Fonction quadratique \( P(x) = -2x^2 + 120x - 1000 \).

["Understanding the Fonction Quadratique ( P(x) = -2x^2 + 120x - 1000 ): Key Features, Graph, and Applications", "The function ( P(x) = -2x^2 + 120x - 1000 ) is a quadratic equation that models many real-world situations involving parabolic relationships. Whether you're analyzing profit, motion, or optimization problems, understanding this quadratic function is essential. This article breaks down its mathematical properties, graph shape, key points, and practical applications to help students, educators, and professionals fully grasp this important concept.", "---", "### What Is a Fonction Quadratique?", "In mathematics, a fonction quadratique (quadratic function) is any function of the form:\n[\nP(x) = ax^2 + bx + c\n]\nwhere ( a ), ( b ), and ( c ) are constants and ( a <br/>\ne 0 ). The presence of the ( x^2 ) term gives the graph a characteristic U-shape (parabola), which can open upwards or downwards depending on the sign of ( a ).", "For ( P(x) = -2x^2 + 120x - 1000 ):\n- ( a = -2 ) (negative coefficient)\n- ( b = 120 )\n- ( c = -1000 )", "---", "### Step 1: Analyzing the Graph of the Quadratic Function", "Since ( a < 0 ), the parabola opens downward. This means the function has a maximum value at its vertex—ideal for modeling scenarios where a peak outcome is desired.", "#### Key Features of the Graph:\n- Direction: Downward-opening parabola\n- Vertex: The highest point (maximum)\n- Axis of Symmetry: A vertical line through the vertex\n- Y-intercept: Value when ( x = 0 )\n- X-intercepts: Solutions to ( P(x) = 0 ) (if any)", "---", "### Step 2: Finding the Vertex", "The vertex of any quadratic function ( P(x) = ax^2 + bx + c ) occurs at:\n[\nx = -\frac{b}{2a}\n]\nSubstituting ( a = -2 ) and ( b = 120 ):\n[\nx = -\frac{120}{2(-2)} = \frac{120}{4} = 30\n]\nSo the vertex is at ( x = 30 ). To find the maximum value:\n[\nP(30) = -2(30)^2 + 120(30) - 1000\n]\nCalculate step-by-step:\n- ( 30^2 = 900 )\n- ( -2 \cdot 900 = -1800 )\n- ( 120 \cdot 30 = 3600 )\n- Add: ( -1800 + 3600 - 1000 = 800 )", "Vertex: ( (30, 800) )\nThe maximum profit or value occurs when ( x = 30 ), yielding ( P = 800 ).", "---", "### Step 3: Calculating the Y-Intercept", "The y-intercept occurs when ( x = 0 ):\n[\nP(0) = -2(0)^2 + 120(0) - 1000 = -1000\n]\nSo the graph crosses the y-axis at ( (0, -1000) ).", "---", "### Step 4: Finding the X-Intercepts (Roots)", "To find where the function crosses the x-axis, solve ( P(x) = 0 ):\n[\n-2x^2 + 120x - 1000 = 0\n]\nDivide through by -2 to simplify:\n[\nx^2 - 60x + 500 = 0\n]\nUse the quadratic formula:\n[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{60 \pm \sqrt{(-60)^2 - 4(1)(500)}}{2}\n]\n[\nx = \frac{60 \pm \sqrt{3600 - 2000}}{2} = \frac{60 \pm \sqrt{1600}}{2} = \frac{60 \pm 40}{2}\n]\nSo:\n- ( x = \frac{60 + 40}{2} = 50 )\n- ( x = \frac{60 - 40}{2} = 10 )", "X-intercepts: ( x = 50 ) and ( x = 10 )", "These represent the points where the modeled quantity becomes zero—critical values in financial or physical models.", "---", "### Step 5: Understanding the Axis of Symmetry", "For a quadratic function, the axis of symmetry is the vertical line passing through the vertex:\n[\nx = 30\n]\nThis means values equidistant from 30 on the x-axis will yield equal function values.", "---", "### Real-World Applications of ( P(x) = -2x^2 + 120x - 1000 )", "Quadratic functions like this commonly model situations with diminishing returns:", "- Profit Maximization: If ( P(x) ) represents profit from selling ( x ) units, the maximum profit occurs at 30 units sold, yielding ( $800 ). Selling fewer or more decreases profit.\n- Projectile Motion: As a simplified model of height vs. horizontal distance, the downward curve reflects gravity pulling objects downward.\n- Revenue Modeling: Stable revenue with a peak point—flexible pricing or advertising campaigns often fit this pattern.", "---", "### Summary of Key Points", "| Property | Value/Explanation |\n|-----------------------|--------------------------------------|\n| Function Form | ( P(x) = -2x^2 + 120x - 1000 ) |\n| Direction | Opens downward (concave down) |\n| Vertex (maximum) | ( (30,\ 800) ) |\n| Y-intercept | ( (0,\ -1000) ) |\n| X-intercepts | ( x = 10 ) and ( x = 50 ) |\n| Axis of Symmetry | ( x = 30 ) |\n| Real-world Use | Profit, motion, optimization models |", "---", "### Final Thoughts", "Understanding the fonction quadratique ( P(x) = -2x^2 + 120x - 1000 ) unlocks insights into systems where growth or gain is nonlinear. By identifying the vertex, intercepts, and shape, one can solve optimization problems, interpret economic data, and analyze trajectories. Mastering such functions is fundamental for students and professionals in math, economics, engineering, and the sciences.", "---", "Keywords: fonction quadratique, ( P(x) = -2x^2 + 120x - 1000 ), parabola, vertex, optimization, quadratic formula, graph analysis, real-world applications, high school math, algebra, calculus prep, profit model.", "---", "Reference:\nThis resource explains key concepts in quadratic functions, suitable for learners focusing on algebra, calculus, and applied mathematics. For further exercises, use graphing calculators or online quadratic function analyzers to visualize transformations and behaviors.", "---", "Unlock the power of quadratic mathematics—start analyzing ( P(x) ) today!"]