The vertex occurs at \( x = -\frac{b}{2a} = -\frac{12}{2(-2)} = 3 \).
["# Understanding the Vertex of a Quadratic Function: At ( x = 3 )", "When analyzing quadratic functions in the standard form ( f(x) = ax^2 + bx + c ), one of the most essential concepts is the vertex—the highest or lowest point on the parabola, depending on the parabola's orientation. This critical point determines whether the function reaches a maximum or minimum value, making it indispensable in calculus, optimization problems, and graphing.", "## The Formula Behind the Vertex", "For any quadratic function ( ax^2 + bx + c ), the x-coordinate of the vertex is found using the formula:", "$$\nx = -\frac{b}{2a}\n$$", "This formula arises from finding the axis of symmetry of the parabola, which splits the curve into two mirror-image halves. Plugging in the values ( a ) and ( b ) gives precise location data vital for plotting and interpreting the function.", "In a classic example, consider the quadratic function:", "$$\nf(x) = -2x^2 + 12x - 20\n$$", "Here, ( a = -2 ) and ( b = 12 ). Applying the vertex formula:", "$$\nx = -\frac{b}{2a} = -\frac{12}{2(-2)} = -\frac{12}{-4} = 3\n$$", "This calculation reveals that the vertex lies at ( x = 3 ).", "## What Does ( x = 3 ) Represent?", "With ( x = 3 ) as the x-coordinate, the vertex completes the point ( (3, f(3)) )—the lowest or highest point of the parabola, depending on the sign of ( a ). In this case, since ( a = -2 < 0 ), the parabola opens downward, meaning the vertex at ( x = 3 ) is actually the maximum point.", "Computing ( f(3) ):", "$$\nf(3) = -2(3)^2 + 12(3) - 20 = -18 + 36 - 20 = -2\n$$", "Thus, the vertex is the point ( (3, -2) ), representing the peak value of the function.", "## Why Knowing the Vertex Matters", "The vertex is key for:", "- Optimization: Finding maximum or minimum values in economics, engineering, and physics.\n- Graphing: Accurately sketching the parabola’s shape and position.\n- Solving Equations: Identifying turning points and roots more efficiently.", "## Final Thoughts", "The vertex’s location ( x = -\frac{b}{2a} = 3 ) is more than just a numerical result—it’s a gateway to deeper insight into quadratic functions. Whether maximizing profit, modeling projectile motion, or analyzing data trends, recognizing and calculating the vertex empowers smarter decision-making and clearer problem-solving.", "---", "Keywords: vertex x-coordinate, vertex formula ( x = -\frac{b}{2a} ), quadratic vertex, maximizing functions, quadratic graph, vertex calculation, ( f(x) = -2x^2 + 12x - 20 )", "Use this formula confidently—your understanding of parabolas will reach new heights, with the vertex always sitting confidently at ( x = 3 )."]
