IB Mathematics SL Core Content Review | IB 数学 SL 核心内容复习

📚 IB Mathematics SL Core Content Review | IB 数学 SL 核心内容复习

This article provides a comprehensive review of the core topics in IB Mathematics SL, designed to help you consolidate key concepts and prepare effectively for the final exam. Whether you are taking Analysis and Approaches or Applications and Interpretation, many of these topics form the foundation of your course.

本文全面复习IB数学SL的核心主题，旨在帮助你巩固关键概念，有效备战期末考试。无论你选择分析与方法还是应用与解释课程，这些主题中很多都是你课程的基础。

1. Number Systems and Sequences | 数系与数列

Number sets form the foundation of all algebra. The natural numbers N are {1,2,3,…}, integers Z include negatives and zero, rationals Q are fractions a/b with b≠0, and reals R include irrationals like √2 and π.

数集是所有代数的基础。自然数N是{1,2,3,…}，整数Z包括负数和零，有理数Q是形如a/b（b≠0）的分数，实数R包括无理数如√2和π。

An arithmetic sequence has a common difference d between successive terms. The nth term is uₙ = u₁ + (n-1)d, and the sum of the first n terms is given by Sₙ = n/2 [2u₁ + (n-1)d] or Sₙ = n/2 (u₁ + uₙ). Always check if a sequence is arithmetic by examining uₙ₊₁ – uₙ.

等差数列在连续项之间有公差d。第n项为uₙ = u₁ + (n-1)d，前n项和公式为Sₙ = n/2 [2u₁ + (n-1)d] 或 Sₙ = n/2 (u₁ + uₙ)。一定要通过计算uₙ₊₁ – uₙ来判断数列是否为等差数列。

A geometric sequence has a common ratio r (r ≠ 0). The nth term is uₙ = u₁ rⁿ⁻¹. The sum of the first n terms is Sₙ = u₁(1 – rⁿ)/(1 – r) for r ≠ 1. If |r| < 1, the infinite sum converges: S∞ = u₁/(1 - r).

等比数列有公比r（r ≠ 0）。第n项为uₙ = u₁ rⁿ⁻¹。前n项和公式为当r ≠ 1时，Sₙ = u₁(1 – rⁿ)/(1 – r)。如果|r| < 1，无穷级数收敛：S∞ = u₁/(1 - r)。

Sigma notation Σ is used to express sums compactly. For example, Σ(k=1 to n) (2k+1) means the sum of the first n terms of 2k+1. Learn to translate between sigma notation and explicit sequences.

求和符号Σ用于简洁地表示求和。例如，Σ(k=1到n) (2k+1)表示2k+1的前n项之和。学会在求和符号和显式数列之间进行转换。

2. Algebraic Techniques and Equations | 代数技巧与方程

Manipulating algebraic expressions confidently is essential. Master expanding brackets: (a+b)(c+d) = ac+ad+bc+bd, and factorising common factors, difference of squares a² – b² = (a-b)(a+b), and trinomials like x² + bx + c.

熟练进行代数表达式的操作至关重要。掌握展开括号：(a+b)(c+d) = ac+ad+bc+bd，以及提取公因式、平方差a² – b² = (a-b)(a+b)以及形如x² + bx + c的三项式因式分解。

Laws of exponents must be automatic: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ, and a^(1/n) = ⁿ√a.

指数定律必须熟练：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1，a⁻ⁿ = 1/aⁿ，以及a^(1/n) = ⁿ√a。

Solve linear equations by isolating the variable. For systems of two linear equations in two unknowns, use substitution or elimination. Always check your solutions by substituting back into the original equations.

通过分离变量求解线性方程。对于二元一次方程组，使用代入法或消元法。一定要将解代回原方程进行验证。

The quadratic equation ax² + bx + c = 0 can be solved by factorising, completing the square, or using the quadratic formula:

二次方程 ax² + bx + c = 0 可通过因式分解、配方法或使用求根公式求解：

x = [-b ± √(b² – 4ac)] / (2a)

The discriminant Δ = b² – 4ac determines the nature of the roots: Δ > 0 → two distinct real roots, Δ = 0 → one repeated real root, Δ < 0 → no real roots.

判别式Δ = b² – 4ac决定了实根的性质：Δ > 0 → 两个不等的实根，Δ = 0 → 一个重实根，Δ < 0 → 无实根。

3. Functions: Domain, Range, and Inverses | 函数：定义域、值域与反函数

A function f maps each input x to exactly one output f(x). The domain is the set of all allowable inputs; the range is the set of all possible outputs. For example, f(x)=√(x-2) has domain x ≥ 2 and range y ≥ 0.

函数f将每个输入x对应到唯一的输出f(x)。定义域是所有允许输入的集合；值域是所有可能输出的集合。例如，f(x)=√(x-2)的定义域为x ≥ 2，值域为y ≥ 0。

The vertical line test helps decide if a graph represents a function: a vertical line must intersect the graph at most once. The horizontal line test tells if the function is one-to-one and thus has an inverse that is also a function.

竖直线检验可以用来判断一个图像是否表示函数：任何一条竖直线与图像至多交于一点。水平线检验则说明该函数是否是一一映射，从而其反函数也是函数。

To find the inverse f⁻¹(x), swap x and y in the equation y = f(x) and solve for y. The domain of f⁻¹ is the range of f, and the graphs of f and f⁻¹ are reflections in the line y = x.

求反函数f⁻¹(x)时，在方程y = f(x)中交换x和y，然后解出y。f⁻¹的定义域就是f的值域，且f和f⁻¹的图像关于直线y = x对称。

Composite functions (f ∘ g)(x) = f(g(x)) are read as “f of g of x”. Work from the inside out, and be mindful of domain restrictions: the domain of f ∘ g is the set of x in the domain of g such that g(x) is in the domain of f.

复合函数(f ∘ g)(x) = f(g(x))读作“f of g of x”。要由内向外计算，并注意定义域限制：f ∘ g的定义域是使得g(x)在f的定义域内的所有x（且x在g的定义域内）。

4. Transformations of Graphs | 图像变换

Transformations allow you to sketch related functions quickly. The four essential transformations are: translation, reflection, vertical stretch, and horizontal stretch.

变换使得你可以快速绘制相关函数的草图。四种基本的变换是：平移、反射、垂直伸缩和水平伸缩。

For a basic graph y = f(x):

y = f(x) + d → vertical translation by d (up if d>0)
y = f(x + c) → horizontal translation by -c (left if c>0)
y = -f(x) → reflection in the x-axis
y = f(-x) → reflection in the y-axis
y = a f(x) → vertical stretch by factor a (a>1 stretches, 0
y = f(bx) → horizontal stretch by factor 1/b (b>1 compresses, 0

对于基本图像y = f(x)：

y = f(x) + d → 垂直平移d（d>0上移）
y = f(x + c) → 水平平移 -c（c>0左移）
y = -f(x) → 关于x轴反射
y = f(-x) → 关于y轴反射
y = a f(x) → 垂直伸缩，伸缩因子为a（a>1拉伸，0
y = f(bx) → 水平伸缩，伸缩因子为1/b（b>1压缩，0

When applying multiple transformations, the order matters. For y = a f(b(x + c)) + d, a typical order is: horizontal translation, horizontal stretch, vertical stretch, vertical translation. However, always follow any instructions given or think about the effect of each step.

当应用多个变换时，顺序很重要。对于y = a f(b(x + c)) + d，通常的顺序是：水平平移、水平伸缩、垂直伸缩、垂直平移。然而，应始终遵循给定的指示或考虑每一步的效果。

5. Quadratic Functions | 二次函数

The quadratic function f(x) = ax² + bx + c (a ≠ 0) produces a parabola. The sign of a determines whether the parabola opens upwards (a>0) or downwards (a<0). The vertex can be found by completing the square to the form f(x) = a(x - h)² + k, where (h,k) is the vertex.

二次函数 f(x) = ax² + bx + c (a ≠ 0) 的图像是一条抛物线。a的符号决定了抛物线是开口向上（a>0）还是向下（a<0）。可以通过配方法将其化为形式 f(x) = a(x - h)² + k，其中(h,k)为顶点坐标。

The axis of symmetry is the vertical line x = h. The y-intercept is (0,c). The x-intercepts, if they exist, are found by solving ax² + bx + c = 0. The minimum or maximum value of the function is k, occurring at x = h.

对称轴是竖直线x = h。y截距为(0,c)。如果存在x截距，则可通过求解ax² + bx + c = 0得出。函数的最小值或最大值是k，在x = h处取得。

For a quadratic given in factored form a(x – p)(x – q), the x-intercepts are p and q, and the axis of symmetry is x = (p+q)/2. The vertex lies on this line, so substitute this x-value into the function to find the vertex coordinates.

对于以因式分解形式给出的二次函数a(x – p)(x – q)，其x截距为p和q，对称轴为x = (p+q)/2。顶点位于该直线上，因此将该x值代入函数即可求得顶点坐标。

6. Exponential and Logarithmic Functions | 指数与对数函数

An exponential function is of the form f(x) = aˣ, where a > 0 and a ≠ 1. Its domain is all real numbers, the range is y > 0, and the horizontal asymptote is y = 0. The graph passes through (0,1) and (1,a). The natural exponential function has base e ≈ 2.718.

指数函数的形式为 f(x) = aˣ，其中 a > 0 且 a ≠ 1。其定义域为全体实数，值域为 y > 0，水平渐近线为 y = 0。图像经过点(0,1)和(1,a)。自然指数函数的底数为 e ≈ 2.718。

The logarithmic function g(x) = logₐx is the inverse of the exponential function aˣ. It is defined only for x > 0, has vertical asymptote x = 0, and passes through (1,0) and (a,1). The natural logarithm is ln x = logₑx.

对数函数 g(x) = logₐx 是指数函数 aˣ 的反函数。它仅在 x > 0 时有定义，具有竖直线渐近线 x = 0，并且经过点(1,0)和(a,1)。自然对数 ln x = logₑx。

Key logarithm properties:

logₐ1 = 0, logₐa = 1
logₐ(MN) = logₐM + logₐN
logₐ(M/N) = logₐM – logₐN
logₐ(Mⁿ) = n logₐM
Change of base: logₐb = logₓb / logₓa

关键的对数性质：

logₐ1 = 0, logₐa = 1
logₐ(MN) = logₐM + logₐN
logₐ(M/N) = logₐM – logₐN
logₐ(Mⁿ) = n logₐM
换底公式：logₐb = logₓb / logₓa

Solving exponential equations often requires taking the logarithm of both sides. For example, 3ˣ = 7 becomes x = log₃7 or using common logarithms x = log 7 / log 3.

求解指数方程通常需要对两边取对数。例如，3ˣ = 7 变为 x = log₃7，或使用常用对数 x = log 7 / log 3。

7. Trigonometry: Triangles and Circular Functions | 三角学：三角形与圆函数

In a right-angled triangle, define sinθ = opposite/hypotenuse, cosθ = adjacent/hypotenuse, and tanθ = opposite/adjacent. Learn exact values for 0°, 30°, 45°, 60°, 90° (and their radian equivalents: 0, π/6, π/4, π/3, π/2).

在直角三角形中，定义 sinθ = 对边/斜边，cosθ = 邻边/斜边，tanθ = 对边/邻边。要记住0°, 30°, 45°, 60°, 90°（及其弧度制对应值：0, π/6, π/4, π/3, π/2）的精确值。

For non-right triangles, the sine rule and cosine rule are essential:

a/sin A = b/sin B = c/sin C

a² = b² + c² – 2bc cos A

The area of a triangle can be found using ½ ab sin C.

对于非直角三角形，正弦定理和余弦定理是必不可少的：

a/sin A = b/sin B = c/sin C

a² = b² + c² – 2bc cos A

三角形面积可使用公式 ½ ab sin C 求得。

The unit circle extends definitions to all real angles. For any point on the circle, cosθ = x-coordinate, sinθ = y-coordinate. The fundamental identity is sin²θ + cos²θ = 1. Radians are used extensively; 180° = π rad.

单位圆将三角函数的定义扩展到所有实数角。对于圆上的任意一点，cosθ = x坐标，sinθ = y坐标。基本恒等式为 sin²θ + cos²θ = 1。弧度制被广泛使用；180° = π 弧度。

Trigonometric equations like sinθ = k usually have multiple solutions within a given interval. Use the quadrants or the graph to find all solutions. For instance, if sinθ = ½, the principal solution is π/6, and other solutions are π – π/6, plus multiples of 2π.

诸如 sinθ = k 的三角方程在一个给定区间内通常有多个解。利用象限或图像找出所有解。例如，如果 sinθ = ½，主解为 π/6，其他解为 π – π/6，再加上 2π 的整数倍。

8. Introduction to Calculus: Differentiation | 微积分导论：微分

Differentiation finds the instantaneous rate of change of a function. The derivative f'(x) is defined as the limit of the difference quotient. In practice, we use rules. The power rule: d/dx (xⁿ) = n xⁿ⁻¹.

微分用于求函数的瞬时变化率。导数 f'(x) 被定义为差商的极限。实际应用中，我们使用法则来求导。幂法则：d/dx (xⁿ) = n xⁿ⁻¹。

For a constant c, d/dx (c) = 0. The derivative of a sum is the sum of derivatives. The derivative of a constant multiple: d/dx [c f(x)] = c f'(x). The chain rule: if y = f(g(x)), then dy/dx = f'(g(x)) g'(x).

对于常数 c，d/dx (c) = 0。和的导数是导数的和。常数倍法则：d/dx [c f(x)] = c f'(x)。链式法则：如果 y = f(g(x))，则 dy/dx = f'(g(x)) g'(x)。

The product and quotient rules are not required in SL but may be useful. Instead, for rational functions, rewrite as a power before differentiating. For example, y = 1/x = x⁻¹, so dy/dx = -x⁻².

乘积法则和商法则在SL课程中不作要求，但可能有用。相反，对于有理函数，可先改写成幂的形式再求导。例如，y = 1/x = x⁻¹，于是 dy/dx = -x⁻²。

Derivatives help find the slope of a tangent line at a point. The equation of the tangent at (x₀,y₀) is y – y₀ = f'(x₀)(x – x₀). The normal line has slope -1/f'(x₀).

导数有助于求出某点处切线的斜率。在点 (x₀,y₀) 处的切线方程为 y – y₀ = f'(x₀)(x – x₀)。法线的斜率为 -1/f'(x₀)。

Stationary points occur where f'(x) = 0. Use the sign of f'(x) or the second derivative test to classify them as local maxima, local minima, or points of inflection. Also, the derivative can be used in optimisation problems to find maximum or minimum values within a domain.

驻点出现在 f'(x) = 0 处。利用 f'(x) 的符号或二阶导数检验将其分类为局部极大值、局部极小值或拐点。此外，导数可用于最优化问题，在一个定义域内找到最大值或最小值。

9. Introduction to Integration | 积分导论

Integration is the reverse process of differentiation. The indefinite integral ∫ f(x) dx = F(x) + C, where F'(x) = f(x). The power rule for integration: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, valid for n ≠ -1.

积分是微分的逆过程。不定积分 ∫ f(x) dx = F(x) + C，其中 F'(x) = f(x)。积分的幂法则：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，适用于 n ≠ -1。

For integrating sums and constant multiples, use ∫ [a f(x) + b g(x)] dx = a ∫ f(x) dx + b ∫ g(x) dx. The integral of 1/x is ln|x| + C, and ∫ eˣ dx = eˣ + C.

对于求和与常数倍的积分，使用 ∫ [a f(x) + b g(x)] dx = a ∫ f(x) dx + b ∫ g(x) dx。1/x 的积分是 ln|x| + C，且 ∫ eˣ dx = eˣ + C。

The definite integral ∫[a to b] f(x) dx gives the signed area between the curve and the x-axis from x=a to x=b. It is evaluated using the FTC: ∫[a to b] f(x) dx = F(b) – F(a), where F is any antiderivative of f.

定积分 ∫[a to b] f(x) dx 表示曲线与 x 轴之间从 x=a 到 x=b 的有向面积。它可使用微积分基本定理计算：∫[a to b] f(x) dx = F(b) – F(a)，其中 F 是 f

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IB Mathematics SL Core Content Review | IB 数学 SL 核心内容复习