📚 Year 12 SQA Mathematics: Core Concepts Review | Year 12 SQA 数学：核心知识点梳理
This article provides a structured overview of the essential topics covered in the Year 12 SQA Mathematics course at Higher level. Each section pairs an English explanation with its Chinese equivalent to support bilingual learners and reinforce key definitions, formulas, and techniques required for exam success.
本文为 Year 12 SQA 数学（高等）课程的核心主题提供了结构化的梳理。每个部分将英文解释与中文对应，以帮助双语学习者巩固考试所必需的关键定义、公式和方法。
1. The Straight Line | 直线
The gradient between two points (x₁, y₁) and (x₂, y₂) is given by m = (y₂ − y₁)/(x₂ − x₁).
直线是坐标几何中的基本单元，贯穿于高等数学的多个主题。两点 (x₁, y₁) 和 (x₂, y₂) 之间的梯度由 m = (y₂ − y₁)/(x₂ − x₁) 给出。
The equation of a line can be written in point– x − a), where (a, b) is a point on the line, or as y = mx + c, where c is the y-intercept.
直线的方程可以写为点斜式 y − b = m(x − a)，其中 (a, b) 为直线上一点；或写为斜截式 y = mx + c，其中 c 为 y 轴截距。
× m₂ = −1. The distance between two points is found using d = √[(x₂ − x₁)² + (y₂ − y₁)²].
对于梯度分别为 m₁ 和 m₂ 的两条直线，它们垂直当且仅当 m₁ × m₂ = −1。两点间的距离用 d = √[(x₂ − x₁)² + (y₂ − y₁)²] 计算。
The angle θ between two lines with gradients m₁ and m₂ can be determined from tanθ = |(m₁ − m₂)/(1 + m₁m₂)|, and medians, altitudes, and perpendicular bisectors are common applications.
两条直线的夹角 θ 可以通过 tanθ = |(m₁ − m₂)/(1 + m₁m₂)| 求得。中线、高线和垂直平分线是常见的应用。
2. Functions and Graphs | 函数与图像
A function f maps elements from a domain to a codomain. The range is the set of all output values. Composite functions such as f(g(x)) are formed by applying one function after another, with the domain restricted so that the inner function's output lies in the outer function's domain.
函数 f 将定义域中的元素映射到值域。值域是所有输出值的集合。复合函数如 f(g(x)) 是通过将一个函数作用于另一个之后形成的，其定义域需限制以使内层函数的输出落在外层函数的定义域内。
o-one on the chosen domain. Graphs of inverse functions are reflections in the line y = x.
函数的反函数通过交换 x 和 y 并解出 y 得到，前提是原函数在所选定义域上是单射的。反函数的图像关于直线 y = x 对称。
(kx)). Understanding these transformations allows rapid sketching without plotting points.
函数图像的变换包括平移 (y = f(x) + a 和 y = f(x + a))、反射 (y = −f(x) 和 y = f(−x)) 以及伸缩 (y = kf(x) 和 y = f(kx))。理解这些变换可以在不逐点描图的情况下快速绘制草图。
3. Trigonometric Functions and Equations | 三角函数与方程
n²θ + cos²θ = 1 and tanθ = sinθ/cosθ are used to simplify expressions and solve equations.
必须掌握 sin, cos, tan 在 0°, 30°, 45°, 60°, 90° 及其弧度对应值的精确值。恒等式 sin²θ + cos²θ = 1 和 tanθ = sinθ/cosθ 用于化简表达式和求解方程。
Trigonometric equations often require rearrangement, use of identities, or factoring. Solutions are given in radian measure within a specified interval, using the CAST diagram to determine all quadrants.
三角方程通常需要重新整理、使用恒等式或因式分解。解以弧度制在指定区间内给出，借助 CAST 图确定所有象限的解。
The graphs of y = a sin(bx) + c and y = a cos(bx) + c have amplitude |a| period 2π/b, and vertical shift c. These parameters allow modelling of periodic phenomena.
y = a sin(bx) + c 和 y = a cos(bx) + c 的图像具有振幅 |a|、周期 2π/b 和垂直平移 c。这些参数可用于对周期现象建模。
4. Differentiation | 微分
ve at a point. The derivative of xⁿ is nxⁿ⁻¹. For a function y = f(x), we write dy/dx or f'(x). Standard results include d/dx (sin x) = cos x, d/dx (cos x) = −sin x, d/dx (eˣ) = eˣ, and d/dx (ln x) = 1/x.
微分给出曲线上某点的梯度。xⁿ 的导数为 nxⁿ⁻¹。对函数 y = f(x)，记作 dy/dx 或 f'(x)。标准结果包括 d/dx (sin x) = cos x, d/dx (cos x) = −sin x, d/dx (eˣ) = eˣ 以及 d/dx (ln x) = 1/x。
The chain rule is used for composite functions: if y = f(g(x)), then dy/dx = f'(g(x))·g'(x). The product rule states that if y = u(x)v(x), then dy/dx = u'v + uv'. The quotient rule gives dy/dx = (u'v − uv')/v² for y = u/v.
链式法则用于复合函数：若 y = f(g(x))，则 dy/dx = f'(g(x))·g'(x)。积法则指出若 y = u(x)v(x)，则 dy/dx = u'v + uv'。商法则给出 y = u/v 时的导数 dy/dx = (u'v − uv')/v²。
Derivatives are used to find equations of tangents and normals, rates of change, and to classify stationary points (maximum, minimum, or point of inflection) using nature tables or the second derivative.
导数可用于求切线和法线方程、变化率，以及利用符号表或二阶导数对驻点进行分类（极大值、极小值或拐点）。
5. Integration | 积分
Integration reverses differentiation. The indefinite integral ∫ f' is the constant of integration. The power rule gives ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C for n ≠ −1.
积分是微分的逆运算。不定积分 ∫ f'(x) dx = f(x) + C，其中 C 为积分常数。幂函数的积分公式为 ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C（n ≠ −1）。
Standard integrals include ∫ sin x dx = −cos x + C, ∫ cos x dx = sin x + C, ∫ eˣ dx = eˣ + C, and ∫ (1/x) dx = ln |x| + C. These are combined with constant multiples and sums.
标准积分包括 ∫ sin x dx = −cos x + C、∫ cos x dx = sin x + C、∫ eˣ dx = eˣ + C 以及 ∫ (1/x) dx = ln |x| + C。这些可与常数倍及和结合使用。
und by integrating the difference of the functions over the interval where they intersect.
定积分 ∫ₐᵇ f(x) dx 表示曲线 y = f(x) 与 x 轴之间从 x = a 到 x = b 的有向面积。两曲线之间的面积通过对它们在交点区间上函数差的积分求得。
6. Exponentials and Logarithms | 指数与对数
The exponential function f(x) = eˣ has derivative eˣ and integral eˣ by functions of the form A eᵏᵗ. Taking natural logarithms helps solve equations such as e²ˣ = 5.
指数函数 f(x) = eˣ 的导数仍为 eˣ，积分也是 eˣ + C。指数增长和衰减用形如 A eᵏᵗ 的函数建模。取自然对数有助于求解 e²ˣ = 5 这类方程。
− logₐ y, and logₐ (xⁿ) = n logₐ x. The natural logarithm ln x is the logarithm to base e.
对数法则可用于处理表达式：logₐ (xy) = logₐ x + logₐ y，logₐ (x/y) = logₐ x − logₐ y，logₐ (xⁿ) = n logₐ x。自然对数 ln x 是以 e 为底的对数。
The derivative of aˣ is aˣ is 1/(x ln a). The relationship logₐ b = ln b / ln a allows change of base when required.
aˣ 的导数是 aˣ ln a，logₐ x 的导数是 1/(x ln a)。关系式 logₐ b = ln b / ln a 可以在需要时进行底数的转换。
7. Vectors | 向量
ai + bj + ck. The position vector of a point gives its coordinates relative to the origin.
向量具有大小和方向。在二维和三维中，向量用分量 i, j, k 表示，例如 v = ai + bj + ck。点的位置向量给出其相对于原点的坐标。
The magnitude (length) of v = ai + bj is |v| = √a² + b². For v = ai + bj + ck, |v| = √a² + b² + c². The dot product of two vectors a and b is a·b = |a||b|cosθ = a₁b₁ + a₂b₂ + a₃b₃, where θ is the angle between them.
向量 v = ai + bj 的模（长度）为 |v| = √a² + b²。对于 v = ai + bj + ck，|v| = √a² + b² + c²。两向量 a 和 b 的点积为 a·b = |a||b|cosθ = a₁b₁ + a₂b₂ + a₃b₃，其中 θ 为它们之间的夹角。
Collinearity is established if one vector is a scalar multiple of another, or if AB and AC are parallel and share a point. Vectors are used to prove geometric relationships and find coordinates of section points.
若一个向量是另一个向量的标量倍数，或者 AB 与 AC 平行且共点，则可判定共线。向量用于证明几何关系并求分点坐标。
8. Polynomials and the Factor Theorem | 多项式与因式定理
ainder is f(h). The Factor Theorem follows: if f(h) = 0, then (x − h) is a factor of f(x).
余数定理指出，当多项式 f(x) 除以 (x − h) 时，余数为 f(h)。因式定理则说明：若 f(h) = 0，则 (x − h) 是 f(x) 的一个因式。
Polynomial long division or synthetic division is used to factorise fully and to solve polynomial equations. Repeated factors lead to touching or inflection at the root. Knowing the shape of cubic and quartic graphs aids sketching.
多项式长除法或综合除法用于完全因式分解和求解多项式方程。重复因式导致在根处相切或出现拐点。了解三次和四次函数的图形有助于绘制草图。
xₙ₊₁ = g(xₙ) may be used to locate an approximate root when exact algebra is not possible.
求 f(x) = k 的根也相当于解 f(x) − k = 0，可在整理后应用因式定理。当无法精确求解时，可使用迭代形式 xₙ₊₁ = g(xₙ) 来逼近根。
9. Recurrence Relations | 递推关系
relation of the form uₙ₊₁ = auₙ + b with a given initial value u₀ describes a sequence. If |a| < 1, the sequence converges to a limit L which satisfies L = aL + b, giving L = b/(1 − a).
形如 uₙ₊₁ = auₙ + b 的递推关系以及给定的初值 u₀ 描述了一个序列。如果 |a| < 1，该序列收敛于极限 L，L 满足 L = aL + b，从而 L = b/(1 − a)。
Recurrence relations are applied in modelling contexts, such as population growth, medication dosages, and finance. Understanding the long‑term behaviour and limit is a key objective.
递推关系被应用于建模情境，如人口增长、药物剂量和金融。理解长期行为和极限是核心目标。
10. The Circle | 圆
l equation of a circle is x² + y² + 2gx + 2fy + c = 0, with centre (−g, −f) and radius r = √g² + f² − c, provided g² + f² − c > 0. Completing the square is the standard technique to convert from general to centre–radius form.
圆的一般方程为 x² + y² + 2gx + 2fy + c = 0，圆心为 (−g, −f)，半径 r = √g² + f² − c，要求 g² + f² − c > 0。配方是将一般形式化为标准形式的标准方法。
uses the fact that the radius is perpendicular to the tangent. The discriminant of a line– can determine if a line intersects (Δ > 0), touches (Δ = 0), or misses (Δ < 0) the circle.
求圆上一点 (x₁, y₁) 处的切线方程，可利用半径与切线垂直的性质。联立直线和圆的方程所得二次式的判别式可判断直线与圆是相交 (Δ > 0)、相切 (Δ = 0) 还是相离 (Δ < 0)。
Intersections and tangency conditions link circles with the straight line chapter and are frequently examined. In many problems, substituting the line equation into the circle equation yields a quadratic where the discriminant gives the required condition.
相交和相切的条件将圆与直线的章节联系起来，常出现在考题中。在许多问题中，把直线方程代入圆的方程会得到一个二次式，利用其判别式可得出所需的条件。
11. Optimisation and Rates of Change | 优化与变化率
Optimisation problems require expressing a quantity as a function of a single variable, then differentiating to find stationary points. First or second derivative tests are used to confirm maximum or minimum values.
优化问题需要将某个量表示为单一变量的函数，然后通过微分找到驻点。用一阶或二阶导数检验来确认最大值或最小值。
Typical contexts include minimising surface area for a fixed volume, maximising area, or finding the most profitable production level. Always check the domain of the problem and that the answer makes practical sense.
典型的背景包括固定体积下最小化表面积、最大化面积，或寻找最盈利的生产水平。务必要检查问题的定义域，并确认答案在实际中合理。
. For example, if a volume expands and the radius increases at a known rate, the rate of change of volume can be found by linking derivatives.
相关变化率涉及链式法则：dy/dt = (dy/dx)(dx/dt)。例如，若体积膨胀且半径以已知速率增加，可以通过联系各个导数求出体积的变化率。
12. Using the Discriminant | 判别式的应用
al roots if Δ > 0, one repeated real root (tangency) if Δ = 0, and no real roots if Δ < 0.
对于二次方程 ax² + bx + c = 0，判别式 Δ = b² − 4ac 决定了根的性质：若 Δ > 0，有两个不等实根；若 Δ = 0，有一个重根（相切）；若 Δ < 0，无实根。
Discriminant analysis is used when investigating intersections of lines with parabolas, lines with circles, or the existence of solutions to equations involving quadratics. It provides a powerful algebraic method to find tangents or conditions for intersection without solving the full system.
当研究直线与抛物线、直线与圆的相交情况，或判断含二次式的方程是否有解时，常使用判别式分析。它提供了一种强大的代数方法，无需解出整个方程组即可求出切线或相交的条件。
conditions for tangency. This unites algebraic manipulation with geometric interpretation.
建立交点方程、化简为二次式，并令 Δ = 0 即可得出相切的必要条件。这将代数运算与几何解释统一了起来。
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