Vertex of parabola: \( x = -\frac{b}{2a} = -\frac{40}{2 \cdot (-2)} = \frac{40}{4} = 10 \).

["# Understanding the Vertex of a Parabola: How to Calculate It Easily", "When studying quadratics, one of the most important concepts is identifying the vertex of a parabola. The vertex represents the peak (a maximum point) or trough (a minimum point) of the parabola, depending on its orientation. For a quadratic function in standard form,\n[ f(x) = ax^2 + bx + c, ]\nthe x-coordinate of the vertex is given by the formula:\n[ x = -\frac{b}{2a} ]\nThis formula is simple yet powerful, allowing you to pinpoint the exact location of this key point without graphing the entire curve.", "## Example Calculation: Finding the Vertex When ( x = -\frac{b}{2a} = -\frac{40}{2 \cdot (-2)} = \frac{40}{4} = 10 )", "Let’s explore a concrete example. Consider the quadratic function:\n[ f(x) = -2x^2 - 40x + c ]\nwhere the coefficient ( a = -2 ) and ( b = -40 ).", "To find the vertex’s x-coordinate, plug ( a ) and ( b ) into the vertex formula:\n[\nx = -\frac{b}{2a} = -\frac{-40}{2 \cdot (-2)} = \frac{40}{-4} = -10\n]\nOops! There’s a miscalculation in the original setup—this shows how slight errors can flip the result. Correcting:\n[\nx = -\frac{-40}{2 \cdot (-2)} = \frac{40}{-4} = -10\n]\nWait—this still seems inconsistent with the claimed vertex at ( x = 10 ). Let’s recheck the example values carefully.", "Suppose instead the function is:\n[\nf(x) = -2x^2 + 40x\n]\nHere, ( a = -2 ), ( b = 40 ). Then:\n[\nx = -\frac{b}{2a} = -\frac{40}{2 \cdot (-2)} = -\frac{40}{-4} = 10\n]\nPerfect! This matches our target result of ( x = 10 ).", "### Why Is This Vertex Coordinate Important?", "The vertex ( (10, f(10)) ) tells us:\n- If ( a < 0 ), the parabola opens downward and the vertex is the maximum point — so at ( x = 10 ), ( y ) is maximized.\n- If ( a > 0 ), it opens upward and the vertex is the minimum.", "### Finding the Full Vertex", "To find the full vertex coordinates, substitute ( x = 10 ) back into the function:\n[\nf(10) = -2(10)^2 + 40(10) = -200 + 400 = 200\n]\nSo the vertex is exactly at ( (10, 200) ).", "### Real-World Applications", "Understanding the vertex helps in physics (projectile motion), optimization problems (maximizing profit or minimizing cost), and engineering designs where peak efficiency is crucial. For instance, the trajectory of a ball follows a parabolic path, with the vertex marking the highest point.", "---", "Summary:\nUsing ( x = -\frac{b}{2a} ), we efficiently locate the vertex of a parabola. For the example ( f(x) = -2x^2 + 40x ),\n[\nx_{\ ext{vertex}} = -\frac{40}{2(-2)} = 10\n]\nThen ( f(10) = 200 ), so the vertex is ( (10, 200) ). Mastering this step unlocks deeper insight into quadratic functions and their real-world significance.", "---", "Keywords: vertex of a parabola, parabola vertex formula, how to find x-coordinate vertex, vertex formulas, quadratic functions, calculate vertex, vertex calculation example, vertex of ( -2x^2 + 40x ), maximize minimum, quadratic optimization."]