📚 Algebra and Functions: Key Exam Topics for OCR A-Level Maths | 代数与函数考点精讲
Algebra and Functions form the backbone of the OCR A-Level Mathematics course. Mastery of these topics is essential not only for the pure mathematics papers but also for applied units such as mechanics and statistics. This revision guide covers the core concepts, common pitfalls, and key techniques that regularly appear in examinations. By working through each section, you will deepen your understanding of algebraic manipulation, functions, and their graphical behaviour, building the confidence needed to tackle both routine and problem-solving questions.
代数与函数是OCR A-Level数学课程的基石。掌握这些主题不仅对纯数学试卷至关重要，对力学和统计等应用单元也同样关键。本复习指南涵盖了考试中频繁出现的核心概念、常见错误和关键技巧。通过逐一学习每个部分，你将加深对代数操作、函数及其图像行为的理解，从而建立应对常规题和综合题所需的信心。
1. Laws of Indices and Surds | 指数与根式的基本律
Indices are used to express repeated multiplication in a compact form. The fundamental laws include am × an = am+n, am ÷ an = am−n, and (am)n = amn. For rational exponents, a1/n = ⁿ√a and am/n = (ⁿ√a)m. You must also be able to handle negative exponents, where a−n = 1 / an. Surds involve irrational roots and require simplification skills such as expressing √12 as 2√3. Questions often test your ability to simplify expressions like (2 + √3)(2 − √3) using the difference of two squares, or to rationalise denominators such as 5/(2 + √3).
指数用于将重复乘法表达为紧凑形式。基本定律包括 am × an = am+n，am ÷ an = am−n 以及 (am)n = amn。对于有理指数，a1/n = ⁿ√a 且 am/n = (ⁿ√a)m。你还需要掌握负指数，即 a−n = 1 / an。根式涉及无理根，需要进行化简，例如将 √12 表达为 2√3。题目经常考察利用平方差公式简化 (2 + √3)(2 − √3) 这类表达式，或有理化分母，例如 5/(2 + √3)。
2. Algebraic Manipulation and Expansion | 代数式操作与展开
Expanding brackets accurately is a prerequisite for many advanced problems. You should be comfortable expanding expressions such as (x + a)(x + b)(x + c) and using the binomial theorem for small positive integer powers, e.g., (1 + x)5. Collecting like terms and simplifying algebraic fractions by factorising and cancelling common factors are also tested regularly. Remember that when expanding a bracket preceded by a negative sign, the signs of all inside terms must be reversed. A common mistake is to forget to multiply every term, especially when dealing with three or more brackets.
准确展开括号是解决许多高阶问题的前提。你应当能够轻松展开 (x + a)(x + b)(x + c) 这样的表达式，并会使用二项式定理处理较小的正整数幂，例如 (1 + x)5。合并同类项，以及通过因式分解和约去公因式来化简代数分式，也是经常考查的内容。请记住，当括号前有负号时，括号内所有项的符号都必须改变。一个常见错误是忘记乘以每一项，尤其是在处理三个或更多括号时。
3. Factorisation and Completing the Square | 因式分解与配方法
Factorising quadratics of the form ax² + bx + c is a core skill. For simple cases where a = 1, find two numbers that multiply to c and add to b. When a ≠ 1, use the method of splitting the middle term or the ‘ac’ technique. Completing the square transforms a quadratic into the form a(x + p)² + q, which is crucial for finding the vertex of a parabola and for solving equations where factorisation is not straightforward. For example, x² − 6x + 5 becomes (x − 3)² − 4. This method is also used when deriving the quadratic formula and when integrating rational functions later in the course.
对形如 ax² + bx + c 的二次式进行因式分解是一项核心技能。当 a = 1 时，寻找两个数使其乘积为 c、和为 b。当 a ≠ 1 时，可使用拆分中项法或“ac”法。配方法将二次式转化为 a(x + p)² + q 的形式，这在求抛物线顶点以及解不易直接因式分解的方程时至关重要。例如，x² − 6x + 5 可化为 (x − 3)² − 4。后续推导求根公式以及积分有理函数时，也都会用到这种方法。
4. Quadratic Equations and the Discriminant | 二次方程与判别式
The solutions to ax² + bx + c = 0 are given by the quadratic formula:
x = [−b ± √(b² − 4ac)] / (2a)
The discriminant, Δ = b² − 4ac, determines the nature of the roots. If Δ > 0, the equation has two distinct real roots; if Δ = 0, there is one repeated real root; and if Δ < 0, there are no real roots. In OCR exam questions, you may be asked to find the range of values of a constant k for which a quadratic has equal roots, or to show that a quadratic is always positive for all real x by completing the square and interpreting the discriminant.
ax² + bx + c = 0 的解由二次求根公式给出：
x = [−b ± √(b² − 4ac)] / (2a)
判别式 Δ = b² − 4ac 决定了根的性质。若 Δ > 0，方程有两个相异的实根；若 Δ = 0，有一个重根；若 Δ < 0，则无实根。在OCR考试中，你可能会被要求求常数 k 的取值范围，使某个二次方程有等根；或者通过配方法并结合判别式思想，证明某个二次式对所有实数 x 恒正。
5. Simultaneous Equations and Inequalities | 联立方程与不等式
Solving simultaneous equations often involves one linear and one quadratic equation. Substitute the linear expression into the quadratic and solve the resulting single-variable equation. Remember to find both corresponding values after solving for the first variable. Graphical interpretation may be required, such as finding the intersection points of a line and a parabola. For inequalities, you will solve linear and quadratic inequalities, representing solution sets on a number line or using interval notation. When multiplying or dividing an inequality by a negative number, the direction of the inequality sign must reverse. Quadratic inequalities are best tackled by sketching the parabola and identifying the intervals where the graph is above or below the x-axis.
解联立方程常涉及一个一次方程和一个二次方程。将一次表达式代入二次方程，解所得的单变量方程。务必在求出第一个变量后接着求对应的另一个变量的值。有时还需要进行图形解释，例如求直线与抛物线的交点。对于不等式，你需要解一次和二次不等式，并在数轴上或用区间记法表示解集。当不等式两边同乘或除以一个负数时，不等号方向必须反转。解二次不等式的最佳方法是画出抛物线草图，然后确定图像在 x 轴上侧或下侧的区间。
6. Polynomial Division and the Factor Theorem | 多项式除法与因式定理
Algebraic long division allows you to divide a polynomial by a linear or quadratic divisor. The factor theorem states that (x − a) is a factor of polynomial f(x) if and only if f(a) = 0. This theorem is used to factorise cubic and higher-degree polynomials. Typically you will test possible integer factors (using the constant term) until you find one that gives zero, then use long division or synthetic division to find the remaining quadratic factor, which can then be factorised further. The remainder theorem is closely related: when dividing f(x) by (x − a), the remainder is f(a). Questions frequently combine these theorems with solving equations and sketching graphs.
代数长除法可用于将多项式除以一次或二次除式。因式定理指出：(x − a) 是多项式 f(x) 的因式当且仅当 f(a) = 0。该定理用于分解三次及更高次多项式。通常做法是代入可能的整数因式（利用常数项）进行检验，直至找到使多项式值为零的一个，然后用长除法或综合除法求出剩下的二次因式，并进一步分解。余数定理与之紧密相关：当 f(x) 除以 (x − a) 时，余数为 f(a)。考题常将这些定理与解方程、画图像相结合。
7. Functions: Domain, Range and Notation | 函数的定义域、值域与符号
A function maps each element of its domain to exactly one element in the codomain. The domain is the set of all input values for which the function is defined, while the range is the set of all output values that the function actually produces. Function notation, f(x), is used to name the output when the input is x. You must be able to identify the maximal domain, for example excluding values that would make a denominator zero or cause a negative value under a square root. Questions often ask for f(a) for a specific a, or to solve f(x) = k, interpreting the function correctly. Understanding the difference between f(x) and f⁻¹(x) is essential.
函数将其定义域中的每个元素映射到值域中的一个确定元素。定义域是使函数有意义的所有输入值的集合，而值域是函数实际产生的所有输出值的集合。函数记法 f(x) 用于表示输入为 x 时的输出。你必须能够确定最大定义域，例如剔除会使分母为零或使根号下为负的值。考题常要求计算特定值 f(a)，或正确解释函数并解 f(x) = k。理解 f(x) 与 f⁻¹(x) 的区别也至关重要。
8. Composite and Inverse Functions | 复合函数与反函数
A composite function, denoted by fg(x) or f(g(x)), means applying g first and then f to the result. The order is crucial: fg(x) is generally not the same as gf(x). The inverse function f⁻¹(x) reverses the effect of f, so that f⁻¹(f(x)) = x. To find an inverse, write y = f(x), rearrange to make x the subject, and then swap x and y. The domain of f⁻¹ is the range of f, and vice versa. Graphically, the inverse function is a reflection of the original function in the line y = x. A function must be one‑to‑one to have an inverse on its whole domain; sometimes the domain is restricted to make this possible.
复合函数记作 fg(x) 或 f(g(x))，表示先作用 g，再将结果代入 f。顺序至关重要：fg(x) 一般不等于 gf(x)。反函数 f⁻¹(x) 的作用是将 f 的效果逆转过来，因此满足 f⁻¹(f(x)) = x。求反函数的方法是：令 y = f(x)，重新整理使 x 成为 y 的表达，然后互换 x 与 y。f⁻¹的定义域是 f 的值域，反之亦然。在图像上，反函数是原函数关于直线 y = x 的反射。一个函数必须是单射才在其整个定义域上存在反函数；有时需要限制定义域以使反函数存在。
9. Graph Transformations | 函数图像的变换
Transformations of graphs are tested extensively. The basic rules are: f(x + a) translates the graph horizontally by −a units; f(x) + a translates it vertically by a units; f(ax) stretches the graph horizontally by factor 1/a; a f(x) stretches it vertically by factor a. Reflections include −f(x) reflecting in the x‑axis and f(−x) reflecting in the y‑axis. Combinations of transformations must be applied in the correct order: think about whether the change is inside the function argument (affecting x) or outside (affecting y). For example, y = 2f(3x − 1) + 4 involves a horizontal translation, a horizontal stretch, a vertical stretch, and a vertical translation – you need to handle the sequence carefully.
图像变换是考试重点。基本规则为：f(x + a) 将图像水平平移 −a 个单位；f(x) + a 垂直平移 a 个单位；f(ax) 水平拉伸至 1/a 倍；a f(x) 垂直拉伸 a 倍。对称变换包括 −f(x) 关于 x 轴对称，以及 f(−x) 关于 y 轴对称。组合变换必须按正确顺序进行：要考虑变化发生在函数参数内部（影响 x）还是外部（影响 y）。例如 y = 2f(3x − 1) + 4 涉及水平平移、水平伸缩、垂直伸缩和垂直平移——需要仔细处理先后顺序。
10. Exponentials and Logarithms | 指数与对数
The exponential function eˣ and the natural logarithm ln x are inverse functions. Key logarithm laws include: logₐ(xy) = logₐx + logₐy, logₐ(x/y) = logₐx − logₐy, and logₐ(xk) = k logₐx. The change of base formula is often needed: logₐb = log_cb / log_ca. In equations, you may need to take logs of both sides to solve for an exponent, such as 2ˣ = 5. Be careful with the domain of logarithmic functions: the argument must always be positive. Modelling exponential growth and decay is a common application: problems involving population, radioactive decay, or compound interest require you to set up and solve equations of the form A = Pekt.
指数函数 eˣ 与自然对数 ln x 互为反函数。关键的对数定律包括：logₐ(xy) = logₐx + logₐy，logₐ(x/y) = logₐx − logₐy，以及 logₐ(xk) = k logₐx。换底公式也经常需要：logₐb = log_cb / log_ca。解方程时，你可能需要两边取对数以解出指数，例如 2ˣ = 5。注意对数函数的定义域：其自变量必须始终为正。指数增长与衰减建模是常见应用：涉及人口、放射性衰变或复利的问题，需要建立并求解形如 A = Pekt 的方程。
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