 2.", "### What does this mean?", "- The vertex of the parabola described by ( f(x) = 3x^2 - 12x + c ) lies at ( x = 2 ).\n- This is the point where the parabola reaches its maximum or minimum value—since ( a = 3 > 0 ), the parabola opens upward and ( x = 2 ) is the minimum point.\n- This calculation helps locate key features of the graph without fully plotting every point.", "## Why This Formula Matters", "- Graphing parabolas: Knowing the x-coordinate of the vertex helps sketch the curve symmetrically around ( x = 2 ).\n- Real-world applications: In physics, this represents the time at which a projectile reaches its highest point when trajectory modeled by a quadratic equation.\n- Maximization/Minimization: The vertex provides the optimal ( x )-value for fuel efficiency, cost reduction, or profit maximization in applied math.", "## Practice Problem", "Try finding the x-coordinate of the vertex for the function:", "[\nf(x) = 5x^2 - 20x + 4\n]", "Using ( a = 5 ), ( b = -20 ):", "[\nx = -\frac{-20}{2 \ imes 5} = \frac{20}{10} = 2\n]", "So, the x-coordinate of the vertex is 2.", "## Conclusion", "Mastering the formula ( x = -\frac{b}{2a} ) is essential for understanding quadratic functions and their graphs. With just ( a = 3 ) and ( b = -12 ), we computed the vertex’s x-position in just one step. Use this knowledge to interpret parabolas confidently and apply it to real-life scenarios in science, engineering, and economics.", "Keywords: vertex formula, parabola x-coordinate, quadratic vertex calculation, ( x = -\frac{b}{2a} ), algebra tutorial, graphing parabolas, optimization with quadratics."]