📚 Quadratic Functions and Equations | 二次函数与方程
Quadratic functions and equations form the backbone of IGCSE algebra. Understanding their graphs, transformations, and solution methods is essential for achieving top marks in Paper 2 and Paper 4.
二次函数与方程是 IGCSE 代数部分的核心内容。掌握它们的图像、变换及求解方法，是在 Paper 2 和 Paper 4 中获得高分的必要基础。
1. Standard Form and Vertex Form | 标准形式与顶点形式
A quadratic function is usually written in standard form:
二次函数通常写成标准形式：
f(x) = ax² + bx + c , a ≠ 0
h)² + k, where (h, k) is the vertex of the parabola. The value of a determines the direction and width of the curve.
每一个二次函数也可以写成顶点形式：f(x) = a(x − h)² + k，其中 (h, k) 是抛物线的顶点。系数 a 决定曲线的开口方向与宽窄。
2. The Graph of a Quadratic | 二次函数的图像
The graph of a quadratic function is always a parabola. If a > 0, the parabola opens upwards and has a minimum point; if a < 0, it opens downwards and has a maximum point.
二次函数的图像始终是一条抛物线。若 a > 0，抛物线开口向上，存在最小值点；若 a < 0，抛物线开口向下，存在最大值点。
The axis of symmetry is the vertical line x = −b/(2a).
对称轴是竖直线 x = −b/(2a)。
The y-intercept is the point (0, c).
y 截距是点 (0, c)。
The x-intercepts, if they exist, are the real roots of the equation f(x) = 0.
x 截距（若存在）是方程 f(x) = 0 的实数根。
3. Completing the Square | 配方法
Completing the square converts standard form into vertex form. This technique is frequently tested in IGCSE, especially for finding turning points and solving equations.
配方法将标准形式转化为顶点形式。这是 IGCSE 的高频考点，常用于求极值点和解方程。
ax² + bx + c = a(x + b/(2a))² + c − b²/(4a)
r example, f(x) = 2x² + 8x + 5 becomes f(x) = 2(x + 2)² − 3, so the vertex is at (−2, −3).
例如，f(x) = 2x² + 8x + 5 可化为 f(x) = 2(x + 2)² − 3，因此顶点为 (−2, −3)。
4. Solving by Factorisation | 因式分解法
When the quadratic expression can be factorised into two linear factors, the equation is solved by setting each factor equal to zero.
当二次式可以分解为两个一次因式时，将每个因式等于零即可解出方程。
x² − 5x + 6 = 0 → (x − 2)(x − 3) = 0 → x = 2 or 3
Always check whether the coefficient a is not 1, and factor out any common factor first.
注意，当 a ≠ 1 时需格外小心，并应首先提取公因子。
5. The Quadratic Formula | 求根公式
For any quadratic equation ax² + bx + c = 0, the roots are given by the quadratic formula:
对于任意一元二次方程 ax² + bx + c = 0，其根由求根公式给出：
x = (−b ± √b² − 4ac) / (2a)
This formula works for all cases, including irrational and complex roots. In IGCSE, you must be able to substitute values correctly and simplify surds.
该公式适用于一切情形，包括无理根与复数根。在 IGCSE 中，你须能正确代入数值并化简根式。
6. The Discriminant | 判别式
The discriminant, denoted by Δ, is the expression under the square root in the quadratic formula:
判别式用 Δ 表示，即求根公式中根号内的表达式：
Δ = b² − 4ac
| Δ > 0 | Two distinct real roots | 两个不等实根 |
| Δ = 0 | One repeated real root | 一个重根 |
| Δ < 0 | No real roots | 无实根 |
The discriminant also tells us whether the parabola intersects the x-axis at two points, touches it, or never crosses it.
判别式还决定抛物线与 x 轴有两个交点、相切还是无交点。
7. Sum and Product of Roots | 根与系数的关系
If α and β are the roots of ax² + bx + c = 0, then:
若 α 和 β 是方程 ax² + bx + c = 0 的两个根，则：
α + β = −b/a and αβ = c/a
This relationship is frequently used in questions that ask you to find symmetric expressions such as α² + β², without solving the equation.
这一关系常用于求出对称式（如 α² + β²）的值，而不需要具体解方程。
8. Solving Simultaneous Equations with Quadratics | 二次方程组求解
IGCSE often tests linear-quadratic simultaneous equations. The standard method is substitution: replace y in the linear equation, forming a single quadratic equation in one variable.
IGCSE 经常考查一次-二次联立方程组。标准方法是代入法：将一次式中的 y 代入二次式，得到关于单一变量的一元二次方程。
For example, solve y = 2x + 1 and y = x² − 1. Substituting gives:
例如，解方程组 y = 2x + 1 和 y = x² − 1。代入得：
2x + 1 = x² − 1 → x² − 2x − 2 = 0
This quadratic can then be solved using the formula, giving two possible x-values, each with a corresponding y-value.
随后用求根公式解该二次方程，得到两个 x 值，并分别求出对应的 y 值。
9. Applications in Word Problems | 实际应用题
Quadratic equations are used to model projectile motion, area problems, and number puzzles. For example, the product of two consecutive positive integers is 132. The equation is:
二次方程常用于建模抛体运动、面积问题和数字谜题。例如，两个连续正整数的乘积为 132。方程为：
n(n + 1) = 132 → n² + n − 132 = 0
(n + 12)(n − 11) = 0 → n = 11
Always discard negative or physically impossible roots in context.
在具体情境中要注意剔除负数或无实际意义的根。
10. Connecting Graphs and Equations | 图像与方程的呼应
Every question about quadratic equations corresponds to a graphical interpretation. Roots are x-intercepts, the discriminant controls the number of intersections, and the vertex corresponds to the maximum or minimum value.
每一个关于二次方程的问题都对应着图像解释。根是 x 截距，判别式决定交点个数，顶点则对应最大值或最小值。
In the exam, when given a graph, you should be able to read off approximate roots, the y-intercept, and the turning point.
考试中，面对图像题时，你应能读出近似根、y 截距及极值点。
Summary | 总结: Master standard form, completing the square, the quadratic formula, and the discriminant. These four tools unlock every quadratic question in IGCSE Mathematics.
总结：熟练掌握标准形式、配方法、求根公式与判别式，这四大工具能帮你解锁 IGCSE 数学中所有的二次函数与方程问题。
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