📚 Precalculus: Core Concepts and Study Guide | 预备微积分：核心知识点与学习指导
Precalculus is the essential course that consolidates algebra, geometry and trigonometry, preparing students for the rigorous analytical thinking required in calculus. It deepens understanding of functions, equations, and mathematical reasoning, laying a solid groundwork for advanced topics like limits, derivatives, and integrals.
预备微积分是巩固代数、几何和三角学的重要课程，帮助学生为微积分所需的严格分析思维做好准备。它加深对函数、方程和数学推理的理解，为极限、导数和积分等高级主题奠定坚实基础。
1. What Precalculus Covers and Why It Matters | 预备微积分涵盖什么及其重要性
Precalculus brings together algebraic manipulation, coordinate geometry, and trigonometric reasoning into one coherent preparation course. It is not merely a review of earlier algebra; it is the stage where students learn to think about functions as objects, graphs as evidence, and equations as models of real change.
预备微积分把代数运算、坐标几何和三角推理整合为一门连贯的预备课程。它不只是对早期代数的复习；在这个阶段，学生要学会把函数当作对象、把图像当作证据、把方程当作现实变化的模型。
The central topics include function behavior, polynomial and rational expressions, exponential and logarithmic relationships, trigonometric identities, sequences and series, and an intuitive introduction to limits. These ideas reappear constantly in calculus, so fluency here directly improves later performance.
核心主题包括函数行为、多项式与有理式、指数与对数关系、三角恒等式、数列与级数，以及极限的直观入门。这些内容在微积分中反复出现，因此这里的熟练程度会直接影响后续表现。
Precalculus also builds mathematical reasoning habits: checking domain restrictions, interpreting transformations, verifying identities, and moving between symbolic, graphical, and numerical representations. These habits are examined in both AP Precalculus and A-Level mathematics contexts.
预备微积分还培养数学推理习惯：检查定义域限制、解释图像变换、验证恒等式，并在符号、图像和数值表示之间转换。这些习惯在 AP 预备微积分和 A-Level 数学考试中都会考查。
2. Functions, Domain, Range, and the Vertical Line Test | 函数、定义域、值域与垂直线检验
A function f maps each element x of the domain to exactly one element f(x) in the range. The domain is the set of all valid inputs, while the range is the set of all actual outputs. If one input could produce two different outputs, the relation is not a function.
函数 f 将定义域中的每个元素 x 映射到值域中唯一的 f(x)。定义域是所有有效输入的集合，值域是所有实际输出的集合。如果一个输入可能产生两个不同输出，该关系就不是函数。
The vertical line test helps determine if a graph represents a function: any vertical line must intersect the graph at most once. If a vertical line crosses the graph twice, then a single x-value corresponds to two y-values, so the graph fails the test.
垂直线检验可用于判断图形是否表示函数：任何垂直线与图形最多相交一次。如果一条垂直线与图形相交两次，那么同一个 x 值对应两个 y 值，该图形就不通过检验。
Domain restrictions often come from division by zero, even roots of negative numbers, and logarithmic inputs that must be positive. When solving equations or simplifying expressions, always record these restrictions before cancelling or transforming.
定义域限制通常来自除以零、负数偶次方根，以及对数输入必须为正。在解方程或化简表达式时，一定要在约分或变形之前记录这些限制。
3. Function Notation and the Difference Quotient | 函数记号与差商
Function notation such as f(x), g(t), and h(θ) emphasizes the input-output relationship. Evaluating f(a) means substituting a for every x in the rule. Piecewise functions use different rules on different intervals and are common in exam questions about continuity and graph sketching.
函数记号如 f(x)、g(t) 和 h(θ) 强调输入与输出的关系。计算 f(a) 意味着把规则中的每个 x 都替换为 a。分段函数在不同区间使用不同规则，在关于连续性和图像绘制的考题中很常见。
The difference quotient measures the average rate of change of a function over an interval. It is defined as:
差商衡量函数在一个区间上的平均变化率。它定义为：
[f(x + h) - f(x)] / h, h ≠ 0
ge rate of change into the instantaneous rate of change. Practising difference quotients with quadratic, rational, and radical functions prepares students for differentiation rules.
这个表达式是通向导数的代数桥梁。在微积分中，令 h → 0 取极限，就把平均变化率变成瞬时变化率。用二次函数、有理函数和根式函数练习差商，能为微分法则做好准备。
isfy f(-x) = -f(x). Recognizing symmetry reduces graphing effort and reveals structure in equations.
零点、截距、对称性和端行为也是函数的基本特征。偶函数满足 f(-x) = f(x)，奇函数满足 f(-x) = -f(x)。识别对称性可以减少绘图工作量，并揭示方程中的结构。
4. Transformations of Graphs | 图形变换
Transformations describe how a parent graph changes when its equation is modified. For a base function f(x), adding a constant outside the function shifts the graph vertically, while subtracting a constant inside the input shifts it horizontally. Reflections and stretches follow similar inside-outside logic.
变换描述当方程被修改时，母函数图像如何变化。对于基础函数 f(x)，在函数外部加常数会使图像垂直平移，而在输入内部减常数会使图像水平平移。对称与伸缩也遵循类似的内部与外部逻辑。
Transformation types: f(x) + k gives a vertical shift; f(x - h) gives a horizontal shift; -f(x) reflects across the x-axis; f(-x) reflects across the y-axis. Stretching: a f(x) gives a vertical stretch or compression; f(bx) gives a horizontal stretch or compression.
变换类型：f(x) + k 垂直平移，f(x - h) 水平平移，-f(x) 关于 x 轴对称，f(-x) 关于 y 轴对称。伸缩变换：a f(x) 垂直伸缩；f(bx) 水平伸缩。
| Transformation | 变换 | Effect on Graph | 图像效果 | Key Point | 要点 |
|---|---|---|
| f(x) + k | Vertical shift up or down | 上下平移 | Outside change affects y | 外部变化影响 y |
| f(x - h) | Horizontal shift right or left | 左右平移 | Inside change affects x, sign is opposite | 内部变化影响 x，符号相反 |
| -f(x) | Reflection across x-axis | 关于 x 轴对称 | y-values change sign | y 值变号 |
| f(-x) | Reflection across y-axis | 关于 y 轴对称 | x-values change sign | x 值变号 |
| a f(x) | Vertical stretch or compression | 垂直伸缩 | |a| > 1 stretches, 0 < |a| < 1 compresses | |a| > 1 拉伸，0 < |a| < 1 压缩 |
| f(bx) | Horizontal stretch or compression | 水平伸缩 | |b| > 1 compresses, 0 < |b| < 1 stretches | |b| > 1 压缩，0 < |b| < 1 拉伸 |
izontal scale is 1/b, the horizontal shift is h, the vertical scale is a, and the vertical shift is k. Always test key points to confirm the final graph.
当多个变换同时出现时，顺序很重要。一个有用形式是 y = a f(b(x - h)) + k，其中水平缩放为 1/b，水平平移为 h，垂直缩放为 a，垂直平移为 k。始终用关键点检验最终图像。
5. Composition and Inverse Functions | 复合函数与反函数
Composition of functions (f ∘ ge of g lies within the domain of f. The inner function g is evaluated first, and its output becomes the input of the outer function f. Domain analysis for compositions is a frequent exam trap.
函数复合 (f ∘ g)(x) = f(g(x)) 要求 g 的值域在 f 的定义域内。先计算内层函数 g，其输出成为外层函数 f 的输入。复合函数的定义域分析是常见考试陷阱。
oss the line y = x and satisfy f(f⁻¹(x)) = x and f⁻¹(f(x)) = x within appropriate domains.
反函数逆转原函数的作用。如果 f(a) = b，那么 f⁻¹(b) = a。反函数关于直线 y = x 反射，并在适当定义域内满足 f(f⁻¹(x)) = x 和 f⁻¹(f(x)) = x。
Not every function has an inverse that is also a function. The horizontal line test determines whether a function is one-to-one on its domain. If a horizontal line intersects the graph more than once, the function is not one-to-one and must be restricted before finding an inverse.
并非每个函数都有反函数且反函数仍是函数。水平线检验判断函数在其定义域上是否一一对应。如果一条水平线与图像相交超过一次，函数就不是一一对应，必须先限制定义域再求反函数。
To find an inverse algebraically, replace f(x) with y, swap x and y, solve for y, and then write y as f⁻¹(x). Checking the domain and range is essential because inverse relationships are symmetric: the domain of f becomes the range of f⁻¹, and vice versa.
代数求反函数的步骤是：用 y 替换 f(x)，交换 x 和 y，解出 y，再写成 f⁻¹(x)。检查定义域和值域至关重要，因为反函数关系是对称的：f 的定义域成为 f⁻¹ 的值域，反之亦然。
6. Polynomial and Rational Functions | 多项式与有理函数
Polynomial functions are sums of power terms with non-negative integer exponents. Their graphs are smooth and continuous, and their end behavior is controlled by the leading term. The degree and leading coefficient predict whether the graph rises or falls on the left and right.
多项式函数是带有非负整数指数的幂项之和。它们的图像光滑且连续，端行为由首项控制。次数和首项系数可以预测图像在左右两端上升还是下降。
Quadratic functions can be written in vertex form:
二次函数可以写成顶点式：
f(x) = a(x - h)² + k
w many real roots a quadratic has: Δ > 0 gives two, Δ = 0 gives one, and Δ < 0 gives none.
顶点为 (h, k)，对称轴为 x = h。判别式 Δ = b² - 4ac 说明二次方程有多少实根：Δ > 0 有两个，Δ = 0 有一个，Δ < 0 没有实根。
Rational functions are ratios of polynomials and often have vertical asymptotes where the denominator is zero, provided the factor does not cancel. Horizontal or slant asymptotes describe long-run behavior. Holes occur when a common factor cancels from numerator and denominator.
有理函数是多项式之比，通常在分母为零且因子不抵消处有垂直渐近线。水平或斜渐近线描述长期行为。当分子和分母的公因子抵消时，会出现洞。
Key skills include factoring, finding zeros and multiplicities, solving polynomial inequalities, and decomposing rational expressions. These techniques are prerequisites for limits, curve sketching, and integration in calculus.
关键技能包括因式分解、求零点和重数、解多项式不等式，以及分解有理表达式。这些技巧是微积分中极限、曲线绘制和积分的前提。
7. Exponential and Logarithmic Functions | 指数与对数函数
Exponential functions have the form f(x) = a · bˣ ecay. The natural base e ≈ 2.71828 appears in continuous growth and calculus-based models.
指数函数形式为 f(x) = a · bˣ，其中 a 是初始值，b 是底数。如果 b > 1，函数模拟增长；如果 0 < b < 1，函数模拟衰减。自然底数 e ≈ 2.71828 出现在连续增长和基于微积分的模型中。
Logarithms are inverses of exponential functions. The statement y = logₐ x means aʸ ubtraction, and powers into coefficients, which makes them powerful for solving equations.
对数是指数函数的反函数。y = logₐ x 意味着 aʸ = x。对数法则把乘法变为加法、除法变为减法、幂变为系数，因此是解方程的有力工具。
logₐ(xy) = logₐ x + logₐ y
logₐ(x / y) = logₐ x - logₐ y
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