📚 Mastering Algebra and Functions for IB & CIE Mathematics | IB CIE 数学：代数和函数 考点精讲
Algebra and functions form the backbone of the IB and CIE Mathematics syllabus. Whether you are tackling polynomial equations, sketching complicated graphs, or untangling inverse functions, a deep conceptual understanding combined with fluent technical skill will set you up for top marks. This revision guide walks you through the essential topics with paired explanations, examples, and practical tips.
代数和函数是 IB 与 CIE 数学课程的核心支柱。无论你面对的是多项式方程、复杂图像绘制，还是反函数的推理，深刻的概念理解加上熟练的运算技巧都能帮助你冲击高分。本篇考点精讲通过中英对照的讲解、示例和实用建议，带你系统梳理所有必考内容。
1. Algebraic Fundamentals: Expansion & Factorisation | 代数基础：展开与因式分解
Mastering algebraic manipulation begins with accurate expansion and factorisation. For brackets, use the distributive law: a(b + c) = ab + ac. To factorise, look for common factors first, then apply techniques like difference of two squares: a² – b² = (a – b)(a + b), and trinomial factorisation, e.g., x² + 5x + 6 = (x + 2)(x + 3).
掌握代数运算的第一步是准确的展开和因式分解。对于括号，运用分配律：a(b + c) = ab + ac。因式分解时，先提取公因子，然后运用平方差公式：a² – b² = (a – b)(a + b)，以及三项式分解，例如 x² + 5x + 6 = (x + 2)(x + 3)。
When coefficients become larger or include fractions, keep your working tidy. For instance, 6x² – 13x + 6 factorises to (2x – 3)(3x – 2). Checking your answer by expanding is always wise.
当系数较大或包含分数时，要保持步骤清晰。例如，6x² – 13x + 6 可分解为 (2x – 3)(3x – 2)。始终通过展开来检验答案是否准确。
Also recognise perfect squares: (a + b)² = a² + 2ab + b² and (a – b)² = a² – 2ab + b². These patterns speed up both expansion and factorisation.
同时掌握完全平方公式：(a + b)² = a² + 2ab + b² 与 (a – b)² = a² – 2ab + b²。这些模式能大大加快展开和因式分解的速度。
2. Quadratic Equations & the Discriminant | 二次方程与判别式
Quadratic equations of the form ax² + bx + c = 0 can be solved by factorising, completing the square, or using the quadratic formula: x = [–b ± √(b² – 4ac)] / (2a). The discriminant Δ = b² – 4ac tells you about the nature of the roots.
形如 ax² + bx + c = 0 的二次方程可通过因式分解、配方法或求根公式 x = [–b ± √(b² – 4ac)] / (2a) 求解。判别式 Δ = b² – 4ac 揭示了根的性质。
- Δ > 0: Two distinct real roots (两个不等实根)
- Δ = 0: One repeated real root (一个重根)
- Δ < 0: No real roots, two complex conjugates (无实根，两个共轭复根)
For CIE and IB, you may need to find conditions on a parameter to ensure a certain number of roots, or to show a line is tangent to a curve (Δ = 0). Practise setting up the discriminant inequality and solving it.
在 CIE 和 IB 考试中，你可能需要针对参数求条件，以确保特定数量的根，或证明某直线与曲线相切（Δ = 0）。要熟练建立判别式不等式并求解。
3. Functions & Mappings | 函数与映射
A function f maps each element x of its domain to exactly one element f(x) in its range. The notation f : x → f(x) clearly specifies the rule. Understanding the difference between ‘one-to-one’, ‘many-to-one’, and ‘one-to-many’ mappings is crucial—only one-to-one and many-to-one are functions.
函数 f 将其定义域中的每个元素 x 唯一地映射到值域中的一个元素 f(x)。记号 f : x → f(x) 明确规定了对应规则。理解“一对一”、“多对一”和“一对多”映射的区别至关重要——只有一对一和多对一才是函数。
Remember the vertical line test: if any vertical line crosses a graph more than once, the relation is not a function. Learn to switch between set notation, mapping diagrams, and algebraic formulas fluently.
记住垂直线检验：若任意垂线与图像有多于一个交点，该关系就不是函数。要能流利地在集合记号、映射图和代数公式之间切换。
4. Domain & Range | 定义域与值域
The domain of a function is the set of all possible input values x, while the range is the set of all possible output values f(x). For polynomial functions, the natural domain is all real numbers (ℝ), but for square roots, denominators, and logarithms, restrictions apply.
函数的定义域是所有可能输入值 x 的集合，而值域是所有可能输出值 f(x) 的集合。多项式函数的自然定义域是全体实数 (ℝ)，但对于平方根、分母和对数函数，则存在限制。
Denominators cannot be zero, so x ≠ a for f(x) = 1/(x – a). Square roots require the radicand ≥ 0, e.g., domain of √(x – 2) is x ≥ 2. Logarithmic functions require the argument > 0.
分母不能为零，因此对于 f(x) = 1/(x – a)，x ≠ a。平方根要求被开方式 ≥ 0，例如 √(x – 2) 的定义域为 x ≥ 2。对数函数要求真数大于 0。
To find the range, examine the behaviour of the function: use completion of the square for quadratics, consider asymptotes for rational functions, and apply known ranges for trig, exponential, and log functions.
求值域时，要分析函数的行为：对二次函数使用配方法，对有理函数考虑渐近线，并借助三角、指数和对数函数的已知值域。
5. Composite Functions | 复合函数
The composite function f(g(x)) or (f ◦ g)(x) means apply g first, then f to the result. The domain of f ◦ g is x in domain of g such that g(x) is in domain of f. This double-condition can be tricky and often appears in exams.
复合函数 f(g(x)) 或 (f ◦ g)(x) 表示先作用 g，再将结果代入 f。f ◦ g 的定义域是满足“x 在 g 的定义域内，且 g(x) 在 f 的定义域内”的 x 的集合。这一双重要求常有陷阱，会在考试中出现。
For example, if f(x) = √x and g(x) = x – 2, then (f ◦ g)(x) = √(x – 2), and the domain of the composite is x – 2 ≥ 0, i.e., x ≥ 2, even though g is defined for all real x.
例如，若 f(x) = √x，g(x) = x – 2，则 (f ◦ g)(x) = √(x – 2)，复合函数的定义域为 x – 2 ≥ 0，即 x ≥ 2，尽管 g 对所有实数都有定义。
When simplifying composites, always keep domain restrictions in mind; an algebraic simplification may hide restrictions that were present in the original expression.
化简复合函数时，务必时刻谨记定义域限制；代数化简可能掩盖原始表达式中的限制条件。
6. Inverse Functions | 反函数
An inverse function f⁻¹ reverses the original mapping: if f(a) = b, then f⁻¹(b) = a. For f⁻¹ to exist, f must be one-to-one (injective). Graphically, the inverse is the reflection of f in the line y = x.
反函数 f⁻¹ 是对原映射的逆转：若 f(a) = b，则 f⁻¹(b) = a。反函数存在的前提是 f 必须是一对一的（单射）。从图像上看，反函数是 f 关于直线 y = x 的反射。
To find an inverse: write y = f(x), swap x and y, then solve for y. The domain of f⁻¹ is the range of f, and the range of f⁻¹ is the domain of f. These swapped domains and ranges are often tested in context of restricted functions, e.g., f(x) = x² for x ≥ 0.
求反函数的步骤：写出 y = f(x)，交换 x 和 y，然后解出 y。f⁻¹ 的定义域就是 f 的值域，f⁻¹ 的值域就是 f 的定义域。这种定义域和值域的互换常见于限制定义域的函数，例如 f(x) = x² (x ≥ 0)。
7. Function Transformations | 函数变换
Transformations allow you to sketch related graphs quickly. For y = f(x):
变换让你能够快速画出相关函数的图像。对于 y = f(x)：
- y = f(x) + a: vertical translation up by a (向上平移 a 个单位)
- y = f(x + a): horizontal translation left by a (向左平移 a 个单位, 注意方向)
- y = af(x): vertical stretch by factor a (垂直方向伸展 a 倍)
- y = f(ax): horizontal stretch by factor 1/a (水平方向伸展 1/a 倍)
- y = –f(x): reflection in the x‑axis (关于 x 轴反射)
- y = f(–x): reflection in the y‑axis (关于 y 轴反射)
Combining transformations requires careful order: apply horizontal changes (inside the bracket) before vertical changes (outside), and when stretching and translating horizontally, factorise first. Example: y = 2f(3x – 1) can be seen as horizontal translation by 1/3 to the right, then horizontal stretch by factor 1/3, then vertical stretch by factor 2.
组合变换需要注意顺序：先处理水平方向的变化（括号内），再处理垂直方向的变化（括号外）；水平方向同时存在伸缩和平移时，要先提取系数。例如 y = 2f(3x – 1) 可理解为先向右平移 1/3，再水平方向伸缩 1/3 倍，最后垂直方向伸缩 2 倍。
8. Exponentials & Logarithms | 指数与对数
Exponential functions have the form f(x) = aˣ for a > 0, a ≠ 1, and logarithms are their inverses: logₐ(y) = x ⇔ aˣ = y. The natural exponential eˣ and natural log ln x = logₑ x are particularly important in calculus contexts.
指数函数形式为 f(x) = aˣ (a > 0, a ≠ 1)，对数函数是其反函数：logₐ(y) = x ⇔ aˣ = y。自然指数 eˣ 和自然对数 ln x = logₑ x 在微积分中尤为重要。
Key laws of logs: log(xy) = log x + log y; log(x/y) = log x – log y; log(xⁿ) = n log x. The change-of-base formula: logₐ b = log꜀ b / log꜀ a. These are essential for solving exponential equations.
关键对数法则：log(xy) = log x + log y；log(x/y) = log x – log y；log(xⁿ) = n log x。换底公式：logₐ b = log꜀ b / log꜀ a。这些是解指数方程的基础。
To solve equations like 2ˣ = 5, take logs: x ln 2 = ln 5, so x = ln 5 / ln 2. Always check that arguments of logs remain positive.
求解如 2ˣ = 5 的方程时，两边取对数：x ln 2 = ln 5，因此 x = ln 5 / ln 2。始终检验对数真数为正。
9. Polynomial Division & Remainder Theorem | 多项式除法与余式定理
Polynomial long division and synthetic division help factorise higher-degree polynomials. The Remainder Theorem states: when a polynomial P(x) is divided by (x – a), the remainder is P(a). The Factor Theorem follows: (x – a) is a factor of P(x) if and only if P(a) = 0.
多项式长除法和综合除法有助于分解高次多项式。余式定理指出：多项式 P(x) 除以 (x – a) 时，余式为 P(a)。由此得到因式定理：(x – a) 是 P(x) 的因式当且仅当 P(a) = 0。
These tools are used to find unknown coefficients, factorise cubics and quartics, and to solve polynomial equations. For example, given that P(x) = 2x³ – 3x² + kx – 5 has a factor (x – 1), setting P(1) = 0 gives 2 – 3 + k – 5 = 0 → k = 6.
这些工具用于求解未知系数、分解三次和四次多项式以及解多项式方程。例如，已知 P(x) = 2x³ – 3x² + kx – 5 有因式 (x – 1)，设 P(1) = 0 得 2 – 3 + k – 5 = 0 → k = 6。
After division, the quotient can be factorised further if possible, leading to all roots of the polynomial equation P(x) = 0.
除法后，如果可能，对商式进一步分解，即可得到多项式方程 P(x) = 0 的所有根。
10. Rational Functions & Asymptotes | 有理函数与渐近线
Rational functions are ratios of polynomials, e.g., f(x) = (ax + b)/(cx + d). They often have vertical asymptotes where the denominator is zero (cx + d = 0) and horizontal or oblique asymptotes determined by the degrees of numerator and denominator.
有理函数是多项式的比值，例如 f(x) = (ax + b)/(cx + d)。它们通常在分母为零处有垂直渐近线 (cx + d = 0)，并由分子分母的次数决定水平或斜渐近线。
If deg(num) < deg(den), the horizontal asymptote is y = 0. If deg(num) = deg(den), HA is y = leading coefficient ratio. If deg(num) = deg(den) + 1, perform division to find the oblique asymptote.
若分子次数小于分母次数，水平渐近线为 y = 0；若分子次数等于分母次数，水平渐近线为 y = 首项系数之比；若分子比分母高一次，通过长除法求斜渐近线。
Sketching rational functions also requires finding x- and y-intercepts, and checking behaviour on each side of asymptotes. Use sign analysis to determine whether the graph approaches +∞ or –∞ near a vertical asymptote.
绘制有理函数图像还需找出 x 轴和 y 轴截距，并检查渐近线两侧的变化趋势。利用符号分析确定在垂直渐近线附近图像是趋向 +∞ 还是 –∞。
11. Systems of Equations & Inequalities | 方程组与不等式组
Simultaneous equations can be linear–linear, linear–quadratic, or even quadratic–quadratic. Substitution or elimination methods are standard. Always check that your solutions satisfy all original equations.
联立方程组可以是线性—线性、线性—二次，甚至二次—二次。通常使用代入法或消元法求解。务必检验解是否满足所有原方程。
When solving inequalities, be mindful of multiplying or dividing by negative numbers, which reverses the inequality sign. Quadratic inequalities can be solved by sketching the parabola and identifying intervals where the inequality holds. For example, x² – 4 < 0 gives –2 < x < 2.
解不等式时，注意乘或除以负数会反转不等号方向。二次不等式可通过画出抛物线并找出满足不等式的区间来求解。例如，x² – 4 < 0 的解为 –2 < x < 2。
For rational inequalities like (x – 1)/(x + 2) > 0, use a sign table. Identify critical values where numerator or denominator is zero, then test intervals. Always exclude points where the denominator is zero.
对于 (x – 1)/(x + 2) > 0 这类有理不等式，使用符号表。找出分子或分母为零的临界值，然后检验区间。务必排除分母为零的点。
12. Graphical Interpretation of Functions | 函数图像解析
Interpreting graphs is key to many problems involving intersections, inequalities, and number of solutions. If you need to find the number of solutions to f(x) = k, you are essentially finding how many times the horizontal line y = k intersects the curve y = f(x).
图像解读对于许多涉及交点、不等式和解的个数问题至关重要。若要找出方程 f(x) = k 的解的个数，本质上就是找出水平线 y = k 与曲线 y = f(x) 有多少个交点。
Learn to read intervals where a function is increasing or decreasing, where it is positive or negative, and how its gradient changes. These qualitative features often form part of structured exam questions on function analysis.
学会识别函数的增减区间、正负区间以及斜率变化。这些定性特征常常构成函数分析类考试题的组成部分。
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