The function \( Y = -2x^2 + 24x + 100 \) is a quadratic in standard form, opening downward (since coefficient of \( x^2 \) is negative), so the maximum occurs at the vertex.

["Understanding the Quadratic Function ( Y = -2x^2 + 24x + 100 ): How to Find Its Maximum Value Using the Vertex Formula", "When analyzing quadratic functions in standard form, understanding key properties like direction, vertex, and maximum or minimum values is essential for solving optimization problems and graph interpretation. Take the quadratic function ( Y = -2x^2 + 24x + 100 ), which is a clear example of a downward-opening parabola due to its negative coefficient on ( x^2 ). This means the function reaches a single maximum point at its vertex—the spot where ( Y ) achieves its highest value.", "### The Standard Form and Direction of the Parabola\nThe given function is in standard form:\n[ Y = ax^2 + bx + c ]\nHere, ( a = -2 ), ( b = 24 ), and ( c = 100 ). Since ( a < 0 ), the parabola opens downward. This downward orientation guarantees the function has a maximum value, not a minimum, occurring precisely at the vertex.", "### The Vertex: Where the Maximum Occurs\nFor any quadratic function ( Y = ax^2 + bx + c ), the ( x )-coordinate of the vertex (and thus the position of the maximum in this case) is found using the formula:\n[\nx = -\frac{b}{2a}\n]\nPlugging in ( a = -2 ) and ( b = 24 ):\n[\nx = -\frac{24}{2(-2)} = -\frac{24}{-4} = 6\n]\nSo, the maximum occurs when ( x = 6 ).", "### Calculating the Maximum Value of ( Y )\nTo find the corresponding maximum value of ( Y ), substitute ( x = 6 ) back into the original equation:\n[\nY = -2(6)^2 + 24(6) + 100\n]\nBreak down the calculation step-by-step:\n[\nY = -2(36) + 144 + 100 = -72 + 144 + 100 = 172\n]", "### Summary: Maximum at ( (6, 172) )\nThe quadratic function ( Y = -2x^2 + 24x + 100 ) has a maximum value of ( 172 ) at ( x = 6 ). This confirms the fundamental property that a downward-opening parabola reaches its peak at the vertex. Such insight is vital when applying quadratics to real-world scenarios like profit modeling, projectile motion, or maximizing area.", "Key Takeaways:\n- Negative leading coefficient signals a downward-opening parabola and a maximum.\n- The vertex formula ( x = -\frac{b}{2a} ) efficiently finds the location of the maximum.\n- Substituting ( x ) into the equation gives the exact peak value.", "By mastering these steps, interpreting quadratic functions becomes structured and reliable—essential skills in algebra, calculus, and data analysis. Whether optimizing performance or studying curves, recognizing the vertex as the maximum is a foundational strategy.", "---", "Keywords (SEO optimized):\nquadratic function maximum, vertex formula, upward vs downward parabola, parabola opening downward, maximum value of quadratic, standard form quadratic, ( Y = -2x^2 + 24x + 100 ), calculate maximum, vertex calculation, analytical geometry."]