📚 IGCSE Edexcel Maths: Quadratic Functions Key Points | IGCSE Edexcel 数学：二次函数考点精讲
Quadratic functions are a fundamental topic in IGCSE Edexcel Mathematics. They appear in various forms, and mastering their graphs, equations, and applications is essential for high scores. This guide highlights every key exam point you need to know.
二次函数是IGCSE Edexcel数学的基础主题。它们以多种形式出现，掌握其图像、方程和应用是取得高分的关键。本指南突出你需要了解的每一个重要考点。
1. Standard Form and Basic Concepts | 标准形式与基本概念
A quadratic function is a polynomial of degree 2. Its standard form is y = ax² + bx + c, where a, b, c are constants and a ≠ 0. The graph of a quadratic function is called a parabola.
二次函数是一个二次多项式。其标准形式为 y = ax² + bx + c，其中 a、b、c 是常数且 a ≠ 0。二次函数的图像称为抛物线。
If a > 0, the parabola opens upwards (U-shaped) and has a minimum point. If a < 0, it opens downwards (∩-shaped) and has a maximum point. The constant c gives the y-intercept (0, c).
如果 a > 0，抛物线开口向上（U形）且有最低点。如果 a < 0，开口向下（∩形）且有最高点。常数 c 给出 y 轴截距 (0, c)。
The value of |a| affects the ‘width’ of the parabola: larger |a| makes it narrower, smaller |a| makes it wider. All parabolas are symmetric about a vertical line.
a 的绝对值影响抛物线的“宽度”：|a| 越大抛物线越窄，|a| 越小越宽。所有抛物线都关于一条竖直线对称。
2. Factored Form and Roots | 因式分解形式与根
A quadratic can sometimes be expressed in factored form: y = a(x − p)(x − q). Here p and q are the x-intercepts, also called the roots or solutions of ax² + bx + c = 0.
二次函数有时可以表示为因式分解形式：y = a(x − p)(x − q)。这里 p 和 q 是 x 轴截距，也称为方程 ax² + bx + c = 0 的根或解。
To find the roots from this form, set y = 0, giving x = p or x = q. The axis of symmetry lies exactly halfway between the roots: x = (p + q)/2.
要从这种形式求根，令 y = 0，得到 x = p 或 x = q。对称轴恰好位于两根中间：x = (p + q)/2。
If the expression does not factorise over integers, you will need to use completing the square or the quadratic formula to find the roots.
如果该表达式在整数范围内不能因式分解，你需要使用配方法或求根公式来求根。
3. Vertex Form and Completing the Square | 顶点形式与配方法
The vertex form is y = a(x − h)² + k, where (h, k) is the vertex of the parabola. This form is extremely useful for identifying the maximum or minimum value and for graph sketching.
顶点形式为 y = a(x − h)² + k，其中 (h, k) 是抛物线的顶点。这种形式在确定最大值或最小值以及绘制图像时极为有用。
To convert standard form to vertex form, complete the square. For y = ax² + bx + c, first factor a from the x-terms, then add and subtract (b/(2a))² inside the bracket.
要将标准形式转化为顶点形式，使用配方法。对于 y = ax² + bx + c，首先从 x 项中提取 a，然后在括号内加上并减去 (b/(2a))²。
When a = 1, the vertex form becomes y = (x + b/2)² + (c − b²/4). The vertex is (−b/2, c − b²/4), and the axis of symmetry is x = −b/2.
当 a = 1 时，顶点形式变为 y = (x + b/2)² + (c − b²/4)。顶点为 (−b/2, c − b²/4)，对称轴为 x = −b/2。
4. Direction of Opening and Shape | 开口方向与形状
The sign of a controls the direction: a > 0 means the parabola has a minimum point and opens upward; a < 0 means it has a maximum point and opens downward.
a 的符号控制方向：a > 0 表示抛物线有最低点并开口向上；a < 0 表示有最高点并开口向下。
The steepness is determined by |a|. When |a| > 1, the parabola is steeper (narrower) than y = x². When 0 < |a| < 1, it is flatter (wider). The graph is always symmetric about the axis of symmetry.
陡峭程度由 |a| 决定。当 |a| > 1 时，抛物线比 y = x² 更陡（更窄）。当 0 < |a| < 1 时，则更平缓（更宽）。图像始终关于对称轴对称。
5. Axis of Symmetry and Vertex | 对称轴与顶点
For any quadratic function y = ax² + bx + c, the axis of symmetry is the vertical line x = −b/(2a). The vertex lies on this line.
对于任何二次函数 y = ax² + bx + c，对称轴是竖直线 x = −b/(2a)。顶点位于此线上。
The x-coordinate of the vertex is −b/(2a). To find the y-coordinate, substitute this x-value back into the function. This gives the optimum (maximum or minimum) value of the function.
顶点的 x 坐标为 −b/(2a)。要找到 y 坐标，将此 x 值代回函数中。这给出了函数的最优值（最大值或最小值）。
Example: For y = 3x² − 12x + 5, x = −(−12)/(2 × 3) = 2. Then y = 3(2)² − 12(2) + 5 = −7. The vertex is (2, −7) and the minimum value is −7.
示例：对于 y = 3x² − 12x + 5，x = −(−12)/(2×3) = 2。然后 y = 3(2)² −12(2)+5 = −7。顶点为 (2, −7)，最小值为 −7。
6. Discriminant and Nature of Roots | 判别式与根的性质
The discriminant, usually denoted by Δ, is Δ = b² − 4ac. It tells you how many real roots the equation ax² + bx + c = 0 has.
判别式，通常记作 Δ，为 Δ = b² − 4ac。它告诉你方程 ax² + bx + c = 0 有多少个实数根。
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If Δ > 0: two distinct real roots. The graph cuts the x-axis at two points.
若 Δ > 0：有两个不同的实根。图像与 x 轴交于两点。
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If Δ = 0: one repeated real root (or two equal roots). The graph touches the x-axis at the vertex.
若 Δ = 0：有一个重实根（或两个相等实根）。图像在顶点处与 x 轴相切。
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If Δ < 0: no real roots. The graph does not meet the x-axis.
若 Δ < 0：无实数根。图像与 x 轴不相交。
When Δ is a perfect square, the roots are rational and the quadratic can be factorised neatly. The discriminant is also found inside the quadratic formula.
当 Δ 是一个完全平方数时，根是有理数且二次式可被整齐地因式分解。判别式也出现在求根公式中。
7. Solving Quadratic Equations | 解二次方程
There are three primary methods: factorising, using the quadratic formula, and completing the square. Choose the most efficient method for the given equation.
有三种主要方法：因式分解法、使用求根公式法以及配方法。为给定方程选择最有效的方法。
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