The vertex \( x \)-coordinate is \( x = - rac{b}{2a} = - rac{150}{-10} = 15 \).
["### Understanding the Vertex of a Parabola: How to Calculate the x-Coordinate", "The vertex of a quadratic function in standard form plays a fundamental role in graphing parabolas and understanding their key properties. For a parabola described by the quadratic equation ( f(x) = ax^2 + bx + c ), the vertex provides the turning point — the maximum or minimum of the function — and its ( x )-coordinate can be determined with a powerful and concise formula.", "---", "### The Vertex Formula: ( x = -\frac{b}{2a} )", "In algebra, the ( x )-coordinate of the vertex for any quadratic equation is given by:\n[\nx = -\frac{b}{2a}\n]\nThis formula arises from completing the square or using calculus to find the axis of symmetry of the parabola. It reflects the point where the function switches direction — from increasing to decreasing or vice versa.", "---", "### Applying the Formula: A Practical Example", "Consider the specific case:\n[\nx = -\frac{b}{2a} = -\frac{150}{-10} = 15\n]\nHere, ( a = -10 ) and ( b = 150 ). Plugging these values into the formula:\n[\nx = -\frac{150}{2 \ imes (-10)} = -\frac{150}{-20} = 15\n]\nThus, the ( x )-coordinate of the vertex is ( 15 ).", "---", "### Why This Value Matters: The Axis of Symmetry", "The most significant insight from the vertex lies in the parabola’s axis of symmetry, which is the vertical line ( x = -\frac{b}{2a} ). This line divides the parabola into two mirror-image halves. Knowing the ( x )-coordinate of the vertex allows you to determine the peak (for ( a < 0 )) or trough (for ( a > 0 )), making it essential for sketching the graph, solving optimization problems, and analyzing quadratic behavior.", "---", "### Writing the Vertex in Vertex Form", "Once you’ve found ( x = -\frac{b}{2a} ), substituting this value back into the original equation gives the full vertex as a point:\n[\n\ ext{Vertex} = \left( -\frac{b}{2a},\ f\left(-\frac{b}{2a}\right) \right)\n]\nIn the example, we calculate ( f(15) ) to find the actual vertex ( (15, f(15)) ), further enriching the graph’s description.", "---", "### Final Thoughts", "The vertex ( x )-coordinate formula ( x = -\frac{b}{2a} ) is a cornerstone of quadratic analysis. For any quadratic function, this simple expression reveals the symmetry center and peak (or valley) of the curve. Using the example ( x = -\frac{150}{-10} = 15 ), we see how algebra elegantly identifies this pivotal point. Mastering this formula empowers deeper graphing skills and problem-solving in algebra and beyond.", "---", "Keywords: vertex formula, x coordinate of vertex, quadratic vertex formula, calculate x-coordinate parabola, vertex axis of symmetry, quadratic functions, ( x = -\frac{b}{2a} ), graphing parabolas, algebra tips, coordinate geometry."]
