📚 Algebra and Functions: AQA A-Level Exam Focus | A-Level AQA 数学：代数和函数考点精讲
Algebra and functions form the backbone of the AQA A-Level Mathematics course, appearing in pure, mechanics and statistics contexts. Mastering these topics is essential for solving equations, analysing graphs and applying mathematical modelling. This intensive revision guide distils key concepts, common pitfalls and efficient techniques to boost your exam performance.
代数和函数是 AQA A-Level 数学课程的核心，贯穿纯数学、力学和统计。掌握这些内容是解方程、分析图像和建立数学模型的基础。本考点精讲浓缩了关键概念、常见错误和高效技巧，帮助你在考试中脱颖而出。
1. Laws of Indices and Surds | 指数律与根式
The product rule: a^m × a^n = a^(m+n). When multiplying like bases, add the exponents.
乘法法则：同底数幂相乘，指数相加：a^m × a^n = a^(m+n)。
The quotient rule: a^m ÷ a^n = a^(m−n). For division, subtract the exponent of the denominator from the exponent of the numerator.
除法法则：同底数幂相除，指数相减：a^m ÷ a^n = a^(m−n)。
A power raised to another power: (a^m)^n = a^(mn). This often catches students when negative or fractional exponents are involved.
幂的乘方：(a^m)^n = a^(mn)。当涉及负指数或分数指数时，此类运算容易出错。
Zero and negative indices: a^0 = 1 (a ≠ 0); a^(−n) = 1/a^n. Fractional indices link roots and powers: a^(1/2) = √a, a^(1/3) = ∛a, and a^(m/n) = (ⁿ√a)^m = ⁿ√(a^m).
零指数与负指数：a^0 = 1（a ≠ 0）；a^(−n) = 1/a^n。分数指数连接根式与幂：a^(1/2) = √a，a^(1/3) = ∛a，a^(m/n) = (ⁿ√a)^m = ⁿ√(a^m)。
To simplify surds, factor out perfect squares: √50 = √(25×2) = 5√2. Rationalise denominators like 1/√a by multiplying numerator and denominator by √a to give √a/a. For a denominator of the form a + b√c, multiply by its conjugate a − b√c.
化简根式时提取完全平方因子：√50 = √(25×2) = 5√2。分母有理化如 1/√a 时，分子分母同乘以 √a 得到 √a/a。若分母为 a + b√c 的形式，则乘以其共轭 a − b√c。
2. Quadratic Functions and the Discriminant | 二次函数与判别式
A quadratic function can be written in three forms: standard y = ax² + bx + c, vertex y = a(x − h)² + k, and factored y = a(x − p)(x − q). The vertex (h, k) is the maximum or minimum point, and the line of symmetry is x = h.
二次函数有三种表达形式：标准式 y = ax² + bx + c，顶点式 y = a(x − h)² + k 和因式分解式 y = a(x − p)(x − q)。顶点 (h, k) 为最大值或最小值点，对称轴为 x = h。
Completing the square transforms standard form into vertex form: x² + bx = (x + b/2)² − (b/2)². This is essential for finding the vertex and solving equations.
配方法可将标准式化为顶点式：x² + bx = (x + b/2)² − (b/2)²。这是求顶点和解方程的基本方法。
Discriminant Δ = b² − 4ac
The discriminant determines the nature of roots: Δ > 0 gives two distinct real roots; Δ = 0 gives exactly one real root (repeated); Δ < 0 gives no real roots (complex conjugates). In sketching, the sign of a together with Δ tells you how the graph interacts with the x-axis.
判别式确定根的性质：Δ > 0 有两个相异实根；Δ = 0 有一个实根（重根）；Δ < 0 无实根（共轭复数根）。画函数草图时，a 与 Δ 的符号共同决定了图像与 x 轴的交点情况。
3. Polynomials and the Factor Theorem | 多项式与因式定理
A polynomial of degree n is an expression of the form aₙxⁿ + aₙ₋₁x⁽ⁿ⁻¹⁾ + … + a₁x + a₀. Algebraic long division is used to divide a polynomial by a linear or quadratic divisor, yielding a quotient and remainder.
n 次多项式形如 aₙxⁿ + aₙ₋₁x⁽ⁿ⁻¹⁾ + … + a₁x + a₀。代数长除法用于将多项式除以线性或二次除式，得到商式和余式。
Factor Theorem: f(p) = 0 ⇔ (x − p) is a factor of f(x)
If substituting x = p into f(x) gives zero, then (x − p) divides the polynomial exactly. This is used to factorise cubics and higher-degree polynomials. The Remainder Theorem states that when f(x) is divided by (x − p), the remainder is f(p).
如果代入 x = p 使得 f(x) = 0，则 (x − p) 是该多项式的因式。此定理用于因式分解三次及更高次多项式。余数定理指出，f(x) 除以 (x − p) 所得余数为 f(p)。
To factorise a cubic, guess a small integer root using factors of the constant term, then long divide to get a quadratic, and factorise the quadratic if possible. Always check whether a factor can be taken out first, such as a common x.
分解三次多项式时，通过常数项的因子猜测一个小整数根，进行长除法得到二次因式，再尝试分解二次因式。务必先检验是否有公因子可提取，如公因子 x。
4. Simultaneous Equations | 联立方程组
Linear simultaneous equations in two variables are solved by elimination or substitution. Multiply equations to align coefficients, then add or subtract to eliminate one variable.
二元一次方程组可用消元法或代入法求解。将方程乘以适当倍数使某变量的系数相等，然后相加或相减消去该变量。
When one equation is linear and the other quadratic, substitution is the standard approach. Rearrange the linear equation for one variable and substitute into the quadratic. You will obtain a quadratic in one variable; solving it gives up to two x-values, and substituting back yields corresponding y-values.
当一个方程为线性、另一个为二次方程时，通常使用代入法。将线性方程改写为某个变量的表达式，代入二次方程，得到关于另一变量的二次方程，解出后再回代求得对应值，最多可得两组解。
Always verify your solutions in both original equations, especially when squaring might introduce extraneous roots. Graphs of a line and a parabola can intersect in 0, 1, or 2 points, reflecting the number of real solutions.
务必在原方程组中检验解，尤其在平方过程中可能引入增根。直线与抛物线的图像可能交于 0、1 或 2 个点，对应方程组解的个数。
5. Inequalities | 不等式
Linear inequalities are solved similarly to equations, but the inequality sign reverses when multiplying or dividing by a negative number. Represent solution sets on a number line with open or closed circles.
线性不等式的解法与方程类似，但当两边同乘或除以负数时，不等号方向需反转。解集可用开区间或闭区间在数轴上表示。
Quadratic inequalities rely on sketching the parabola. Rearrange to ax² + bx + c > 0 or < 0, find the roots of the corresponding quadratic equation, then test intervals. For a > 0, the graph is ∪-shaped, and the inequality > 0 holds outside the roots; for ≤ 0 it holds between the roots.
二次不等式借助抛物线草图求解。将式子整理为 ax² + bx + c > 0 或 < 0 的形式，求出对应二次方程的根，然后检验区间。若 a > 0，图像开口向上，此时 > 0 的解在两根本身之外；≤ 0 的解在两根之间。
Rational inequalities e.g. (x − a)/(x − b) > 0 must be handled by multiplying by the square of the denominator to avoid undefined regions, or by using a sign table. Never multiply across an inequality by an expression whose sign is unknown.
分式不等式如 (x − a)/(x − b) > 0 可通过乘以分母的平方来避免分母为零的情况，也可使用符号表。切勿在不等式两边同乘符号未知的表达式。
6. Functions: Domain, Range and Notation | 函数：定义域、值域与符号
A function f maps each input x (domain) to exactly one output f(x) (range). The domain is the set of all possible input values; for AQA, common restrictions include denominators ≠ 0 and expressions inside square roots ≥ 0.
函数 f 将每个输入 x（定义域）唯一对应到一个输出 f(x)（值域）。定义域是所有可能输入值的集合；对 AQA 考点而言，常见限制包括分母不为零、平方根内表达式非负。
To find the range, consider the shape of the graph, turning points, asymptotes and endpoints. For quadratics, the range is either [k, ∞) or (−∞, k] depending on the sign of a. For rational functions, analyse horizontal asymptotes and excluded values.
求值域时需考虑图像形态、极值点、渐近线和端点。二次函数的值域根据 a 的符号为 [k, ∞) 或 (−∞, k]。有理函数则应分析水平渐近线及被排除的值。
Function notation: f(x) = x² + 3. For piecewise-defined functions, read the condition to decide which rule applies before evaluating. Knowing key features such as one-to-one or many-to-one nature is essential for inverse functions.
函数符号：f(x) = x² + 3。对于分段定义的函数，需根据条件选择正确的表达式再求值。了解函数是一一映射还是多一映射，对反函数学习至关重要。
7. Composite and Inverse Functions | 复合函数与反函数
A composite function fg(x) means apply g first, then f: fg(x) = f(g(x)). The domain of fg is the set of x in the domain of g such that g(x) is in the domain of f. Always check the order, because fg(x) is generally not equal to gf(x).
复合函数 fg(x) 表示先作用 g，再作用 f：fg(x) = f(g(x))。fg 的定义域是 g 的定义域中使得 g(x) 属于 f 的定义域的那些 x 的集合。注意次序，fg(x) 通常不等于 gf(x)。
To find an inverse function f⁻¹(x), write y = f(x), swap x and y, then solve for y. The graph of f⁻¹ is the reflection of y = f(x) in the line y = x. For f⁻¹ to exist, f must be one-to-one; if not, restrict its domain so that it becomes one-to-one.
求反函数 f⁻¹(x) 时，设 y = f(x)，交换 x 与 y，然后解出 y。f⁻¹ 的图像是 y = f(x) 关于直线 y = x 的对称图形。反函数存在的条件是 f 必须是一一映射，否则需限制定义域使之成为一一映射。
Domain and range swap: the domain of f⁻¹ is the range of f, and vice versa. Exam questions often ask you to state these clearly after finding the inverse.
定义域与值域互换：f⁻¹ 的定义域是 f 的值域，反之亦然。试题经常要求在求出反函数后明确写出其定义域和值域。
8. Transformations of Graphs | 图像变换
Recognising transformations lets you sketch related graphs quickly without plotting points. The four basic transformation types for y = f(x) are:
掌握图像变换可以让你无需描点即可快速绘制相关函数图像。y = f(x) 的四类基本变换如下：
| Transformation | Effect on graph | 变换方式 |
| f(x) + a | Vertical translation by a | 纵向平移 a |
| f(x + a) | Horizontal translation by −a | 横向平移 −a |
| a f(x) | Vertical stretch by factor a | 纵向拉伸 a 倍 |
| f(ax) | Horizontal stretch by factor 1/a | 横向压缩 1/a 倍 |
Reflections: −f(x) reflects in the x-axis; f(−x) reflects in the y-axis. When combining transformations, apply horizontal changes (inside the function) before vertical ones, and take care with order: e.g. y = 2f(3(x + 1)) involves a translation −1 in x, a horizontal stretch factor 1/3, and a vertical stretch factor 2.
对称变换：−f(x) 关于 x 轴对称；f(−x) 关于 y 轴对称。组合变换时，先进行横向变换（括号内的变换），再进行纵向变换，并注意次序：例如 y = 2f(3(x + 1)) 表示 x 方向平移 −1、横向压缩 1/3 倍，最后纵向拉伸 2 倍。
9. Exponentials and Logarithms | 指数函数与对数函数
The exponential function y = a^x (with a > 0, a ≠ 1) models growth and decay. For a > 1, the graph increases and passes through (0,1); for 0 < a < 1, it decreases. The special base e ≈ 2.718 gives y = e^x, whose gradient at any point equals the function value.
指数函数 y = a^x（a > 0, a ≠ 1）可用于模拟增长与衰减。当 a > 1 时，图像递增且过点 (0,1)；当 0 < a < 1 时，图像递减。以 e ≈ 2.718 为底的函数 y = e^x，其曲线上任一点处的梯度等于函数值。
Logarithms are inverses of exponentials: logₐ(x) = y ⇔ a^y = x. The natural logarithm ln x means logₑ x. Key laws: logₐ(xy) = logₐ x + logₐ y; logₐ(x/y) = logₐ x − logₐ y; logₐ(xⁿ) = n logₐ x.
对数是指数函数的反函数：logₐ(x) = y ⇔ a^y = x。自然对数 ln x 表示 logₑ x。重要法则：logₐ(xy) = logₐ x + logₐ y；logₐ(x/y) = logₐ x − logₐ y；logₐ(xⁿ) = n logₐ x。
Solving exponential equations often involves taking logs on both sides. For example, 3^x = 7 gives x = log₃7 = ln 7 / ln 3. When the unknown appears in the power of e, take natural logs: e^(2x) = 5 ⇒ 2x = ln 5, so x = (1/2) ln 5.
解指数方程时常对方程两边取对数。例如 3^x = 7 可化为 x = log₃7 = ln 7 /
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