📚 IB OCR Mathematics: Algebra and Functions – Key Concept Revision | IB OCR 数学：代数和函数考点精讲
Algebra and functions form the backbone of IB and OCR mathematics courses, bridging pure abstract reasoning with applied problem solving. A strong command of polynomials, quadratics, exponentials, logarithms, and transformations is essential for success in both Standard and Higher Level papers. This revision guide distills the core topics, techniques, and common pitfalls, helping you build fluency and confidence for your exams.
代数和函数是 IB 与 OCR 数学课程的主干，连接着纯抽象推理与应用问题求解。无论是标准级别还是高级别考试，扎实掌握多项式、二次函数、指数、对数以及函数变换都是取得高分的关键。本复习指南提炼了核心主题、解题技巧和常见易错点，帮助你提升熟练度，增强应考信心。
1. Polynomials and Factorisation | 多项式与因式分解
A polynomial in one variable x is an expression of the form aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀, where n is a non-negative integer and aᵢ are constants. Factorisation is the process of writing a polynomial as a product of its factors. The Factor Theorem states that if f(a) = 0, then (x – a) is a factor of f(x).
一元多项式是形如 aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀ 的表达式，其中 n 为非负整数，aᵢ 为常数。因式分解就是将一个多项式写成其因式的乘积。因式定理指出：若 f(a) = 0，则 (x – a) 是 f(x) 的一个因式。
Common factorisation techniques include taking out the highest common factor, factoring by grouping, recognising the difference of two squares a² – b² = (a – b)(a + b), and decomposing quadratic trinomials. For cubic and higher-degree polynomials, synthetic division or long division is used alongside the Rational Root Theorem.
常见的因式分解方法包括：提取最大公因式、分组分解、应用平方差公式 a² – b² = (a – b)(a + b)，以及十字相乘分解二次三项式。对于三次及更高次多项式，通常结合有理根定理使用综合除法或长除法进行分解。
2. Quadratic Functions and the Discriminant | 二次函数与判别式
The standard form of a quadratic function is f(x) = ax² + bx + c, with a ≠ 0. The discriminant, Δ = b² – 4ac, determines the nature and number of real roots. Its value provides immediate information about the intersection of the parabola with the x-axis.
二次函数的标准形式为 f(x) = ax² + bx + c，且 a ≠ 0。判别式 Δ = b² – 4ac 决定了实根的性质与个数，其数值直接给出了抛物线与 x 轴相交情况的信息。
If Δ > 0, the quadratic has two distinct real roots and the graph crosses the x-axis at two points. If Δ = 0, there is one repeated real root (a double root), so the graph touches the x-axis at the vertex. If Δ < 0, the quadratic has no real roots; the graph lies entirely above or below the x-axis depending on the sign of a.
若 Δ > 0，方程有两个不等实根，图像与 x 轴交于两点；若 Δ = 0，有一个重实根（二重根），图像在顶点处与 x 轴相切；若 Δ < 0，方程无实根，图像完全位于 x 轴上方或下方，具体取决于 a 的正负。
Δ = b² – 4ac
3. Completing the Square and Vertex Form | 配方法与顶点式
Completing the square transforms the expression ax² + bx + c into the form a(x – h)² + k. The coordinates (h, k) give the vertex of the parabola. Starting from f(x) = ax² + bx + c, factor out a from the first two terms, then add and subtract (b/(2a))² inside the bracket.
配方法将 ax² + bx + c 变形为 a(x – h)² + k 的形式，其中 (h, k) 即为抛物线的顶点坐标。对于 f(x) = ax² + bx + c，先从前两项中提取 a，然后在括号内加上并减去 (b/(2a))² 即可完成配方。
This form makes it straightforward to determine the maximum or minimum value of the quadratic function. If a > 0, the parabola opens upwards and k is the minimum value; if a < 0, it opens downwards and k is the maximum value. The axis of symmetry is the vertical line x = h.
通过顶点式可以方便地判断二次函数的最大值或最小值。若 a > 0，抛物线开口向上，k 为最小值；若 a < 0，开口向下，k 为最大值。对称轴为直线 x = h。
f(x) = a(x – h)² + k
4. Exponential and Logarithmic Functions | 指数函数与对数函数
Exponential functions take the form f(x) = aˣ with a > 0 and a ≠ 1. The most important base in calculus is Euler’s number e ≈ 2.718. The graph of y = aˣ passes through (0, 1), has the x-axis as a horizontal asymptote, and is either strictly increasing (a > 1) or strictly decreasing (0 < a < 1).
指数函数具有 f(x) = aˣ 的形式，其中 a > 0 且 a ≠ 1。微积分中最重要的底数是欧拉数 e ≈ 2.718。y = aˣ 的图像通过点 (0, 1)，以 x 轴为水平渐近线，当 a > 1 时严格递增，当 0 < a < 1 时严格递减。
The logarithmic function y = logₐx is the inverse of the exponential function y = aˣ. It is defined only for x > 0. The natural logarithm, denoted ln x, uses base e. The graph of y = logₐx passes through (1, 0) and has a vertical asymptote at x = 0.
对数函数 y = logₐx 是指数函数 y = aˣ 的反函数，仅对 x > 0 有定义。以 e 为底的对数称为自然对数，记作 ln x。y = logₐx 的图像通过点 (1, 0)，并有垂直渐近线 x = 0。
5. Laws of Exponents and Logarithms | 指数律与对数律
Manipulating exponential and logarithmic expressions requires fluency with their fundamental laws. For exponents, the key identities are aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, and a⁻ⁿ = 1/aⁿ. These hold for all real exponents when a > 0.
灵活处理指数与对数表达式需要熟练掌握它们的基本运算法则。指数的核心恒等式为 aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，以及 a⁻ⁿ = 1/aⁿ。当 a > 0 时，这些法则对所有实数指数均成立。
For logarithms, the corresponding laws are logₐ(xy) = logₐx + logₐy, logₐ(x/y) = logₐx – logₐy, and logₐ(xⁿ) = n logₐx. The change-of-base formula logₐx = (log_b x) / (log_b a) is particularly useful when evaluating logarithms with uncommon bases.
对数的相应法则为 logₐ(xy) = logₐx + logₐy，logₐ(x/y) = logₐx – logₐy，以及 logₐ(xⁿ) = n logₐx。换底公式 logₐx = (log_b x) / (log_b a) 在计算非常用底数的对数时尤其有用。
logₐ(xy) = logₐx + logₐy
6. Function Transformations: Translations and Stretches | 函数变换：平移与伸缩
Understanding how to transform the graph of a parent function y = f(x) is essential for sketching and modelling. A vertical translation is given by y = f(x) + k, moving the graph k units upwards if k > 0. A horizontal translation y = f(x – h) shifts the graph h units to the right.
理解如何对基本函数 y = f(x) 进行图像变换对于作图和建模至关重要。垂直平移用 y = f(x) + k 表示，当 k > 0 时图像向上移动 k 个单位；水平平移 y = f(x – h) 则将图像向右移动 h 个单位。
Stretches alter the shape of the graph. y = a f(x) represents a vertical stretch by factor a (a > 1 stretches, 0 < a < 1 compresses). y = f(bx) represents a horizontal stretch by factor 1/b. Reflections are special cases: y = –f(x) reflects in the x-axis, and y = f(–x) reflects in the y-axis.
伸缩变换会改变图像的形状。y = a f(x) 表示垂直方向拉伸 a 倍（a > 1 拉伸，0 < a < 1 压缩）。y = f(bx) 表示水平方向拉伸 1/b 倍。反射是伸缩的特殊情况：y = –f(x) 关于 x 轴对称，y = f(–x) 关于 y 轴对称。
When multiple transformations are combined, the order matters. Always apply horizontal transformations (inside the bracket) before vertical transformations and stretches before translations, ensuring the correct result.
当多种变换组合时，顺序至关重要。通常先进行水平变换（括号内的变换），再进行垂直变换，且伸缩在平移之前执行，以确保结果正确。
7. Composite Functions | 复合函数
A composite function, denoted (f ∘ g)(x) = f(g(x)), applies one function to the output of another. The domain of the composite consists of all x in the domain of g such that g(x) is in the domain of f. Always start with the inside function g and then substitute into f.
复合函数记作 (f ∘ g)(x) = f(g(x))，它将一个函数的输出作为另一函数的输入。复合函数的定义域由所有满足 g(x) 属于 f 的定义域的 x 组成，其中 x 必须属于 g 的定义域。计算时总是先内层函数 g，再代入 f。
For example, if f(x) = √x and g(x) = 2x – 1, then f(g(x)) = √(2x – 1). The domain of this composite requires 2x – 1 ≥ 0, so x ≥ 1/2. Note that the domain of the composite is often a subset of the domain of the inner function.
例如，设 f(x) = √x，g(x) = 2x – 1，则 f(g(x)) = √(2x – 1)。此复合函数的定义域要求 2x – 1 ≥ 0，即 x ≥ 1/2。注意复合函数的定义域通常是内层函数定义域的一个子集。
8. Inverse Functions | 反函数
An inverse function f⁻¹ reverses the effect of f, meaning f⁻¹(f(x)) = x for all x in the domain of f. To find an inverse, write y = f(x), swap x and y, then solve for y. The resulting expression y = f⁻¹(x) is the inverse function.
反函数 f⁻¹ 的作用是逆转 f 的运算，即对 f 定义域内的所有 x 有 f⁻¹(f(x)) = x。求反函数的步骤为：设 y = f(x)，交换 x 与 y，再解出 y。得到的表达式 y = f⁻¹(x) 即为反函数。
Graphically, the inverse function is a reflection of the original function in the line y = x. Not every function has an inverse; a function must be one-to-one (pass the horizontal line test) on its restricted domain to have an inverse.
在图像上，反函数是原函数关于直线 y = x 的反射。并非每个函数都有反函数；函数在其限制定义域上必须是一一对应的（通过水平线检验）才存在反函数。
The domain of f⁻¹ is the range of f, and the range of f⁻¹ is the domain of f. When writing the inverse, always state its domain explicitly based on the range of the original function.
f⁻¹ 的定义域是 f 的值域，f⁻¹ 的值域是 f 的定义域。在写出反函数时，务必根据原函数的值域明确给出其定义域。
9. Rational Functions and Asymptotes | 有理函数与渐近线
A rational function is a ratio of two polynomials, f(x) = P(x)/Q(x). Its graph can have vertical asymptotes where Q(x) = 0 and P(x) ≠ 0. Horizontal asymptotes describe the end behaviour as x → ±∞, determined by comparing the degrees of the numerator and denominator.
有理函数是两个多项式的比，即 f(x) = P(x)/Q(x)。其图像在 Q(x) = 0 且 P(x) ≠ 0 处存在垂直渐近线。水平渐近线描述当 x → ±∞ 时的函数趋势，由分子与分母的次数比较决定。
If degree(P) < degree(Q), the horizontal asymptote is y = 0. If the degrees are equal, the asymptote is y = a/b, where a and b are the leading coefficients. If degree(P) > degree(Q), there is no horizontal asymptote; instead, an oblique asymptote may exist, found by polynomial division.
若 deg(P) < deg(Q)，水平渐近线为 y = 0；若次数相等，渐近线为 y = a/b，其中 a, b 分别为分子、分母的首项系数；若 deg(P) > deg(Q)，则没有水平渐近线，但可能存在斜渐近线，需通过多项式除法求得。
10. Key Features of Graphs: Intercepts, Symmetry | 图像关键特征：截距与对称性
When sketching functions, always identify the x-intercepts by solving f(x) = 0 and the y-intercept by evaluating f(0). For many functions, factoring or using the quadratic formula reveals x-intercepts. Turning points and stationary points can be found using calculus or, for quadratics, by completing the square.
在绘制函数图像时，务必先确定 x 截距（解 f(x) = 0）和 y 截距（求 f(0)）。对于许多函数，通过因式分解或使用求根公式可以得到 x 截距。驻点与转向点可通过微积分求得，而对于二次函数，配方后可确定顶点。
Symmetry helps reduce sketching work. Even functions satisfy f(–x) = f(x) and are symmetric about the y-axis. Odd functions satisfy f(–x) = –f(x) and are rotationally symmetric about the origin. Identifying these properties early confirms the shape of the graph.
对称性有助于简化作图。偶函数满足 f(–x) = f(x)，图像关于 y 轴对称；奇函数满足 f(–x) = –f(x)，图像关于原点旋转对称。尽早识别这些性质可以帮助验证图像形状是否正确。
Additionally, consider the behaviour as x → ±∞ and any asymptotic tendencies. A sign diagram showing where the function is positive or negative is a powerful tool for sketching rational and polynomial functions.
此外，还要考察当 x → ±∞ 时的趋势以及渐近行为。绘制符号表，标明函数在哪些区间为正或为负，是勾勒有理函数和多项式函数图像的有力工具。
11. Piecewise Functions and Absolute Value | 分段函数与绝对值
A piecewise function is defined by different rules on different parts of its domain. The absolute value function f(x) = |x| is a classic example, defined as x for x ≥ 0 and –x for x < 0. Its graph is a V-shape with vertex at the origin.
分段函数在其定义域的不同部分由不同的规则定义。绝对值函数 f(x) = |x| 是最经典的例子，当 x ≥ 0 时定义为 x，当 x < 0 时定义为 –x，其图像是顶点在原点的“V”形。
To sketch a piecewise function, divide the x-axis into intervals according to the domain restrictions, then plot each sub-function separately. Pay close attention to open and closed dots at breakpoints to indicate whether the endpoint is included.
绘制分段函数图像时，应根据定义域限制将 x 轴划分为若干区间，然后分别绘制各子函数的图像。在分段点处要特别注意使用空心点或实心点，以准确表示端点是否包含在内。
Equations involving absolute values, such as |2x – 1| = 3, are solved by setting up two separate linear equations: 2x – 1 = 3 and 2x – 1 = –3. Always check solutions satisfy the original equation.
含有绝对值的方程，例如 |2x – 1| = 3，应拆分为两个独立的线性方程：2x – 1 = 3 和 2x – 1 = –3。解出后务必检验它们是否满足原方程。
12. Polynomial Long Division and Remainder Theorem | 多项式长除法与余数定理
Polynomial long division is used to divide a polynomial P(x) by a divisor D(x), producing a quotient Q(x) and a remainder R(x). The process mirrors numeric long division, repeatedly matching the leading term of the dividend with that of the divisor.
多项式长除法用于将多项式 P(x) 除以除式 D(x)，得到商式 Q(x) 和余式 R(x)。其过程与数的长除法类似，反复将被除式的最高次项与除式的最高次项对齐相除。
The Remainder Theorem states that when a polynomial f(x) is divided by (x – a), the remainder is f(a). This is extremely efficient for finding remainders without performing full division. The Factor Theorem is a direct consequence: if f(a) = 0, then (x – a) is a factor.
余数定理指出，当多项式 f(x) 除以 (x – a) 时，余数为 f(a)。这极大地方便了求余数，无需进行完整除法。因式定理正是其直接推论：若 f(a) = 0，则 (x – a) 为因式。
For example, to divide 2x³ – 3x² + 4x – 5 by x – 2, the remainder is f(2) = 2(8) – 3(4) + 4(2) – 5 = 16 – 12 + 8 – 5 = 7. The quotient can then be found by synthetic division or continued long division.
例如，将 2x³ – 3x² + 4x – 5 除以 x – 2，余数为 f(2) = 2(8) – 3(4) + 4(2) – 5 = 7。随后可通过综合除法或继续长除法求出商式。
P(x) = (x – a)Q(x) + R, where R = P(a)
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