📚 Mastering Quadratic Functions for IGCSE Mathematics | 掌握IGCSE数学中的二次函数
Quadratic functions are the backbone of IGCSE Mathematics (0580). They appear in algebra, graphs, coordinate geometry, and even in problem-solving questions. Understanding how to manipulate and solve quadratics is essential for achieving a top grade.
二次函数是IGCSE数学（0580）的基石。它们出现在代数、图像、坐标几何甚至应用题中。掌握二次函数的变形与求解，是斩获高分的关键。
1. Standard Form and Key Features | 标准形式与基本特征
The standard form of a quadratic function is y = ax² + bx + c, where a, b, and c are constants and a ≠ 0. The value of a determines whether the parabola opens upward (a > 0) or downward (a < 0).
二次函数的标准形式为 y = ax² + bx + c，其中 a、b、c 为常数且 a ≠ 0。a 的正负决定了抛物线的开口方向：a > 0 开口向上，a < 0 开口向下。
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The graph of a quadratic is called a parabola. It is symmetric about a vertical line called the axis of symmetry.
二次函数的图像称为抛物线。它关于一条竖直直线对称，这条直线称为对称轴。
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The axis of symmetry has the equation x = -b/(2a). This is also the x-coordinate of the vertex (turning point).
对称轴的方程为 x = -b/(2a)。它同时也是顶点（驻点）的横坐标。
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The y-intercept is found by substituting x = 0, giving the point (0, c).
令 x = 0 可求得 y 截距，即点 (0, c)。
2. Factorising and Roots | 因式分解与求根
To solve x² – 5x + 6 = 0 by factorising, look for two numbers that multiply to 6 and add up to -5. These are -2 and -3. Thus, (x – 2)(x – 3) = 0, so x = 2 or x = 3.
用因式分解求解 x² – 5x + 6 = 0，寻找两个数，它们相乘得 6，相加得 -5。这两个数是 -2 和 -3。因此 (x – 2)(x – 3) = 0，所以 x = 2 或 x = 3。
The roots are the x-coordinates where the graph crosses the x-axis. If the quadratic cannot be factorised easily, use the quadratic formula.
根就是图像与 x 轴交点的横坐标。如果二次多项式不易分解，可以使用求根公式。
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Remember to set the equation to 0 first. For example, x² = 3x should be rewritten as x² – 3x = 0.
记住要先将方程化为零的形式。例如，x² = 3x 必须先改写为 x² – 3x = 0。
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A quadratic can have at most two real roots. If the graph does not touch the x-axis, there are no real roots.
二次方程最多有两个实根。若图像不与 x 轴相交，则没有实根。
3. The Quadratic Formula and the Discriminant | 求根公式与判别式
For any quadratic ax² + bx + c = 0, the solutions are given by the formula:
对于任意一元二次方程 ax² + bx + c = 0，其解由以下公式给出：
x = (-b ± √(b² – 4ac)) / 2a
The discriminant, Δ = b² – 4ac, determines the nature of the roots. If Δ > 0, there are two distinct real roots. If Δ = 0, there is exactly one real root (a repeated root). If Δ < 0, there are no real roots.
判别式 Δ = b² – 4ac 决定了根的性质：若 Δ > 0，方程有两个不等的实根；若 Δ = 0，方程有一个实根（重根）；若 Δ < 0，方程无实根。
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Use the discriminant to quickly check your answers when sketching graphs. For example, y = x² – 4x + 4 has Δ = 16 – 16 = 0, so the graph touches the x-axis at exactly one point.
在绘图时，可借助判别式快速验证答案。例如，y = x² – 4x + 4 的 Δ = 16 – 16 = 0，所以图像与 x 轴仅有一个切点。
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When solving word problems, discard any negative roots that do not make sense in the context (e.g., length
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