Standard Functions and Techniques | 标准函数与技术

📚 Standard Functions and Techniques | 标准函数与技术

In IB Mathematics, a solid grasp of standard functions and the techniques used to analyse them forms the backbone of the syllabus. From linear relationships to trigonometric identities, these tools allow you to model real-world situations, solve equations, and begin calculus. This article walks through the essential function types, their properties, and the foundational techniques you need to handle them confidently in both Analysis & Approaches and Applications & Interpretation courses.

在IB数学中，扎实掌握标准函数及其分析技术是课程大纲的核心。从线性关系到三角恒等式，这些工具让你能够建立现实世界模型、求解方程并开始微积分学习。本文梳理了必备的函数类型、它们的性质以及你需要熟练掌握的基础技术，无论你是学习分析与方法还是应用与解释课程。

1. Definition and Notation of Functions | 函数定义与记号

ossible outputs is the range. For example, f(x) = x² + 3 maps the input 2 to the output 7.

函数 f 是一条规则，它将定义域集合中的每个输入 x 都对应到唯一的输出 f(x)。常用的记号有 f : x → f(x) 或直接写成 f(x)。所有可能输出的集合就是值域。例如，f(x) = x² + 3 将输入 2 映射到输出 7。

2. Domain and Range | 定义域与值域

domain of a function is the set of x-values for which the function is defined. For f(x) = √x t of resulting y-values. Determining the domain and range often involves avoiding division by zero, negative values under even roots, or undefined logarithms.

函数的定义域是使得函数有定义的所有 x 值的集合。对于 f(x) = √x，定义域为 x ≥ 0。值域则是所有可能的 y 值集合。确定定义域和值域时，通常需要避免分母为零、偶次根号下为负或对数真数小于等于零等情况。

3. Linear Functions | 线性函数

change. Two lines are parallel if their gradients are equal and perpendicular if m₁ × m₂ = –1.

线性函数具有 f(x) = mx + c 的形式，其中 m 是斜率，c 是 y 轴截距。它的图像是一条直线。斜率 m = Δy/Δx 表示变化率。如果两条直线的斜率相等，则它们平行；如果 m₁ × m₂ = –1，则它们垂直。

4. Quadratic Functions | 二次函数

written as f(x) = ax² + bx + c, a ≠ 0. Their graphs are parabolas. The discriminant Δ = b² – ermines the number of x-intercepts. Completing the square gives the vertex form f(x) = a(x – h)² + k, where (h, k) is the vertex. The axis of symmetry is x = h = –b/(2a).

二次函数写作 f(x) = ax² + bx + c，a ≠ 0。它们的图像是抛物线。判别式 Δ = b² – 4ac 决定了 x 轴截距的个数。配方法可得到顶点式 f(x) = a(x – h)² + k，其中 (h, k) 为顶点坐标。对称轴为 x = h = –b/(2a)。

5. Polynomials and Factorisation | 多项式与因式分解

A polynomial of degree n has the form aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₀. The Remainder Theorem states that when P(x) is divided by (x – r), the remainder is P(r). The Factor Theorem follows: (x – . These results help factorise cubics and higher-degree polynomials for solving equations.

n 次多项式具有 aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₀ 的形式。余式定理指出，当 P(x) 除以 (x – r) 时，余式为 P(r)。由此得出因式定理：若 P(r) = 0，则 (x – r) 是一个因式。这些结论有助于对三次及更高次多项式进行因式分解，从而求解方程。

6. Rational Functions and Asymptotes | 有理函数与渐近线

Rational functions are ratios of polynomials, e.g. f(x) = (2x+1)/(x– (leading coefficients). Oblique asymptotes appear when the numerator degree is one higher.

有理函数是多项式的比值，例如 f(x) = (2x+1)/(x–3)。它们可能在分母为零而分子不为零处存在垂直渐近线。水平渐近线通过比较分子与分母的次数来确定：若次数相等，渐近线为 y = a/b（首项系数之比）。当分子次数比分母高一次时会出现斜渐近线。

广告 / Sponsored

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

Exponential functions f(x) = aˣ ay. Their domain is all real numbers, and the range is y > 0. The natural exponential is eˣ to which a must be raised to get x. Key laws: logₐ(xy) = logₐx + logₐy; logₐ(x/y) = logₐx – logₐy; logₐ(xⁿ) = n logₐx.

指数函数 f(x) = aˣ（a > 0, a ≠ 1）用于模拟增长和衰减。其定义域为所有实数，值域为 y > 0。自然指数函数为 eˣ。对数是指数的逆运算：logₐx 表示以 a 为底得到 x 所需的指数。重要法则：logₐ(xy) = logₐx + logₐy；logₐ(x/y) = logₐx – logₐy；logₐ(xⁿ) = n logₐx。

8. Trigonometric Functions and Identities | 三角函数与恒等式

+ 1 = sec²x. The sine and cosine rules extend trigonometry to non-right-angled triangles.

三个基本三角函数是 sin x、cos x 以及 tan x = sin x / cos x。它们的图像具有周期性。特殊角（0, π/6, π/4, π/3, π/2）的精确值至关重要。基本恒等式包括 sin²x + cos²x = 1 以及 tan²x + 1 = sec²x。正弦定理和余弦定理将三角学推广到非直角三角形。

9. Graphical Transformations | 图像变换

: y = f(x) + k shifts the graph vertically by k units; y = f(x + h) shifts horizontally by – vertically by factor a; y = f(bx) compresses horizontally by factor 1/b. Reflections: y = –f(x) reflects in the x-axis; y = f(–x) reflects in the y-axis.

变换让你能快速绘制相关函数的草图。对于 y = f(x)：y = f(x) + k 将图像垂直平移 k 个单位；y = f(x + h) 水平平移 –h 个单位；y = a f(x) 以因子 a 垂直拉伸；y = f(bx) 以因子 1/b 水平压缩。对称变换：y = –f(x) 关于 x 轴对称；y = f(–x) 关于 y 轴对称。

10. Composite and Inverse Functions | 复合函数与反函数

The composite function (f ∘ g)(x) = f(g(x)) means applying g first, then f. The domain of f ∘ , swap x and y and solve for y. The graph of f⁻¹ is the reflection of f in the line y = x.

复合函数 (f ∘ g)(x) = f(g(x)) 表示先作用 g，再作用 f。f ∘ g 的定义域限制在使得 g 和 f 都有定义的 x 值上。反函数 f⁻¹ 逆转 f 的效果，满足 f⁻¹(f(x)) = x。求反函数时，交换 x 与 y 并解出 y。f⁻¹ 的图像是 f 关于直线 y = x 的对称图形。

11. Using Functions to Solve Equations | 利用函数解方程

can often be reinterpreted as finding intersections of functions. For example, solving x² – 4 = ln x is equivalent to locating where f(x) = x² – pplying the quadratic formula, or using logarithmic properties remain crucial in IB exams.

方程常常可以重新解释为寻找函数的交点。例如，求解 x² – 4 = ln x 就等价于找出 f(x) = x² – 4 与 g(x) = ln x 相交的位置。图形计算器能提供帮助，但在 IB 考试中，分析方法仍然是关键，比如令因式为零、应用二次公式或运用对数性质。

12. Calculus with Standard Functions | 微积分与标准函数

ions are assumed. Key derivatives: d/dx (xⁿ) = nxⁿ⁻¹, d/dx (sin x) = cos x, d/dx (cos x) = –sin x, d/dx (eˣ) = eˣ, d/dx (ln x) = 1/x. Integration reverses these: ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C (n ≠ –1), ∫ cos x dx = sin x + C, ∫ eˣ dx = eˣ + C, ∫ 1/x dx = ln|x| + C. The chain, product, and quotient rules extend differentiation to composites.

标准函数的基本微分与积分是必备知识。关键导数：d/dx (xⁿ) = nxⁿ⁻¹，d/dx (sin x) = cos x，d/dx (cos x) = –sin x，d/dx (eˣ) = eˣ，d/dx (ln x) = 1/x。积分则是逆运算：∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C（n ≠ –1），∫ cos x dx = sin x + C，∫ eˣ dx = eˣ + C，∫ 1/x dx = ln|x| + C。链式法则、乘法法则和除法法则将微分推广到复合函数。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）