📚 CCEA Year 12 Maths Terminology Quick Memorisation Guide | CCEA Year 12 数学词汇术语速记指南
Mastering mathematical terminology is not just about passing exams – it is the key that unlocks problem-solving confidence. For CCEA Year 12 (AS Level) Mathematics, the precise use of language separates a good answer from a vague one. This guide breaks down essential terms into manageable groups, pairs each with a clear English definition and a Chinese explanation, and offers memory hooks rooted in sound mathematics. Work through each section steadily, and your command of the core vocabulary will become second nature.
掌握数学术语不仅是为了通过考试——更是打开解题自信的钥匙。对于 CCEA Year 12（AS 阶段）数学，语言的精确运用是把一个模棱两可的答案变成精准答案的关键。这份指南将核心术语分解成易于消化的组块，每个术语都配有清晰的英文定义和中文解释，并提供建立在扎实数学基础上的记忆钩子。循序渐进地学习每一节，你对核心词汇的驾驭将变为本能反应。
1. Foundation Algebraic Terminology | 代数基础术语
A term is a single mathematical entity, such as 5x² or 7, separated by addition or subtraction. In the expression 3x² + 2y − 5, there are three terms.
项 是一个单独的数学实体，比如 5x² 或 7，由加减号分隔。在表达式 3x² + 2y − 5 中，有三个项。
A coefficient is the numerical factor multiplying a variable. For −4x³, the coefficient is −4. The constant term is a number that stands alone, like −5 above. To remember, think of the coefficient as the ‘co-pilot’ that rides alongside the variable, while the constant remains ‘constant’ – it never changes.
系数 是乘以变量的数值因子。对于 −4x³，系数是 −4。常数项 是单独出现的数字，比如上面的 −5。为方便记忆，可将系数想象成伴随变量的“副驾驶”，而常数始终保持“不变”。
The degree of a polynomial is the highest power of the variable. In 2x⁴ + x³ − 7, the degree is 4. A monomial has one term (e.g., 8y²), a binomial two (e.g., x+3), and a trinomial three (e.g., x²−5x+6). Link ‘mono’ to ‘one’, ‘bi’ to ‘bicycle (two wheels)’, and ‘tri’ to ‘tricycle’.
多项式的次数 是变量的最高次幂。在 2x⁴ + x³ − 7 中，次数为 4。单项式 有一项（如 8y²），二项式 有两项（如 x+3），三项式 有三项（如 x²−5x+6）。将 ‘mono’ 联系到“单一”，’bi’ 联系到“自行车（两个轮子）”，’tri’ 联系到“三轮车”。
A linear expression has degree 1 (ax + b), and a quadratic expression has degree 2 (ax² + bx + c). The word ‘quad’ hints at ‘square’ (think quadrilateral – a four‑sided shape), even though quadratics involve a square of the variable, not four.
线性表达式 次数为 1（ax + b），二次表达式 次数为 2（ax² + bx + c）。“quad” 一词暗指“平方”（想想 quadrilateral 四边形），虽然二次式涉及变量的平方，而非四。
2. Solving Equations and Inequalities | 解方程与不等式术语
An equation states that two expressions are equal, such as 2x+3 = 11. The goal is to solve it to find the value(s) of the unknown that make the statement true. An identity, however, is true for all values of the variable and uses the ≡ symbol, e.g., (x+1)² ≡ x²+2x+1. Think of identity as ‘identical’ – the two sides are identical for every input.
方程 陈述两个表达式相等，比如 2x+3 = 11。求解的目的是解方程，找到使等式成立的未知数值。而恒等式 对所有变量的取值都成立，使用符号 ≡，例如 (x+1)² ≡ x²+2x+1。可将恒等式视为“总是相同”——对于任何输入两边都完全相同。
Inequalities use symbols <, >, ≤, ≥ to show order relations. ‘≤’ means ‘less than or equal to’. When multiplying or dividing by a negative number, you must reverse the inequality sign. A helpful memory trick: ‘flip the sign when you multiply by a minus – the number line flips too.’
不等式 使用符号 <, >, ≤, ≥ 表示大小关系。’≤’ 表示“小于或等于”。当乘以或除以负数时，必须反转不等号。一个助记窍门是：“乘负号，不等号翻转——数轴也跟着翻转。”
The discriminant (Δ) of ax²+bx+c=0 is b²−4ac. It determines the nature of the roots: Δ>0 gives two distinct real roots, Δ=0 gives one repeated real root, and Δ<0 gives no real roots. Relate ‘discriminant’ to ‘discern’ – it helps you discern what kind of roots you have.
二次方程 ax²+bx+c=0 的判别式（Δ）是 b²−4ac。它决定根的性质：Δ>0 有两个不等实根，Δ=0 有一个重实根，Δ<0 无实根。将 ‘discriminant’ 联系到 ‘discern’（辨别）——它帮你辨别根的类型。
3. Functions and Graphs | 函数与图像术语
A function f maps each input (x) from its domain to exactly one output f(x) in its range. The notation f : x → x² or f(x) = x² is read ‘f of x equals x squared’. To remember domain vs. range: ‘Domain is x, which goes in; Range is y, the output you get.’
函数 f 将其定义域 中的每个输入（x）映射到其值域 中唯一的输出 f(x)。符号 f : x → x² 或 f(x) = x² 读作“f of x 等于 x 平方”。记忆定义域和值域：定义域是输入的 x，值域是得到的输出 y。
A composite function f(g(x)) applies g first, then f. First ‘inner’, then ‘outer’. The inverse function f⁻¹(x) reverses the effect of f, so that f⁻¹(f(x)) = x. Note: f⁻¹ is not the same as a power of −1; it denotes the inverse. To find the inverse, swap x and y and rearrange – ‘swap and solve’.
复合函数 f(g(x)) 先执行 g，再执行 f。先“内”后“外”。反函数 f⁻¹(x) 逆转 f 的作用，满足 f⁻¹(f(x)) = x。注意：f⁻¹ 并非 −1 次方，它表示反函数。求反函数的方法是交换 x 和 y 并整理——“交换再解出”。
The root or zero of a function is an x-value that makes f(x)=0. Graphically, this is where the curve crosses the x‑axis. A turning point is where the gradient changes from positive to negative (a maximum) or negative to positive (a minimum). CCEA often asks for the nature of a turning point using the second derivative.
函数的根 或 零点 是使 f(x)=0 的 x 值。图像上为曲线与 x 轴的交点。驻点 是梯度由正变负（极大值）或由负变正（极小值）的位置。CCEA 经常要求利用二阶导数判断驻点类型。
4. Coordinate Geometry of Straight Lines | 直线坐标几何术语
The gradient (slope) of a line between (x₁, y₁) and (x₂, y₂) is m = (y₂−y₁) / (x₂−x₁). A positive gradient slopes upward, a negative gradient slopes downward. A horizontal line has gradient zero; a vertical line has undefined gradient.
连接 (x₁, y₁) 和 (x₂, y₂) 的直线梯度（斜率）为 m = (y₂−y₁) / (x₂−x₁)。正梯度向上倾斜，负梯度向下倾斜。水平线梯度为零；垂直线梯度无定义。
The equation of a line can take the form y = mx + c, where m is the gradient and c is the y‑intercept (where the line crosses the y‑axis). Alternatively, the point‑gradient form is y − y₁ = m(x − x₁). The phrase ‘rise over run’ helps you recall that gradient equals vertical change divided by horizontal change.
直线方程 可写成 y = mx + c 的形式，其中 m 为梯度，c 为 y 轴截距（直线与 y 轴的交点纵坐标）。也可用点斜式：y − y₁ = m(x − x₁)。“升高比平移”（rise over run）助记梯度等于纵向变化除以横向变化。
Parallel lines have the same gradient: m₁ = m₂. Perpendicular lines have gradients that multiply to −1: m₁ × m₂ = −1, often described as ‘negative reciprocals’. Think: ‘parallel means same slope; perpendicular means flip the fraction and change the sign.’
平行线 有相同的梯度：m₁ = m₂。垂直线 的梯度乘积为 −1：m₁ × m₂ = −1，常描述为“负倒数”。记法：“平行是同斜率，垂直是分子分母互换再变号。”
The midpoint of the segment joining (x₁, y₁) and (x₂, y₂) is ((x₁+x₂)/2, (y₁+y₂)/2). The distance between the same two points is √[(x₂−x₁)² + (y₂−y₁)²]. This distance formula comes from Pythagoras’ theorem – every straight segment is the hypotenuse of a right‑angled triangle.
连接 (x₁, y₁) 和 (x₂, y₂) 的线段中点 是 ((x₁+x₂)/2, (y₁+y₂)/2)。两点间的距离 为 √[(x₂−x₁)² + (y₂−y₁)²]。距离公式源于勾股定理——每条直线段都是某个直角三角形的斜边。
5. Circles in Coordinate Geometry | 圆的坐标几何术语
The standard equation of a circle with centre (a, b) and radius r is (x−a)² + (y−b)² = r². A key skill is completing the square to convert a general expanded form x²+y²+2gx+2fy+c=0 into the centre (-g, -f) and radius √(g²+f²−c). Remember: ‘centre is negative g, negative f’ – the signs flip.
圆心为 (a, b)、半径为 r 的标准圆方程 是 (x−a)² + (y−b)² = r²。一项关键技能是配方，将一般展开式 x²+y²+2gx+2fy+c=0 转化为圆心 (-g, -f) 和半径 √(g²+f²−c)。记住：圆心为负 g，负 f——符号要改变。
A tangent to a circle is a line that touches it at exactly one point. The tangent is perpendicular to the radius at the point of contact. A chord is a line segment connecting two points on the circle; the perpendicular from the centre to a chord bisects the chord. These two properties together form the circle geometry that CCEA examines frequently.
圆的切线 是与圆恰好有一个公共点的直线。切线与半径在切点处垂直。弦 是圆上连接两点的线段；从圆心到弦的垂线平分该弦。这两条性质合起来构成 CCEA 常考的圆的几何。
6. Sequences and Series | 数列与级数术语
A sequence is an ordered list of numbers following a rule, e.g., 2, 5, 8, 11,… . A series is the sum of the terms of a sequence. The nth term (general term) of a sequence is often denoted uₙ. In an arithmetic progression (AP), the difference between consecutive terms is constant – the common difference d.
数列 是按规则排列的有序数字列表，如 2, 5, 8, 11, …。级数 是数列各项之和。数列的第 n 项（通项）常记作 uₙ。在等差数列 (AP) 中，相邻两项之差为常数——公差 d。
In a geometric progression (GP), the ratio between consecutive terms is constant – the common ratio r. The formula for the nth term of a GP is arⁿ⁻¹, where a is the first term. To distinguish AP from GP: AP ‘adds’, GP ‘multiplies’.
在等比数列 (GP) 中，相邻两项的比值为常数——公比 r。等比数列第 n 项公式为 arⁿ⁻¹，a 为首项。区分等差和等比：等差数列做加法，等比数列做乘法。
The sum of the first n terms of an AP is Sₙ = n/2 [2a + (n−1)d] or n/2 (a + l), where l is the last term. For a GP, the finite sum is Sₙ = a(1−rⁿ)/(1−r) for r ≠ 1. An infinite geometric series converges to a sum S∞ = a/(1−r) only when |r| < 1. The notation Σ (sigma) is used for summation.
等差数列前 n 项和为 Sₙ = n/2 [2a + (n−1)d] 或 n/2 (a + l)，其中 l 是末项。等比数列有限和为 Sₙ = a(1−rⁿ)/(1−r) （r ≠ 1）。无穷等比级数 仅当 |r| < 1 时收敛，和为 S∞ = a/(1−r)。求和符号 Σ（西格玛）用于表示求和。
7. Differentiation – Core Concepts | 微分核心概念
Differentiation is the process of finding the derivative, which measures the instantaneous rate of change of a function with respect to its variable. The derivative of f(x) is written f′(x) or dy/dx. Leibniz notation dy/dx reads ‘the derivative of y with respect to x’ and emphasises that the derivative is a rate, not a quotient, even though it looks like a fraction.
微分 是求导数的过程，导数衡量函数关于其变量的瞬时变化率。f(x) 的导数 写作 f′(x) 或 dy/dx。莱布尼茨符号 dy/dx 读作“y 对 x 的导数”，强调导数是一种变化率而非商，尽管它看起来像分数。
The power rule for differentiating xⁿ is d/dx (xⁿ) = nxⁿ⁻¹. For example, the derivative of x⁵ is 5x⁴. To remember: ‘bring the power down and reduce the power by one.’ A constant function differentiates to zero because it has no rate of change.
微分 xⁿ 的幂法则 为 d/dx (xⁿ) = nxⁿ⁻¹。例如，x⁵ 的导数是 5x⁴。记忆：“把指数拉下来乘以系数，再将指数减 1。”常函数 的导数为零，因为它没有变化率。
A stationary point occurs where f′(x)=0. The second derivative f″(x) tells you whether the point is a minimum (f″(x)>0), maximum (f″(x)<0), or point of inflection (f″(x)=0, with sign change). CCEA expects you to determine the nature of stationary points using the second derivative test or a gradient table.
静止点 出现在 f′(x)=0 处。二阶导数 f″(x) 告诉你该点是极小值（f″(x)>0）、极大值（f″(x)<0）还是拐点（f″(x)=0 且符号改变）。CCEA 要求利用二阶导数检验或梯度表判断驻点类型。
The tangent to a curve at a point has gradient equal to the derivative at that point. The normal is perpendicular to the tangent, so its gradient is the negative reciprocal of the derivative. Think ‘tangent touches; normal is normal (perpendicular).’
曲线上某点的切线 斜率等于该点的导数值。法线 与切线垂直，因此其斜率为导数的负倒数。联想：“切线相切，法线正交（垂直）。”
8. Integration – Reverse Differentiation | 积分——微分的逆运算
Indefinite integration is the reverse of differentiation. The notation ∫ f(x) dx gives the antiderivative, plus a constant of integration C. Since the derivative of a constant is zero, every antiderivative includes +C. The rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ −1.
不定积分 是微分的逆运算。符号 ∫ f(x) dx 给出原函数，加上积分常数 C。因为常数的导数为零，所有原函数都包含 +C。法则：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，n ≠ −1。
Definite integration ∫ₐᵇ f(x) dx computes the net area between the curve and the x‑axis from x=a to x=b. The Fundamental Theorem of Calculus connects differentiation and integration: if F is an antiderivative of f, then ∫ₐᵇ f(x) dx = F(b) − F(a). Remember ‘top minus bottom’ when substituting limits.
定积分 ∫ₐᵇ f(x) dx 计算从 x=a 到 x=b 曲线与 x 轴之间的净面积。微积分基本定理 将微分与积分联系起来：若 F 是 f 的原函数，则 ∫ₐᵇ f(x) dx = F(b) − F(a)。代入上下限时记住“上限减下限”。
Integration can also be used to find areas between a curve and the y‑axis, areas between two curves, and volumes of revolution (though volumes are less common at Year 12). When finding the area between curves, subtract the lower function from the upper: ∫ (upper − lower) dx.
积分也可用于求曲线与 y 轴之间的面积、两曲线之间的面积以及旋转体体积（尽管 Year 12 较少涉及体积）。求两曲线之间面积时，用上方函数减去下方函数：∫ (上方 − 下方) dx。
9. Exponentials and Logarithms | 指数与对数
An exponential function has the form aˣ, most importantly eˣ where e ≈ 2.71828. The function eˣ is its own derivative: d/dx(eˣ) = eˣ, and its integral is eˣ + C. The natural logarithm, written ln x, is the inverse of eˣ: if y = eˣ then x = ln y. The domain of ln x is x>0.
指数函数 具有 aˣ 的形式，最重要的是 eˣ，其中 e ≈ 2.71828。函数 eˣ 的导数等于它自身：d/dx(eˣ) = eˣ，其积分为 eˣ + C。自然对数，记作 ln x，是 eˣ 的反函数：若 y = eˣ，则 x = ln y。ln x 的定义域为 x>0。
Laws of logarithms are essential: logₐ(xy) = logₐx + logₐy; logₐ(x/y) = logₐx − logₐy; logₐ(xⁿ) = n logₐx. A memorable phrase: ‘Logs turn multiplication into addition, division into subtraction, and powers into multiplication.’ The change of base formula: logₐ b = log_c b / log_c a.
对数运算法则 至关重要：logₐ(xy) = logₐx + logₐy；logₐ(x/y) = logₐx − logₐy；logₐ(xⁿ) = n logₐx。一句俏皮话助记：“对数把乘法变加法，除法变减法，乘方变乘法。”换底公式 为：logₐ b = log_c b / log_c a。
The derivative of ln x is 1/x, and ∫ 1/x dx = ln |x| + C. For exponentials with a base other than e, use aˣ = eˣ ln a. CCEA often sets problems where you must model growth or decay using eᵏᵗ or combine logarithmic and exponential equations.
ln x 的导数是 1/x，∫ 1/x dx = ln |x| + C。对于底数不是 e 的指数函数，可使用 aˣ = eˣ ln a。CCEA 常设置用 eᵏᵗ 建立增长或衰减模型，或结合对数与指数方程的问题。
10. Trigonometric Functions and Equations | 三角函数与方程术语
The three primary trigonometric ratios in a right‑angled triangle are sine (sin), cosine (cos), and tangent (tan). SOHCAHTOA is the classic memory tool: Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent. CCEA extends these to all real angles using the unit circle and degrees.
直角三角形中的三个三角比 是正弦（sin）、余弦（cos）和正切（tan）。SOHCAHTOA 是经典助记口诀：正弦 = 对边/斜边，余弦 = 邻边/斜边，正切 = 对边/邻边。CCEA 利用单位圆将这些定义扩展到所有实角（度数）。
Exact trigonometric values for 30°, 45°, and 60° must be memorised: sin30°=½, sin45°=1/√2, sin60°=√3/2; cos30°=√3/2, cos45°=1/√2, cos60°=½; tan30°=1/√3, tan45°=1, tan60°=√3. A visual aid: draw an equilateral triangle of side 2 for 60° and 30°, and a right‑angled isosceles triangle for 45°.
必须熟记 30°、45° 和 60° 的精确三角值：sin30°=½, sin45°=1/√2, sin60°=√3/2；cos30°=√3/2, cos45°=1/√2, cos60°=½；tan30°=1/√3, tan45°=1, tan60°=√3。视觉辅助：画边长为 2 的等边三角形得到 60° 和 30° 的值，画等腰直角三角形得到 45° 的值。
Trigonometric identities such as sin²θ + cos²θ ≡ 1 and tanθ ≡ sinθ/cosθ are fundamental for solving equations. The derivative of sin x is cos x, and of cos x is −sin x. A neat way to remember: ‘Differentiating sine gives cosine (S → C), and cosine gives minus sine (C → −S).’ The period of sin and cos is 360°, and for tan it is 180°.
三角恒等式 如 sin²θ + cos²θ ≡ 1 和 tanθ ≡ sinθ/cosθ 是解三角方程的基础。sin x 的导数 是 cos x，cos x 的导数是 −sin x。巧妙记法：“正弦微分成余弦 (S → C)，余弦微分成负正弦 (C → −S)。”sin 和 cos 的周期 为 360°，tan 的周期为 180°。
11. Vectors in Two Dimensions | 二维向量术语
A vector is a quantity with both magnitude and direction, represented by a directed line segment. A scalar has magnitude only. Vectors are often written in bold (v) or as a column matrix [x, y]ᵀ. The magnitude (modulus) of a vector a = xi + yj is |a| = √(x² + y²). This is essentially Pythagoras again.
向量 是既有大小又有方向的量，用有向线段表示。标量 只有大小。向量常以粗体（v）或列矩阵 [x, y]ᵀ 书写。向量 a = xi + yj 的模（大小）为 |a| = √(x² + y²)。本质上还是勾股定理。
A unit vector has magnitude 1; i and j are the standard unit vectors along the x- and y‑axes. To find a unit vector in the direction of a, divide a by its magnitude: a/|a|. The position vector of a point P is its coordinates expressed as a vector from the origin, often OP.
单位向量 的模为 1；i 和 j 分别是沿 x 轴和 y 轴的标准单位向量。要求 a 方向上的单位向量，将 a 除以其模：a/|a|。点 P 的位置向量 是从原点出发的坐标向量，常记作 OP。
Vector addition is done component‑wise: (a₁i + a₂j) + (b₁i + b₂j) = (a₁+b₁)i + (a₂+b₂)j. The scalar product (dot product) of two vectors is a·b = |a||b|cosθ = a₁b₁ + a₂b₂. The dot product is zero for perpendicular vectors. CCEA expects you to use the dot product to find the angle between two vectors and test for perpendicularity.
向量加法 按分量相加：(a₁i + a₂j) + (b₁i + b₂j) = (a₁+b₁)i + (a₂+b₂)j。两个向量的标量积（点积）为 a·b = |a||b|cosθ = a₁b₁ + a₂b₂。垂直向量的点积为零。CCEA 要求利用点积求两向量之间的夹角并检验垂直性。
12. Statistical Measures and Probability | 统计度量与概率术语
Measures of central tendency include the mean (average), median (middle value), and mode (most frequent). The mean is sum of data values divided by the number of values. The range = maximum − minimum measures spread. For grouped data, the modal class is the class with the highest frequency.
集中趋势度量 包括均值（平均数）、中位数（中间值）和众数（出现频率最高）。均值 = 数据值总和 ÷ 数据个数。极差 = 最大值 − 最小值，衡量离散程度。对于分组数据，众数区间 是频率最高的组。
Standard deviation σ (or s) quantifies the spread around the mean. For a data set {x₁, x₂, …, xₙ}, σ = √[ Σ(xᵢ − x̄)² / n ] (population) or s = √[ Σ(xᵢ − x̄)² / (n−1) ] (sample). The term variance is σ². CCEA expects you to use the formula accurately and interpret standard deviation as a measure of consistency: lower SD = less spread, more reliable.
标准差 σ（或 s）量化围绕均值的离散程度。对于数据集 {x₁, x₂, …, xₙ}，σ = √[ Σ(xᵢ − x̄)² / n ]（总体）或 s = √[ Σ(xᵢ − x̄)² / (n−1) ]（样本）。方差 一词表示 σ²。CCEA 要求准确使用公式，并能解释标准差为一致性度量：标准差越小，离散程度越低，数据越可靠。
In probability, an event is a set of outcomes. P(A) denotes the probability of event A. Mutually exclusive events cannot occur together: P(A ∩ B) = 0. Independent events satisfy P(A ∩ B) = P(A) × P(B). Tree diagrams and Venn diagrams are essential visual tools for solving CCEA probability problems, including conditional probability: P(A|B) = P(A ∩ B) / P(B).
在概率 中，事件 是一组结果。P(A) 表示事件 A 的概率。互斥事件 不能同时发生：P(A ∩ B) = 0。独立事件 满足 P(A ∩ B) = P(A) × P(B)。树形图和韦恩图是解 CCEA 概率题的重要视觉工具，包括条件概率：P(A|B) = P(A ∩ B) / P(B)。
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