📚 Year 10 SQA Advanced Maths: Glossary Quick Memory Guide | Year 10 SQA 进阶数学：词汇术语速记指南
Mastering the precise language of advanced mathematics is just as important as solving equations. A firm grasp of key terms—like 'domain', 'discriminant', 'derivative', and 'integral'—can turn ambiguous exam questions into straightforward pathways. This guide introduces essential SQA vocabulary alongside memory hooks to help you recall definitions instantly and use them confidently in context.
掌握进阶数学的精准语言与解方程同等重要。透彻理解“定义域”、“判别式”、“导数”、“积分”等核心术语，能把模糊的考题转化为清晰的解题思路。本指南精选SQA关键词汇，搭配记忆口诀，助你瞬间记起定义，并在场景中自信运用。
1. Functions and Relations | 函数与关系
A function is a rule that assigns exactly one output for every input. It is often written as f(x), and we say 'f of x'. Two distinct functions can share the same expression but have different domains, so always check the allowed input values.
函数是为每个输入值分配唯一输出值的规则，常写作 f(x)，读作“f of x”。两个不同的函数可以有相同表达式但定义域不同，因此务必留意允许的输入值。
The domain is the set of all possible input values, while the range is the set of all possible outputs. Visualising a function's graph can help: the domain sprawls along the x-axis, the range stretches along the y-axis.
定义域是所有可能输入值的集合，值域是所有可能输出值的集合。想象函数图像有助于记忆：定义域沿 x 轴伸展，值域沿 y 轴延伸。
A function is one-to-one if each output comes from exactly one input, and it is onto if every element of the codomain is an image. The horizontal line test checks one-to-one status. Quick memory: 'domain = x-drama, range = y-range'.
若每个输出对应唯一输入，函数是一一映射；若陪域中每个元素都是像，则是映成。水平线检验可判断是否为一对一。记忆口诀：“定义域看 x 轴，值域看 y 轴”。
2. Quadratic Functions and the Discriminant | 二次函数与判别式
A quadratic function has the form ax² + bx + c, where a ≠ 0. Its graph is a parabola that opens upward if a > 0 and downward if a < 0. The symmetry axis and turning point uncover its minimum or maximum value.
二次函数形如 ax² + bx + c（a ≠ 0）。其图像是抛物线，a > 0 时开口向上，a < 0 时开口向下。对称轴和转折点揭示最小值或最大值。
The discriminant n: Δ > 0 → two distinct real roots; Δ = 0 → one repeated real root; Δ < 0 → no real roots.
判别式 Δ = b² − 4ac 不解方程即可判断根的性质。记住决策链：Δ > 0 → 两个不等实根；Δ = 0 → 一个重根；Δ < 0 → 无实根。
Memory tip: the discriminant 'discriminates' between root types — positive yields a pair, zero gives a twin, negative hides in the complex. You can also underline that Δ sits inside the square root of the quadratic formula, so its sign rules reality.
助记诀窍：判别式“判别”根的类型——正数出双根，零出同根，负数无实根。也可强调 Δ 位于求根公式平方根内部，因此它的符号决定实根存在与否。
3. Exponential and Logarithmic Functions | 指数与对数函数
An exponential function f(x) = aˣ rate proportional to its value. The special base e ≈ 2.718 gives the natural exponential eˣ, which has the neat property that its derivative is itself.
指数函数 f(x) = aˣ（a > 0, a ≠ 1）以正比于自身的速率增长或衰减。特殊底数 e ≈ 2.718 构成自然指数 eˣ，其优美性质是导数等于自身。
A logarithm answers the question: 'to what power must the base be raised to get a given number?' That is, logₐ y = x means aˣ = y. The natural logarithm ln x is logₑ x, and it is the inverse of eˣ.
对数回答：“底数必须升到多少次幂才能得到给定数字？”即 logₐ y = x ⇔ aˣ = y。自然对数 ln x 即 logₑ x，是 eˣ 的反函数。
ormula logₐ b = ln b / ln a is your escape route when calculators only offer ln and log₁₀.
核心法则：logₐ(xy) = logₐ x + logₐ y，logₐ(x/y) = logₐ x − logₐ y，logₐ(xⁿ) = n logₐ x。换底公式 logₐ b = ln b / ln a 是当计算器只有 ln 和 log₁₀ 时的救命稻草。
4. Trigonometry: Ratios and Radians | 三角学：比率与弧度
The three primary trigonometric ratios are sine, cosine and tangent, defined for an angle θ in a right‑ : sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. SOH‑CAH‑TOA is the classic mnemonic.
三个基本三角比为正弦、余弦与正切，对直角三角形中的角 θ 定义为：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。经典口诀 SOH‑CAH‑TOA 永不过时。
Radian measure dian is the angle subtended by an arc equal in length to the radius. Radians make calculus–igonometry relationships clean, e.g. d/dx (sin x) = cos x only works when x is in radians.
弧度制是角的天然单位：180° = π rad。一弧度是弧长等于半径时所对的圆心角。弧度使得微积分与三角的关系变得简洁，例如 d/dx (sin x) = cos x 仅在 x 为弧度时成立。
on angles to memorise: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2. Picture the unit circle — the x‑coordinate is cos, the y‑coordinate is sin.
须熟记的常见角：30° = π/6，45° = π/4，60° = π/3，90° = π/2。脑中构想单位圆——x 坐标为 cos，y 坐标为 sin。
5. Trigonometric Identities | 三角恒等式
A trigonometric identity tal identity sin²θ + cos²θ = 1 is derived from the Pythagorean theorem on the unit circle.
三角恒等式是对变量所有允许值都成立的等式。基本恒等式 sin²θ + cos²θ = 1 源自单位圆上的勾股定理。
ng by cos²θ gives 1 + tan²θ = sec²θ, and dividing by sin²θ gives 1 + cot²θ = csc²θ. Double‑ n 2θ = 2 sin θ cos θ and cos 2θ = cos²θ − sin²θ are shortcuts for simplifying expressions.
除以 cos²θ 得 1 + tan²θ = sec²θ，除以 sin²θ 得 1 + cot²θ = csc²θ。倍角公式如 sin 2θ = 2 sin θ cos θ 和 cos 2θ = cos²θ − sin²θ 是化简表达式的捷径。
Memory aid: 'Sine squared plus cosine squared is one' rhymes; for tan², remember 'tan plus one is sec² — just like a knight's sec‑ret formula'. Chant them aloud when solving trig equations.
记忆辅助：“sin 方加 cos 方等于一”朗朗上口；对 tan² 可记“tan 加一等于 sec 方，如同骑士的秘方”。解三角方程时不妨出声吟诵。
6. Differentiation: Derivatives and Rules | 微分：导数与法则
The derivative f'(x) or dy/dx measures the instantaneous rate of change of a function. Geometrically, it is the gradient of the tangent line at a point. The process of finding derivatives is called differentiation.
导数 f'(x) 或 dy/dx 度量函数的变化率。几何上，它是某点处切线的斜率。求导数的过程称为微分。
Power rule: d/dx (xⁿ) = n xⁿ⁻¹. This is the workhorse — nent by one. For example, d/dx (x⁵) = 5x⁴. Extend it to sums and differences term by term.
幂法则：d/dx (xⁿ) = n xⁿ⁻¹。这是主力法则——把指数挪下来，指数减一。例如 d/dx (x⁵) = 5x⁴。逐项地延用至和与差。
The chain rule handles composite functions: dy/dx = dy/du × du/dx. Think 'derivative of the outside times derivative of the inside'. For h(x) = (2x+3)⁴, the derivative is 4(2x+3)³ × 2.
链式法则处理复合函数：dy/dx = dy/du × du/dx。可联想为“外层导数乘内层导数”。如 h(x) = (2x+3)⁴，导数为 4(2x+3)³ × 2。
A stationary point occurs where f'(x) = 0. It can be a maximum, minimum, or point of inflection. The second derivative f''(x) provides a test: f''(x) > 0 suggests a minimum, f''(x) < 0 a maximum.
平稳点出现在 f'(x) = 0 处。它可以是最大值、最小值或拐点。二阶导数 f''(x) 提供检验：f''(x) > 0 暗示最小值，f''(x) < 0 则最大值。
7. Integration: Antiderivatives and Area | 积分：原函数与面积
Integration integration C. The power rule for integration reads ∫ 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 给出原函数族加上积分常数 C。积分的幂法则为 ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C（n ≠ −1）。
The definite integral ∫ₐᵇ f(x) dx represents the signed area between the curve and the x‑ ifferentiation and integration: you evaluate an antiderivative at the limits and subtract.
定积分 ∫ₐᵇ f(x) dx 表示从 x = a 到 x = b 曲线与 x 轴之间的带符号面积。微积分基本定理将微分与积分联系起来：求出原函数后在上下限取值并相减。
Memory tip: 'Integration accumulates pieces; differentiation slices into slopes.' To remember +C, think 'C is the constant we must always add, just like you always put the lid on a jar after use.' The area under the x‑axis comes out negative, so split the interval if needed.
记忆贴士：“积分是聚沙成塔，微分是切见斜率。”记住 +C 可联想为“恒加常数 C，如同用后必盖瓶盖”。x 轴下方的面积为负值，必要时可分割区间。
8. Vectors: Magnitude and Direction | 向量：大小与方向
A vector has both magnitude and direction, unlike a scalar which has only magnitude. In the plane, it can be written as a column matrix [x; y] or using unit vectors i and j, e.g. 3i + 4j.
向量兼具大小和方向，而标量仅有大小。在平面上，向量可表示为列矩阵 [x; y] 或单位向量 i、j 的组合，如 3i + 4j。
The magnitude (length) of vector a = xi + yj is |a| = √x² + y². A unit vector is a vector of magnitude 1, found by dividing the vector by its magnitude. The zero vector 0 has magnitude zero and undefined direction.
向量 a = xi + yj 的模（长度）为 |a| = √x² + y²。单位向量是模为 1 的向量，用原向量除以其模得到。零向量 0 的模为零且方向无定义。
The scalar (dot) product a · b = |a||b| cos θ yields a scalar. Two non‑ rpendicular if a · b = 0. This is a quick check for right angles without measuring slopes.
数量积（点积） a · b = |a||b| cos θ 结果为标量。若非零向量满足 a · b = 0，则二者垂直。无需测量斜率，即可快速检验直角。
9. Matrices: Operations and Transformations | 矩阵：运算与变换
A matrix , often written as [a b; c d]. Matrices can represent geometric transformations such as rotations, reflections, and enlargements.
矩阵是数字构成的矩形阵列。2×2 矩阵有两行两列，常写作 [a b; c d]。矩阵可表示几何变换，如旋转、反射和缩放。
Matrix multiplication is not element‑wise but row‑by‑column. The product of a 2×2 matrix and a 2×1 column vector gives a new vector. Remember: 'row hits column'. The order matters — AB and BA are usually different.
矩阵乘法并非逐元素相乘，而是行乘列。2×2 矩阵与 2×1 列向量的乘积得到新向量。记住：“行撞列”。顺序至关重要——AB 与 BA 通常不相等。
The determinant of [a b; c d] is ad − bc. A zero determinant signals that the matrix is singular, meaning it has no inverse and the transformation collapses area to zero. The inverse, if it exists, 'undoes' the transformation.
[a b; c d] 的行列式是 ad − bc。行列式为零表明矩阵奇异，无逆矩阵，变换使面积坍缩为零。逆矩阵若存在，可“撤销”变换。
Memory aid: 'Determinant decides invertibility — ad minus bc like a magic number.' For the inverse, swap a and d, change signs of b and c, and divide by the determinant.
记忆口诀：“行列式决定可逆性——ad 减 bc 如同魔法数字。”求逆时，交换 a 和 d，将 b 和 c 变号，再除以行列式。
10. Sequences and Series | 数列与级数
A sequence is an ordered list of numbers; a series is the sum of the terms of a sequence. Two key types are arithmetic and geometric. In an arithmetic sequence, the difference d between consecutive terms is constant. The n‑th term is uₙ = a + (n − 1)d.
数列是有序的数字列表；级数是数列各项之和。两大关键类型是等差数列与等比数列。等差数列中相邻两项的差 d 为常数，第 n 项为 uₙ = a + (n − 1)d。
In a geometric sequence ¹. The sum of the first n terms of a geometric series is Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1.
等比数列中每一项乘以公比 r：uₙ = a rⁿ⁻¹。等比级数前 n 项和为 Sₙ = a(1 − rⁿ)/(1 − r)（r ≠ 1）。
Sigma notation ∑ (capital sigma) provides a compact way to write series, e.g. ∑k=1n (2k+1). The index k runs from the lower limit to the upper limit. You can unpack it to see the pattern.
求和符号 ∑（大写西格玛）提供简洁书写级数的方式，如 ∑k=1n (2k+1)。指标 k 从下限运行至上限。展开后即可看清规律。
Memory hook: 'Arithmetic adds the same step; geometric multiplies the same jump.' For the sum formula, picture 'first term minus last term times ratio over one minus ratio' as a rhyme. Practice translating word problems into sequence language.
记忆捷径：“等差是等步累加，等比是等倍跳跃。”求和公式可构想为“首项减末项乘公比除以一减公比”的口诀。勤练将文字题转换数列语言。
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