📚 IGCSE Maths Core Revision: Quadratic Graphs, Trigonometry and Kinematics | IGCSE 数学核心复习：二次图像、三角与运动学
This revision guide covers three high-yield IGCSE Mathematics topics: quadratic graphs, right-angled triangle trigonometry, and kinematics graphs. These topics frequently appear in both Core and Extended papers, especially in questions that require interpreting graphs, calculating unknown sides or angles, and explaining motion from distance-time or speed-time graphs.
本复习指南涵盖 IGCSE 数学三个高频主题：二次函数图像、直角三角形三角学以及运动学图像。这些主题在核心卷和扩展卷中经常出现，尤其涉及图像解读、未知边角计算以及根据距离-时间或速度-时间图像解释运动的问题。
1. Quadratic Expressions and Standard Form | 二次表达式与标准形式
A quadratic expression has the general form ax² + bx + c, where a, b and c are constants and a ≠ 0. The highest power of x is 2, which gives the quadratic its characteristic U-shaped graph called a parabola.
二次表达式的一般形式为 ax² + bx + c，其中 a、b、c 为常数且 a ≠ 0。x 的最高次数为 2，这使得二次函数图像呈特有的 U 形，称为抛物线。
3x² − 5x + 2 is a quadratic, but y = 5x + 2 is linear because the highest power of x is 1.
你应能从代数形式和图像中识别二次表达式。例如，y = 3x² − 5x + 2 是二次式，而 y = 5x + 2 是一次式，因为 x 的最高次数为 1。
2. Factorising Quadratics | 二次式因式分解
Factorising is the reverse of expanding brackets. For a monic quadratic such as x² + bx + c, look for two numbers that multiply to give c and add to give b. Then write the expression as (x + p)(x + q).
因式分解是去括号的逆运算。对于首一二次式（如 x² + bx + c），寻找两个数，它们相乘等于 c，相加等于 b。然后将表达式写成 (x + p)(x + q)。
on-monic quadratics such as 2x² + 7x + 3, use the ac method or trial and error to obtain (2x + 1)(x + 3).
示例：x² + 7x + 10 可因式分解为 (x + 2)(x + 5)，因为 2 × 5 = 10 且 2 + 5 = 7。对于非首一二次式，如 2x² + 7x + 3，可用 ac 法或尝试法得到 (2x + 1)(x + 3)。
3. Solving Quadratic Equations by Factorising | 用因式分解法解二次方程
n a = 0 or b = 0. Set each bracket equal to zero and solve the resulting linear equations.
一旦二次式被写成因式分解形式，使用零乘积性质：若 a × b = 0，则 a = 0 或 b = 0。令每个括号等于零，并解所得的一次方程。
orise to get (x − 5)(x + 2) = 0. Therefore x − 5 = 0 or x + 2 = 0, giving x = 5 or x = −2.
示例：解 x² − 3x − 10 = 0。因式分解得 (x − 5)(x + 2) = 0。因此 x − 5 = 0 或 x + 2 = 0，解得 x = 5 或 x = −2。
4. The Quadratic Formula | 二次公式
rised easily, use the quadratic formula. For ax² + bx + c = 0, the solutions are given by:
当二次式不易因式分解时，使用二次公式。对于 ax² + bx + c = 0，解为：
x = (−b ± √b² − 4ac) ÷ 2a
The expression b² − 4ac is called the discriminant. If it is positive, there are two distinct real roots; if it is zero, there is one repeated real root; if it is negative, there are no real roots.
表达式 b² − 4ac 称为判别式。若其值为正，则有两个不同实根；若为零，则有一个重复实根；若为负，则无实根。
5. Sketching Quadratic Graphs | 绘制二次函数图像
it opens downwards and has a maximum point. The y-intercept is c, since y = c when x = 0.
二次函数图像 y = ax² + bx + c 是抛物线。若 a > 0，抛物线开口向上且有最低点；若 a < 0，开口向下且有最高点。y 轴截距为 c，因为当 x = 0 时 y = c。
The line of symmetry is vertical and passes through the turning point. Its equation is:
对称轴是垂直的，并经过转折点。其方程为：
x = −b ÷ 2a
To sketch the graph, calculate the y-intercept, the roots if possible, and the turning point. Then draw a smooth curve, not a series of straight line segments.
绘制草图时，计算 y 轴截距、可能存在的根以及转折点。然后画出平滑曲线，而不是一段段直线。
6. Using Graphs to Solve Equations | 利用图像解方程
graph of y = ax² + bx + c crosses the x-axis. These are also called roots or x-intercepts.
方程 ax² + bx + c = 0 的解是 y = ax² + bx + c 的图像与 x 轴交点的 x 坐标。这些也称为根或 x 轴截距。
section points of y = x² + 2x − 3 and y = x + 1 give the solutions to x² + 2x − 3 = x + 1.
你可能还需要用图像法解联立方程。例如，y = x² + 2x − 3 与 y = x + 1 的交点给出方程 x² + 2x − 3 = x + 1 的解。
7. Right-Angled Triangle Trigonometry | 直角三角形三角学
In a right-angled triangle, the three trigonometric ratios relate the angles to the side lengths. Label the sides as opposite, adjacent and hypotenuse relative to a chosen acute angle θ.
在直角三角形中，三个三角比将角度与边长联系起来。相对于选定的锐角 θ，将边标记为对边、邻边和斜边。
sin θ = opposite ÷ hypotenuse
sin θ = 对边 ÷ 斜边
cos θ = adjacent ÷ hypotenuse
cos θ = 邻边 ÷ 斜边
tan θ = opposite ÷ adjacent
tan θ = 对边 ÷ 邻边
A common memory aid is SOH CAH TOA: Sine equals Opposite over Hypotenuse, Cosine equals Adjacent over Hypotenuse, Tangent
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