一、函数的定义：从输入到输出的映射规则 | What Is a Function: Mapping Rules from Input to Output
函数是IB数学中最基础也最核心的概念。简单来说，函数是一种规则，它把每一个输入值（通常记为x）对应到唯一的输出值（通常记为f(x)或y）。想象一台自动售货机：你按下A3按钮（输入），它必定掉出一罐可乐（输出），绝不会同时掉出可乐和雪碧。这就是函数的核心特性 – 对于每一个输入，有且仅有一个输出。
A function is the most fundamental and central concept in IB Mathematics. In simple terms, a function is a rule that maps every input value (usually denoted as x) to exactly one output value (usually denoted as f(x) or y). Imagine a vending machine: you press button A3 (input), it always dispenses a can of cola (output), never both cola and Sprite at the same time. This is the core property of a function – for every input, there is one and only one output.
在数学符号中，我们写作 f: x → f(x)。例如，f(x) = 2x + 3 表示一个函数，它将每个输入x映射到2x + 3。当x = 4时，f(4) = 2(4) + 3 = 11。函数的定义域（domain）是所有可能输入值的集合，而值域（range）是所有可能输出值的集合。在IB考试中，识别一个关系是否是函数是基础题型 – 使用垂直线测试（vertical line test）：如果任何垂直线与图形相交多于一点，则该关系不是函数。
In mathematical notation, we write f: x → f(x). For example, f(x) = 2x + 3 represents a function that maps each input x to 2x + 3. When x = 4, f(4) = 2(4) + 3 = 11. The domain of a function is the set of all possible input values, while the range is the set of all possible output values. In IB exams, identifying whether a relation is a function is a fundamental question type – use the vertical line test: if any vertical line intersects the graph at more than one point, the relation is not a function.
二、一次函数与直线图像：斜率与截距的几何意义 | Linear Functions and Straight-Line Graphs: Geometric Meaning of Slope and Intercept
一次函数（linear function）是IB数学中最简单的函数类型，形式为 f(x) = mx + c，其中m是斜率（gradient），c是y轴截距（y-intercept）。斜率m表示直线的倾斜程度：m > 0时直线向上倾斜，m < 0时向下倾斜，m = 0时是水平线。截距c是直线与y轴交点的y坐标。
A linear function is the simplest type of function in IB Mathematics, taking the form f(x) = mx + c, where m is the gradient (slope) and c is the y-intercept. The gradient m indicates the steepness of the line: when m > 0 the line slopes upward, when m < 0 it slopes downward, and when m = 0 it is a horizontal line. The y-intercept c is the y-coordinate where the line crosses the y-axis.
两个关键公式必须掌握：第一，斜率公式 m = (y₂ – y₁) / (x₂ – x₁)，用于通过两点坐标计算斜率。第二，点斜式方程 y – y₁ = m(x – x₁)，用于已知一点和斜率写出直线方程。在IB的Paper 1和Paper 2中，常常需要根据图上的两点求直线方程，或者求两条直线的交点 – 交点坐标通过解两个方程组成的联立方程组得到。
Two key formulas must be mastered: first, the gradient formula m = (y₂ – y₁) / (x₂ – x₁), used to calculate the gradient from two point coordinates. Second, the point-slope form y – y₁ = m(x – x₁), used to write the equation of a line given one point and the gradient. In IB Paper 1 and Paper 2, you are often asked to find the equation of a line from two points on a graph, or to find the intersection of two lines – the intersection coordinates are obtained by solving the simultaneous equations formed by the two line equations.
平行线（parallel lines）具有相同的斜率（m₁ = m₂），而垂直线（perpendicular lines）的斜率满足 m₁ × m₂ = -1。例如，如果一条直线的斜率是2/3，那么垂直于它的直线的斜率就是-3/2。这个性质在IB考试中频繁出现，特别是涉及几何证明的题目。
Parallel lines share the same gradient (m₁ = m₂), while perpendicular lines satisfy m₁ × m₂ = -1. For example, if a line has gradient 2/3, a line perpendicular to it has gradient -3/2. This property appears frequently in IB exams, especially in questions involving geometric proofs.
三、二次函数与抛物线：顶点、对称轴和开口方向 | Quadratic Functions and Parabolas: Vertex, Axis of Symmetry, and Direction of Opening
二次函数（quadratic function）的标准形式为 f(x) = ax² + bx + c，其中a ≠ 0。它的图像是一条抛物线（parabola）。系数a决定抛物线的开口方向：a > 0时开口向上（形如U），a < 0时开口向下（形如∩）。|a|越大，抛物线越"窄"；|a|越小，抛物线越"宽"。
A quadratic function in standard form is f(x) = ax² + bx + c, where a ≠ 0. Its graph is a parabola. The coefficient a determines the direction of opening: when a > 0 the parabola opens upward (shaped like U), when a < 0 it opens downward (shaped like ∩). The larger |a| is, the "narrower" the parabola; the smaller |a| is, the "wider" the parabola.
顶点（vertex）是二次函数图像的最高点或最低点。将一般式转换为顶点式 f(x) = a(x – h)² + k，可以直接读出顶点坐标(h, k)。转换过程称为配方法（completing the square）：对于 f(x) = x² + 6x + 5，先写成 f(x) = (x² + 6x + 9) – 9 + 5 = (x + 3)² – 4，因此顶点为(-3, -4)。对称轴是穿过顶点的垂直线 x = h。
The vertex is the highest or lowest point on a quadratic function’s graph. By converting the standard form into vertex form f(x) = a(x – h)² + k, we can directly read the vertex coordinates (h, k). The conversion process is called completing the square: for f(x) = x² + 6x + 5, first write f(x) = (x² + 6x + 9) – 9 + 5 = (x + 3)² – 4, so the vertex is (-3, -4). The axis of symmetry is the vertical line x = h passing through the vertex.
求二次方程的根（roots）有三种方法：因式分解法（factorization）、配方法（completing the square）和求根公式（quadratic formula）x = [-b ± √(b² – 4ac)] / 2a。判别式（discriminant）Δ = b² – 4ac 决定了根的数量和性质：Δ > 0时有两个不同实根，Δ = 0时有一个重根（抛物线与x轴相切），Δ < 0时没有实根（抛物线与x轴不相交）。
There are three methods for finding the roots of a quadratic equation: factorization, completing the square, and the quadratic formula x = [-b ± √(b² – 4ac)] / 2a. The discriminant Δ = b² – 4ac determines the number and nature of the roots: Δ > 0 gives two distinct real roots, Δ = 0 gives one repeated root (the parabola touches the x-axis), and Δ < 0 gives no real roots (the parabola does not intersect the x-axis).
四、函数的变换：平移、伸缩与对称的数学操作 | Function Transformations: Translation, Stretching, and Reflection as Mathematical Operations
函数的变换（transformations）是IB数学AA课程中的重点内容。掌握四种基本变换，你就能从已知函数的图像推导出相关函数的图像，而不需要逐点计算。这四种变换分别是：平移（translation）、伸缩（stretch）、对称（reflection）和复合变换（composite transformation）。
Function transformations are a key topic in the IB Mathematics AA course. By mastering four basic transformations, you can derive the graph of a related function from a known function’s graph without plotting point by point. The four transformations are: translation, stretch, reflection, and composite transformation.
1. 平移变换（Translation）：f(x) + d 将图像向上平移d个单位（d > 0）或向下平移|d|个单位（d < 0）。f(x + c) 将图像向左平移c个单位（c > 0时向左，注意这与直觉相反）或向右平移|c|个单位（c < 0时）。例如，f(x) = (x - 2)² + 3 的顶点在(2, 3)，相对于y = x²向右平移2、向上平移3。
1. Translation: f(x) + d shifts the graph up by d units (when d > 0) or down by |d| units (when d < 0). f(x + c) shifts the graph left by c units (when c > 0, note this is counterintuitive) or right by |c| units (when c < 0). For example, f(x) = (x - 2)² + 3 has its vertex at (2, 3), shifted right by 2 and up by 3 relative to y = x².
2. 伸缩变换（Stretch）：a·f(x) 是垂直伸缩（vertical stretch），当|a| > 1时沿y轴方向拉伸，当0 < |a| < 1时沿y轴方向压缩。f(bx) 是水平伸缩（horizontal stretch），当|b| > 1时沿x轴方向压缩，当0 < |b| < 1时沿x轴方向拉伸。注意水平伸缩的效果与系数大小的关系也是反直觉的。
2. Stretch: a·f(x) is a vertical stretch – when |a| > 1 the graph is stretched vertically, when 0 < |a| < 1 it is compressed vertically. f(bx) is a horizontal stretch - when |b| > 1 the graph is compressed horizontally, when 0 < |b| < 1 it is stretched horizontally. Note that the effect of horizontal stretch relative to the coefficient magnitude is also counterintuitive.
3. 对称变换（Reflection）：-f(x) 将图像关于x轴对称（上下翻转）。f(-x) 将图像关于y轴对称（左右翻转）。例如，y = √x 只在第一象限有定义，但 y = √(-x) 在第二象限有定义，两者关于y轴对称。
3. Reflection: -f(x) reflects the graph across the x-axis (flips it upside down). f(-x) reflects the graph across the y-axis (flips it left-to-right). For example, y = √x is defined only in the first quadrant, but y = √(-x) is defined in the second quadrant – the two are symmetric about the y-axis.
五、指数函数与对数函数：互为反函数的增长与衰减模型 | Exponential and Logarithmic Functions: Growth and Decay Models as Inverse Functions
指数函数（exponential function）的形式为 f(x) = a·b^x，其中a ≠ 0，b > 0且b ≠ 1。当b > 1时，函数表示指数增长（exponential growth），比如细菌繁殖、复利计算。当0 < b < 1时，函数表示指数衰减（exponential decay），比如放射性衰变、药物在体内的代谢。
An exponential function takes the form f(x) = a·b^x, where a ≠ 0, b > 0, and b ≠ 1. When b > 1, the function represents exponential growth, such as bacterial reproduction or compound interest. When 0 < b < 1, the function represents exponential decay, such as radioactive decay or drug metabolism in the body.
自然指数函数 f(x) = e^x 是最重要的指数函数，其中e ≈ 2.71828是自然常数。它的特殊性质是：在任意点的斜率等于该点的函数值，即 d/dx(e^x) = e^x。自然对数函数（natural logarithm）ln(x) 是 e^x 的反函数，满足 ln(e^x) = x 且 e^(ln x) = x（x > 0）。
The natural exponential function f(x) = e^x is the most important exponential function, where e ≈ 2.71828 is the natural constant. Its special property is that the slope at any point equals the function value at that point, i.e., d/dx(e^x) = e^x. The natural logarithm function ln(x) is the inverse of e^x, satisfying ln(e^x) = x and e^(ln x) = x (for x > 0).
对数运算的基本规则（laws of logarithms）是IB考试的高频考点：log_a(xy) = log_a(x) + log_a(y)，log_a(x/y) = log_a(x) – log_a(y)，log_a(x^n) = n·log_a(x)。换底公式 log_a(b) = log_c(b) / log_c(a) 在求解涉及不同底数的方程时至关重要。在IB的Paper 1不允许使用计算器的部分，这些法则的灵活运用是得分关键。
The basic laws of logarithms are high-frequency exam topics in IB: log_a(xy) = log_a(x) + log_a(y), log_a(x/y) = log_a(x) – log_a(y), log_a(x^n) = n·log_a(x). The change of base formula log_a(b) = log_c(b) / log_c(a) is crucial when solving equations involving different bases. In the non-calculator Paper 1 section of IB, flexible application of these laws is key to scoring well.
六、三角函数：周期函数的图像、性质和恒等式 | Trigonometric Functions: Graphs, Properties, and Identities of Periodic Functions
三角函数（trigonometric functions）是最重要的周期函数。三个基本三角函数是正弦函数 sin(θ)、余弦函数 cos(θ) 和正切函数 tan(θ)。在单位圆（unit circle）上，点P的坐标为(cos θ, sin θ)，其中θ是从正x轴逆时针旋转的角度。tan θ = sin θ / cos θ。
Trigonometric functions are the most important periodic functions. The three basic trigonometric functions are sine sin(θ), cosine cos(θ), and tangent tan(θ). On the unit circle, point P has coordinates (cos θ, sin θ), where θ is the angle measured counterclockwise from the positive x-axis. tan θ = sin θ / cos θ.
正弦和余弦函数的图像是波浪形的：y = sin x 在x = 0时从0开始，在x = π/2时达到最大值1，在x = π时回到0，在x = 3π/2时达到最小值-1，在x = 2π时完成一个完整周期。振幅（amplitude）为1，周期（period）为2π。y = cos x与y = sin x形状相同，但向左平移了π/2 – 即cos x = sin(x + π/2)。
The graphs of sine and cosine are wave-shaped: y = sin x starts at 0 when x = 0, reaches its maximum of 1 at x = π/2, returns to 0 at x = π, reaches its minimum of -1 at x = 3π/2, and completes one full cycle at x = 2π. The amplitude is 1 and the period is 2π. y = cos x has the same shape as y = sin x but is shifted left by π/2 – that is, cos x = sin(x + π/2).
三角恒等式（trigonometric identities）是IB数学中的核心工具。最重要的恒等式是毕达哥拉斯恒等式（Pythagorean identity）：sin²θ + cos²θ = 1。由此可推出 1 + tan²θ = sec²θ 和 1 + cot²θ = csc²θ。双角公式（double-angle formulas）同样重要：sin(2θ) = 2 sin θ cos θ，cos(2θ) = cos²θ – sin²θ = 2 cos²θ – 1 = 1 – 2 sin²θ。这些恒等式在解三角方程和证明时必不可少。
Trigonometric identities are core tools in IB Mathematics. The most important identity is the Pythagorean identity: sin²θ + cos²θ = 1. From this we can derive 1 + tan²θ = sec²θ and 1 + cot²θ = csc²θ. The double-angle formulas are equally important: sin(2θ) = 2 sin θ cos θ, cos(2θ) = cos²θ – sin²θ = 2 cos²θ – 1 = 1 – 2 sin²θ. These identities are essential for solving trigonometric equations and proofs.
七、微积分入门：极限、导数与函数的变化率 | Introduction to Calculus: Limits, Derivatives, and Rates of Change
微积分（calculus）是IB数学AA HL课程的核心，也是大学理工科的基础。微积分的核心思想是研究变化 – 导数（derivative）描述瞬时变化率，积分（integral）描述累积量。极限（limit）是微积分的理论基石：lim_{x→a} f(x) = L 表示当x无限趋近于a时，f(x)无限趋近于L。
Calculus is the core of the IB Mathematics AA HL course and the foundation for university STEM subjects. Its central idea is the study of change – the derivative describes instantaneous rate of change, while the integral describes accumulation. The limit is the theoretical cornerstone of calculus: lim_{x→a} f(x) = L means that as x approaches a, f(x) approaches L.
导数的定义基于极限：f'(x) = lim_{h→0} [f(x + h) – f(x)] / h。这个定义式的几何意义是：当h越来越小时，割线的斜率趋近于切线的斜率。基本求导法则包括：幂法则 d/dx(x^n) = n·x^(n-1)，常数法则 d/dx(c) = 0，和差法则 d/dx[f(x) ± g(x)] = f'(x) ± g'(x)，乘积法则 d/dx[f(x)·g(x)] = f'(x)·g(x) + f(x)·g'(x)，商法则 d/dx[f(x)/g(x)] = [f'(x)·g(x) – f(x)·g'(x)] / [g(x)]²，以及链式法则 d/dx[f(g(x))] = f'(g(x))·g'(x)。
The definition of the derivative is based on limits: f'(x) = lim_{h→0} [f(x + h) – f(x)] / h. The geometric meaning of this definition is: as h becomes smaller and smaller, the slope of the secant line approaches the slope of the tangent line. Basic differentiation rules include: the power rule d/dx(x^n) = n·x^(n-1), the constant rule d/dx(c) = 0, the sum/difference rule d/dx[f(x) ± g(x)] = f'(x) ± g'(x), the product rule d/dx[f(x)·g(x)] = f'(x)·g(x) + f(x)·g'(x), the quotient rule d/dx[f(x)/g(x)] = [f'(x)·g(x) – f(x)·g'(x)] / [g(x)]², and the chain rule d/dx[f(g(x))] = f'(g(x))·g'(x).
导数的应用非常广泛。一阶导数f'(x)表示函数的增减性：f'(x) > 0时函数递增，f'(x) < 0时函数递减，f'(x) = 0时可能存在极值点（turning point）。二阶导数f''(x)表示函数的凹凸性：f''(x) > 0时图像凹向上（concave up），f”(x) < 0时图像凹向下（concave down）。通过一阶导数测试和二阶导数测试可以确定极值点的性质（极大值或极小值）。
The applications of derivatives are extensive. The first derivative f'(x) indicates the increasing/decreasing behavior of a function: when f'(x) > 0 the function is increasing, when f'(x) < 0 it is decreasing, and when f'(x) = 0 there may be a turning point. The second derivative f''(x) indicates concavity: when f''(x) > 0 the graph is concave up, when f”(x) < 0 it is concave down. The first derivative test and second derivative test can determine the nature of turning points (maximum or minimum).
八、积分：从反导数到定积分与面积计算 | Integration: From Antiderivatives to Definite Integrals and Area Calculation
积分（integration）是微分的逆运算。如果F'(x) = f(x)，则称F(x)是f(x)的一个反导数（antiderivative）或不定积分（indefinite integral），记作 ∫ f(x) dx = F(x) + C，其中C是积分常数。基本积分公式包括：∫ x^n dx = x^(n+1)/(n+1) + C（n ≠ -1），∫ 1/x dx = ln|x| + C，∫ e^x dx = e^x + C，∫ sin x dx = -cos x + C，∫ cos x dx = sin x + C。
Integration is the inverse operation of differentiation. If F'(x) = f(x), then F(x) is called an antiderivative or indefinite integral of f(x), written as ∫ f(x) dx = F(x) + C, where C is the constant of integration. Basic integration formulas include: ∫ x^n dx = x^(n+1)/(n+1) + C (n ≠ -1), ∫ 1/x dx = ln|x| + C, ∫ e^x dx = e^x + C, ∫ sin x dx = -cos x + C, ∫ cos x dx = sin x + C.
定积分（definite integral）∫_a^b f(x) dx 表示函数f(x)在区间[a, b]上与x轴之间的有向面积（signed area）。根据牛顿-莱布尼茨公式（Fundamental Theorem of Calculus）：∫_a^b f(x) dx = F(b) – F(a)，其中F是f的任意一个反导数。如果f(x)在某个区间内为负值，定积分的结果就是”负面积”，即曲线在x轴下方的部分贡献负的面积值。
The definite integral ∫_a^b f(x) dx represents the signed area between the function f(x) and the x-axis over the interval [a, b]. According to the Fundamental Theorem of Calculus: ∫_a^b f(x) dx = F(b) – F(a), where F is any antiderivative of f. If f(x) is negative over part of the interval, the definite integral gives “negative area” – the portion of the curve below the x-axis contributes a negative area value.
求两条曲线之间的面积是IB考试中的经典题型。如果f(x) ≥ g(x)在区间[a, b]上，则两条曲线之间的面积 = ∫_a^b [f(x) – g(x)] dx。如果两条曲线在区间内相交，需要分段计算 – 先找到所有交点（解方程f(x) = g(x)），然后在每个子区间上分别计算面积并取绝对值后再求和。
Finding the area between two curves is a classic question type in IB exams. If f(x) ≥ g(x) on the interval [a, b], then the area between the two curves = ∫_a^b [f(x) – g(x)] dx. If the two curves intersect within the interval, you need to calculate piecewise – first find all intersection points (solve f(x) = g(x)), then calculate the area on each subinterval separately, take absolute values, and sum them.
九、概率与统计：数据分布、概率模型与假设检验 | Probability and Statistics: Data Distributions, Probability Models, and Hypothesis Testing
概率（probability）和统计（statistics）在IB数学AI课程中占据核心地位，在AA课程中也有重要覆盖。概率的基本概念包括样本空间（sample space）、事件（event）、以及概率的公理化定义：0 ≤ P(A) ≤ 1，P(S) = 1（S为样本空间），互斥事件的概率可加。
Probability and statistics occupy a central position in IB Mathematics AI and are also significantly covered in AA. Basic probability concepts include the sample space, events, and the axiomatic definition of probability: 0 ≤ P(A) ≤ 1, P(S) = 1 (where S is the sample space), and mutually exclusive events have additive probabilities.
条件概率（conditional probability）P(A|B) = P(A ∩ B) / P(B) 表示在事件B已经发生的条件下事件A发生的概率。贝叶斯定理（Bayes’ Theorem）P(A|B) = P(B|A)·P(A) / P(B) 是条件概率的重要推论，在医学检测、机器学习等领域有广泛应用。独立事件满足 P(A ∩ B) = P(A)·P(B)。
Conditional probability P(A|B) = P(A ∩ B) / P(B) represents the probability of event A occurring given that event B has already occurred. Bayes’ Theorem P(A|B) = P(B|A)·P(A) / P(B) is an important corollary of conditional probability, with wide applications in medical testing, machine learning, and other fields. Independent events satisfy P(A ∩ B) = P(A)·P(B).
概率分布（probability distributions）描述了随机变量的行为。二项分布（binomial distribution）X ~ B(n, p) 描述n次独立试验中成功次数的分布，其中P(X = k) = C(n,k)·p^k·(1-p)^(n-k)。正态分布（normal distribution）X ~ N(μ, σ²) 是最重要的连续分布，其钟形曲线（bell curve）以均值μ为中心，标准差σ控制散布宽度。在IB考试中，使用GDC（图形计算器）计算正态分布的概率和逆运算是必备技能。
Probability distributions describe the behavior of random variables. The binomial distribution X ~ B(n, p) describes the distribution of the number of successes in n independent trials, where P(X = k) = C(n,k)·p^k·(1-p)^(n-k). The normal distribution X ~ N(μ, σ²) is the most important continuous distribution, with its bell curve centered at the mean μ and its spread controlled by the standard deviation σ. In IB exams, using the GDC (graphical display calculator) to calculate normal distribution probabilities and inverse operations is an essential skill.
十、向量：从几何直观到代数运算的桥梁 | Vectors: Bridging Geometric Intuition and Algebraic Computation
向量（vector）是既有大小（magnitude）又有方向（direction）的量，与只有大小的标量（scalar）相对。在IB数学中，向量主要用分量形式表示：v = (x, y) 在二维空间中，或 v = (x, y, z) 在三维空间中。向量的模（magnitude）|v| = √(x² + y²) 表示向量的长度。
A vector is a quantity that has both magnitude and direction, as opposed to a scalar which has only magnitude. In IB Mathematics, vectors are primarily expressed in component form: v = (x, y) in two dimensions, or v = (x, y, z) in three dimensions. The magnitude of a vector |v| = √(x² + y²) represents the vector’s length.
向量运算包括：加法（addition） – 对应分量相加，几何上满足平行四边形法则；标量乘法（scalar multiplication） – 将向量的每个分量乘以标量，改变向量的长度但保持方向（标量为正）或反转方向（标量为负）；点积（dot product）a·b = |a||b| cos θ = a₁b₁ + a₂b₂，用于计算两向量之间的夹角和一个向量在另一个向量上的投影。点积为零意味着两向量垂直（perpendicular）。
Vector operations include: addition – adding corresponding components, geometrically satisfying the parallelogram law; scalar multiplication – multiplying each component by a scalar, changing the vector’s length while preserving direction (positive scalar) or reversing it (negative scalar); dot product a·b = |a||b| cos θ = a₁b₁ + a₂b₂, used to calculate the angle between two vectors and the projection of one vector onto another. A dot product of zero means the two vectors are perpendicular.
在三维空间中，叉积（cross product）a × b 产生一个垂直于a和b所在平面的向量，其大小为 |a × b| = |a||b| sin θ。叉积在计算三角形和平行四边形的面积、以及求平面的法向量（normal vector）时非常有用。向量方程 r = r₀ + t·d 表示一条通过点r₀、方向为d的直线。两条直线的关系（平行、相交或异面）可以通过分析方向向量和位置向量来确定。
In three dimensions, the cross product a × b produces a vector perpendicular to the plane containing a and b, with magnitude |a × b| = |a||b| sin θ. The cross product is very useful for calculating areas of triangles and parallelograms, and for finding the normal vector of a plane. The vector equation r = r₀ + t·d represents a line passing through point r₀ with direction d. The relationship between two lines (parallel, intersecting, or skew) can be determined by analyzing their direction vectors and position vectors.
十一、IB数学考试策略：时间分配、常见陷阱与图形计算器的使用 | IB Math Exam Strategy: Time Management, Common Pitfalls, and GDC Usage
IB数学考试的成功不仅取决于对数学概念的掌握，还取决于考试策略。Paper 1（不允许使用计算器）通常占40%的分数，考验的是纯数学推理和计算能力。Paper 2（允许使用图形计算器GDC）占40%，侧重于应用和建模。内部评估IA（Internal Assessment）占20%，需要对一个自选数学问题进行深入探究。
Success in IB Mathematics exams depends not only on mastering mathematical concepts but also on exam strategy. Paper 1 (non-calculator) typically accounts for 40% of the marks and tests pure mathematical reasoning and computation skills. Paper 2 (GDC allowed) accounts for 40% and focuses on applications and modeling. The Internal Assessment (IA) accounts for 20% and requires an in-depth exploration of a self-chosen mathematical problem.
常见的时间分配策略：对于一道6分的题目，建议分配约6-7分钟。如果超过分配时间仍无进展，做标记后跳过，在完成其他题目后再回来。常见的失分原因包括：忘记写加常数C在不定积分中、混淆sin²θ和sin(θ²)的含义、在处理函数变换时将水平变换的方向搞反、使用求根公式时代入a、b、c时出现符号错误。在考前仔细检查这些陷阱可以帮助你避免不必要的失分。
A common time management strategy: for a 6-mark question, allocate approximately 6-7 minutes. If you exceed this time without progress, mark it and move on, returning after completing other questions. Common reasons for losing marks include: forgetting to write the constant of integration C in indefinite integrals, confusing the meanings of sin²θ and sin(θ²), reversing the direction of horizontal transformations when dealing with function transformations, and making sign errors when substituting a, b, c into the quadratic formula. Carefully reviewing these pitfalls before the exam can help you avoid unnecessary mark losses.
图形计算器（GDC）在Paper 2中是强大的工具，但需要熟练操作。关键功能包括：绘制函数图像并查找交点、零点、最大值和最小值；计算定积分的数值解；进行统计计算如回归分析和假设检验；求解方程组。建议在考前至少练习使用GDC完成三套完整的Paper 2真题，确保对计算器的各种功能了如指掌。
The GDC is a powerful tool in Paper 2 but requires skilled operation. Key functions include: graphing functions and finding intersections, zeros, maxima, and minima; computing numerical solutions of definite integrals; performing statistical calculations such as regression analysis and hypothesis testing; and solving systems of equations. It is recommended to practice completing at least three full Paper 2 past papers with the GDC before the exam, ensuring familiarity with all calculator functions.
Summary | 总结
本文系统地梳理了IB数学的核心知识体系，从函数的基本定义出发，涵盖了一次函数与直线、二次函数与抛物线、函数的变换、指数函数与对数函数、三角函数、微积分（导数与积分）、概率统计以及向量等核心主题，最后提供了IB数学考试的实用策略。每个主题都包含了关键公式、图像特征和常见考试题型。掌握这些内容将为IB数学考试打下坚实的基础。数学学习的精髓在于理解概念之间的联系 – 函数、图形、代数和微积分不是孤立的知识碎片，而是相互关联的整体。当你能够自如地在图像、方程和实际应用之间进行切换时，你就真正掌握了IB数学的核心。
This article has systematically reviewed the core knowledge framework of IB Mathematics, starting from the basic definition of functions and covering linear functions and straight lines, quadratic functions and parabolas, function transformations, exponential and logarithmic functions, trigonometric functions, calculus (derivatives and integrals), probability and statistics, and vectors, followed by practical strategies for the IB Mathematics exam. Each topic includes key formulas, graphical characteristics, and common exam question types. Mastering this content will build a solid foundation for the IB Mathematics examination. The essence of mathematical learning lies in understanding the connections between concepts – functions, graphs, algebra, and calculus are not isolated fragments of knowledge but an interconnected whole. When you can seamlessly switch between graphs, equations, and real-world applications, you have truly mastered the core of IB Mathematics.
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