📚 Year 9 SQA Advanced Mathematics: Quick Reference Formula & Theorem Handbook | Year 9 SQA 进阶数学：公式定理速查手册
This handbook is designed to give you a rapid but thorough review of the key formulas and theorems you need for Year 9 SQA Advanced Mathematics. Keep it close during revision, and use it to test yourself on each topic.
本手册旨在为你快速而全面地复习 Year 9 SQA 进阶数学所需的关键公式和定理。复习时随身携带，用它来检验自己对每个主题的掌握程度。
1. Laws of Indices | 指数定律
The laws of indices simplify expressions involving powers of the same base. When multiplying, add the exponents.
指数定律用于简化含有相同底数的幂的表达式。同底数幂相乘时，指数相加。
aᵐ × aⁿ = aᵐ⁺ⁿ
When dividing terms of the same base, subtract the exponents.
同底数幂相除时，指数相减。
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
Raising a power to another power means you multiply the exponents.
幂的乘方意味着指数相乘。
(aᵐ)ⁿ = am × n
Any non‑zero number raised to the power of zero equals one. A negative exponent represents the reciprocal of the base raised to the corresponding positive exponent.
任何非零数的零次幂等于 1。负指数表示底数的倒数并取相应正指数。
a⁰ = 1 (a ≠ 0), a⁻ⁿ = 1 / aⁿ
2. Factorising | 因式分解
Factorising reverses expansion. Always look for a common factor first. For two squares, the difference factorises neatly.
因式分解是展开的逆过程。首先总是寻找公因式。两项平方差可以整齐地分解因式。
a² − b² = (a + b)(a − b)
When factorising a quadratic trinomial x² + bx + c, find two numbers that multiply to give c and add to give b.
分解二次三项式 x² + bx + c 时，找出两个数，它们的积等于 c，和等于 b。
x² + bx + c = (x + p)(x + q), where p + q = b and pq = c
Always check your factorisation by expanding the brackets to ensure you obtain the original expression.
始终通过展开括号验证你的因式分解，以确保还原到原表达式。
3. Solving Equations | 解方程
For a linear equation, isolate the variable by performing inverse operations. The solution is the value that balances the equation.
对于一元一次方程，通过逆运算分离变量。方程的解是使等式成立的值。
ax + b = c → x = (c − b) / a
When a quadratic equation is set to zero, you can either factorise or use the quadratic formula. The discriminant b² − 4ac tells you how many real roots there are.
当二次方程等于零时，你可以通过因式分解或使用求根公式求解。判别式 b² − 4ac 告诉你它有几个实根。
ax² + bx + c = 0 → x = (−b ± √b² − 4ac) / 2a
If an equation contains fractions, multiply every term by the lowest common denominator early in your working.
如果方程中含有分数，应在解题初期就每项乘以最小公分母。
4. Equation of a Straight Line | 直线方程
The gradient–intercept form tells you the slope and where the line crosses the y‑axis. You can find the gradient from two points.
斜截式告诉你直线的斜率以及它与 y 轴的交点。你可以根据两点求出斜率。
y = mx + c, m = (y₂ − y₁) / (x₂ − x₁)
The midpoint between two points is the average of their x‑ and y‑coordinates. The distance between them comes from Pythagoras' theorem.
两点之间的中点是它们 x 坐标和 y 坐标的平均值。两点间的距离来自勾股定理。
Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Distance d = √[(x₂ − x₁)² + (y₂ − y₁)²]
ve equal gradients; perpendicular lines have gradients whose product is −1 (m₁ × m₂ = −1).
平行线斜率相等；垂直线的斜率之积为 −1 (m₁ × m₂ = −1)。
5. Pythagoras' Theorem | 勾股定理
In any right‑angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. This relationship is used both to find missing sides and to check whether a triangle is right‑angled.
在任何直角三角形中，斜边的平方等于另外两边的平方之和。这个关系既可用于求未知边，也可用于检验一个三角形是否为直角三角形。
a² + b² = c², where c is the hypotenuse
e satisfy a² + b² = c², the triangle must contain a right angle opposite the longest side.
逆定理表明，如果一个三角形的三边满足 a² + b² = c²，那么它必定包含一个直角，且直角对着最长边。
Always label the sides carefully, and remember that the hypotenuse is always the longest side, opposite the right angle.
始终仔细标记边长，记住斜边总是最长的那条，对着直角。
6. Trigonometric Ratios | 三角比
SOH CAH TOA helps you remember the three basic trigonometric ratios in a right‑angled triangle. Choose the ratio that links the side you know with the side you want to find.
SOH CAH TOA 帮助记忆直角三角形中的三种基本三角比。选择能将已知边与所求边联系起来的那个比。
tenuse cos θ = Adjacent / Hypotenuse tan θ = Opposite / Adjacent
When the unknown is an angle, use the inverse functions (sin⁻¹, cos⁻¹, tan⁻¹) to find its measure.
当未知量是角度时，使用反函数 (sin⁻¹, cos⁻¹, tan⁻¹) 来求角的度数。
In any right‑ riangle, the sine of one acute angle equals the cosine of the other: sin(90° − θ) = cos θ.
在任意直角三角形中，一个锐角的正弦等于另一锐角的余弦：sin(90° − θ) = cos θ。
7. Area and Volume | 面积与体积
These formulae cover common 2D shapes. For a rectangle, multiply length by width. For a triangle, use half the base times the perpendicular height.
这些公式涵盖了常见的二维图形。矩形面积等于长乘宽。三角形面积等于底乘高的一半。
A = ½ × b × h Parallelogram: A = b × h Trapezium: A = ½(a + b)h
For 3D shapes, start with the area of the cross‑section and multiply by the length for a prism. A cylinder is a circular prism.
对于三维图形，先从截面积开始，对棱柱乘以长度。圆柱体就是一个圆形截面的棱柱。
Volume of a prism: V = area of cross‑section × length
Cuboid: V = l × w × h Cylinder: V = πr²h
Surface area is the sum of the areas of all faces. For a cylinder, the curved surface area is 2πrh, and total surface area adds the two circular ends.
表面积是所有面的面积之和。对于圆柱，侧面积是 2πrh，总表面积需加上两个圆形底面。
8. Circles | 圆
The circumference is the distance around the circle, while the area measures the space inside. Both depend on the radius r.
周长是围绕圆一周的距离，而面积度量圆内部的区域。两者都取决于半径 r。
Circumference: C = 2πr or C = πd
Area: A = πr²
When you are given the diameter, halve it to obtain the radius before substituting into the area formula. Leave answers in terms of π unless told otherwise.
给出直径时，先除以 2 得到半径再代入面积公式。除非另有要求，答案保留 π 的形式。
9. Statistics: Averages and Spread | 统计：平均数与离散度
The mean is calculated by summing all data values and dividing by the number of values. The median is the middle value when the data is ordered.
平均数是所有数据值之和除以数据个数。中位数是数据排序后位于中间的那个值。
Mean = (sum of values) / (number of values)
The mode is the most frequent value. The range measures spread: it is the difference between the largest and smallest values.
众数是出现频率最高的值。极差度量离散程度：它是最大值与最小值的差。
Range = Maximum − Minimum
For grouped data, use the midpoint of each class interval to estimate the mean. Always check whether the question asks for the modal class rather than a single mode.
对于分组数据，用每个组区间的中点值来估计平均数。务必确认题目要求的是众数组还是单个众数。
10. Inequalities | 不等式
Inequalities show the relative size of two expressions. The symbols are read as 'less than', 'greater than', 'less than or equal to' and 'greater than or equal to'.
不等式表示两个表达式之间的大小关系。符号分别读作“小于”、“大于”、“小于等于”和“大于等于”。
<, >, ≤, ≥
Solving a linear inequality follows the same steps as solving an equation, but remember: if you multiply or divide by a negative number, you must reverse the inequality sign.
解一元一次不等式的步骤与解方程相同，但要记住：如果乘或除以一个负数，必须反转不等式方向。
e.g. −2x < 6 → x > −3
Solutions can be shown on a number line with an open circle for strict inequalities and a closed circle when the endpoint is included.
解集可在数轴上表示，严格不等式用空心圆，包含端点时用实心圆。
Published by TutorHao | Advanced Mathematics Revision Series | aleveler.com
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