Functions and Graphs in IGCSE Mathematics — IGCSE数学中的函数与图像

什么是函数？| What is a Function?

函数是 IGCSE 数学中最基础的概念之一。简单来说，函数是一种特殊的对应关系：对于每一个输入值（x），都有且只有一个输出值（y）。我们可以把函数想象成一台”机器” – 你从一端放入一个数字，机器按照固定的规则进行处理，然后从另一端输出一个结果。

A function is one of the most fundamental concepts in IGCSE Mathematics. Simply put, a function is a special relationship: for every input value (x), there is exactly one output value (y). You can think of a function as a “machine” – you put a number in at one end, the machine processes it according to a fixed rule, and outputs a result at the other end.

在函数中，输入值 x 被称为”自变量”（independent variable），输出值 y 被称为”因变量”（dependent variable），因为 y 的值取决于 x。函数可以用三种方式表示：代数表达式（如 f(x) = 2x + 3）、图像（在坐标平面上画出的曲线）或表格（列出 x 和对应的 y 值）。

In a function, the input value x is called the “independent variable”, and the output value y is called the “dependent variable”, because the value of y depends on x. Functions can be represented in three ways: algebraic expressions (e.g., f(x) = 2x + 3), graphs (curves plotted on a coordinate plane), or tables (listing x and corresponding y values).

垂直线测试 | The Vertical Line Test

如何判断一个图像是否表示函数？IGCSE 考试中常用”垂直线测试”：在图像上任意位置画一条垂直线，如果这条线与图像相交多于一个点，那么这个图像就不表示函数。这是因为函数的定义要求每个 x 值只能对应一个 y 值。

How do you determine whether a graph represents a function? The IGCSE exam commonly uses the “vertical line test”: draw a vertical line at any position on the graph. If this line intersects the graph at more than one point, then the graph does not represent a function. This is because the definition of a function requires each x-value to correspond to exactly one y-value.

例如，圆的方程 x^2 + y^2 = r^2 的图像不是一个函数，因为一条垂直线会在两个点与圆相交（除了最左和最右的端点）。但抛物线 y = x^2 的图像是一个函数 – 任何垂直线最多与它相交一次。

For example, the graph of the circle equation x^2 + y^2 = r^2 is not a function, because a vertical line will intersect the circle at two points (except at the extreme left and right endpoints). But the graph of the parabola y = x^2 is a function – any vertical line intersects it at most once.

函数符号 | Function Notation

在 IGCSE 数学中，我们使用 f(x) 来表示一个函数，读作”f of x”。这种符号让你可以简洁地表达”将 x 代入函数 f 中”。例如，如果 f(x) = 3x – 5，那么 f(2) = 3(2) – 5 = 1，f(-1) = 3(-1) – 5 = -8。

In IGCSE Mathematics, we use f(x) to denote a function, read as “f of x”. This notation allows you to concisely express “substitute x into the function f”. For example, if f(x) = 3x – 5, then f(2) = 3(2) – 5 = 1, and f(-1) = 3(-1) – 5 = -8.

函数不一定总是叫 f – 你也可以使用 g(x)、h(x) 或其他字母。当题目中有多个函数时，使用不同的字母可以避免混淆。例如：f(x) = x^2，g(x) = 2x + 1。当题目要求计算 f(3) + g(4) 时，你需要分别代入：f(3) = 9，g(4) = 9，总和为 18。

Functions are not always called f – you can also use g(x), h(x), or other letters. When a problem involves multiple functions, using different letters avoids confusion. For example: f(x) = x^2, g(x) = 2x + 1. When asked to calculate f(3) + g(4), you substitute separately: f(3) = 9, g(4) = 9, so the sum is 18.

定义域与值域 | Domain and Range

定义域（domain）是函数可以接受的所有输入值（x 值）的集合。值域（range）是函数可以产生的所有输出值（y 值）的集合。理解定义域和值域是 IGCSE 数学中的重要考点。

The domain is the set of all input values (x-values) that a function can accept. The range is the set of all output values (y-values) that a function can produce. Understanding domain and range is an important topic in IGCSE Mathematics.

如何求定义域 | How to Find the Domain

对于大多数多项式函数（如 f(x) = x^2 + 3x – 2），定义域是所有实数 R，因为任何实数代入都不会产生问题。但对于包含分母或平方根的函数，需要特别注意：分母不能为零，平方根内部不能为负数。

For most polynomial functions (e.g., f(x) = x^2 + 3x – 2), the domain is all real numbers R, because any real number can be substituted without issue. But for functions involving denominators or square roots, you need to be careful: denominators cannot be zero, and expressions inside square roots cannot be negative.

例如：f(x) = 1/(x – 2) 的定义域是 x ≠ 2（因为 x = 2 时分母为零）。f(x) = √(x + 3) 的定义域是 x ≥ -3（因为平方根内部必须大于等于零）。对于 f(x) = 1/√(x – 1)，定义域是 x > 1（分母不能为零且平方根内部必须为正）。

For example: the domain of f(x) = 1/(x – 2) is x ≠ 2 (because the denominator becomes zero when x = 2). The domain of f(x) = √(x + 3) is x ≥ -3 (because the expression inside the square root must be non-negative). For f(x) = 1/√(x – 1), the domain is x > 1 (the denominator cannot be zero and the radicand must be positive).

如何求值域 | How to Find the Range

求值域通常比求定义域更具挑战性。对于线性函数 f(x) = mx + c，值域是所有实数 R。对于二次函数 f(x) = ax^2 + bx + c（a > 0），图像是开口向上的抛物线，最小值在顶点处，因此值域是 y ≥ y_min。如果 a < 0，抛物线开口向下，最大值在顶点处，值域是 y ≤ y_max。

Finding the range is usually more challenging than finding the domain. For linear functions f(x) = mx + c, the range is all real numbers R. For quadratic functions f(x) = ax^2 + bx + c where a > 0, the graph is an upward-opening parabola with a minimum at the vertex, so the range is y ≥ y_min. If a < 0, the parabola opens downward with a maximum at the vertex, and the range is y ≤ y_max.

对于 f(x) = x^2 + 2，最小值为 2（当 x = 0 时），因此值域是 y ≥ 2。对于 f(x) = 2^x（指数函数），值域是 y > 0（指数函数永远不取零或负值）。对于 f(x) = sin x，值域是 -1 ≤ y ≤ 1（正弦函数在 -1 和 1 之间振荡）。

For f(x) = x^2 + 2, the minimum value is 2 (when x = 0), so the range is y ≥ 2. For f(x) = 2^x (exponential function), the range is y > 0 (exponential functions never reach zero or negative values). For f(x) = sin x, the range is -1 ≤ y ≤ 1 (the sine function oscillates between -1 and 1).

线性函数及其图像 | Linear Functions and Their Graphs

线性函数是最简单的函数类型，形式为 f(x) = mx + c 或 y = mx + c，其中 m 是斜率（gradient），c 是 y 轴截距（y-intercept）。它的图像是一条直线。

Linear functions are the simplest type of function, in the form f(x) = mx + c or y = mx + c, where m is the gradient (slope) and c is the y-intercept. Their graph is a straight line.

斜率 m 表示直线的陡峭程度：m > 0 时直线向上倾斜，m < 0 时直线向下倾斜，m = 0 时直线是水平的。斜率可以通过两点 (x1, y1) 和 (x2, y2) 计算：m = (y2 - y1)/(x2 - x1)。y 轴截距 c 是直线与 y 轴相交的点的 y 坐标，即当 x = 0 时的 y 值。

The gradient m indicates the steepness of the line: m > 0 means the line slopes upward, m < 0 means it slopes downward, and m = 0 means the line is horizontal. The gradient can be calculated from two points (x1, y1) and (x2, y2): m = (y2 - y1)/(x2 - x1). The y-intercept c is the y-coordinate of the point where the line crosses the y-axis, i.e., the y-value when x = 0.

平行线具有相同的斜率（m1 = m2）。垂直线的斜率乘积为 -1（m1 × m2 = -1），即它们互为负倒数。例如，如果一条直线的斜率是 2，那么与它垂直的直线的斜率是 -1/2。

Parallel lines have the same gradient (m1 = m2). Perpendicular lines have gradients whose product is -1 (m1 × m2 = -1), meaning they are negative reciprocals of each other. For example, if a line has gradient 2, then a line perpendicular to it has gradient -1/2.

二次函数与抛物线 | Quadratic Functions and Parabolas

二次函数的形式为 f(x) = ax^2 + bx + c，其中 a ≠ 0。它的图像是一条抛物线（parabola），这是 IGCSE 数学中最重要的图像之一。

Quadratic functions are in the form f(x) = ax^2 + bx + c, where a ≠ 0. Their graph is a parabola, one of the most important graphs in IGCSE Mathematics.

当 a > 0 时，抛物线开口向上（U 形），有一个最小值点。当 a < 0 时，抛物线开口向下（倒 U 形），有一个最大值点。抛物线的对称轴（axis of symmetry）是一条穿过顶点的垂直线，方程为 x = -b/(2a)。

When a > 0, the parabola opens upward (U-shaped) and has a minimum point. When a < 0, the parabola opens downward (inverted U-shape) and has a maximum point. The axis of symmetry of the parabola is a vertical line passing through the vertex, with the equation x = -b/(2a).

二次函数的三种形式 | Three Forms of Quadratic Functions

IGCSE 考试要求你熟练掌握二次函数的三种表示形式：（1）标准式：f(x) = ax^2 + bx + c，最容易识别 y 轴截距 c 和开口方向；（2）顶点式：f(x) = a(x – h)^2 + k，直接给出顶点坐标 (h, k)；（3）因式分解式：f(x) = a(x – p)(x – q)，直接给出 x 轴截距（根）p 和 q。

The IGCSE exam requires you to be proficient with three forms of quadratic functions: (1) Standard form: f(x) = ax^2 + bx + c, which makes it easiest to identify the y-intercept c and the direction of opening; (2) Vertex form: f(x) = a(x – h)^2 + k, which directly gives the vertex coordinates (h, k); (3) Factorised form: f(x) = a(x – p)(x – q), which directly gives the x-intercepts (roots) p and q.

在三种形式之间转换是 IGCSE 的常见题型。从标准式转换为顶点式需要”配方法”（completing the square）：f(x) = x^2 + 6x + 5 = (x + 3)^2 – 4，顶点为 (-3, -4)。从标准式转换为因式分解式需要”因式分解”：f(x) = x^2 – 5x + 6 = (x – 2)(x – 3)，根为 x = 2 和 x = 3。

Converting between the three forms is a common IGCSE question type. Converting from standard form to vertex form requires “completing the square”: f(x) = x^2 + 6x + 5 = (x + 3)^2 – 4, giving the vertex (-3, -4). Converting from standard form to factorised form requires “factorisation”: f(x) = x^2 – 5x + 6 = (x – 2)(x – 3), giving the roots x = 2 and x = 3.

三次函数与倒数函数 | Cubic and Reciprocal Functions

三次函数的形式为 f(x) = ax^3 + bx^2 + cx + d，其中 a ≠ 0。它的图像是一条 S 形的曲线，可以有一个、两个或三个 x 轴截距。最简单的三次函数 f(x) = x^3 是一条关于原点对称的曲线，经过 (-1, -1)、(0, 0) 和 (1, 1)。

Cubic functions are in the form f(x) = ax^3 + bx^2 + cx + d, where a ≠ 0. Their graph is an S-shaped curve that can have one, two, or three x-intercepts. The simplest cubic function f(x) = x^3 is a curve symmetric about the origin, passing through (-1, -1), (0, 0), and (1, 1).

倒数函数的形式为 f(x) = k/x 或 f(x) = k/(x – h) + v。它的图像是一条双曲线（hyperbola），有两条渐近线（asymptotes）：一条垂直渐近线在 x = 0（或 x = h），一条水平渐近线在 y = 0（或 y = v）。函数在渐近线处无定义，图像永远不会触及渐近线。

Reciprocal functions are in the form f(x) = k/x or f(x) = k/(x – h) + v. Their graph is a hyperbola with two asymptotes: a vertical asymptote at x = 0 (or x = h) and a horizontal asymptote at y = 0 (or y = v). The function is undefined at the asymptote, and the graph never touches the asymptotes.

对于 f(x) = 1/x，当 x 趋近于 0 从正值一侧时，y 趋近于正无穷大；当 x 趋近于 0 从负值一侧时，y 趋近于负无穷大。图像由两个分支组成，分别位于第一和第三象限。

For f(x) = 1/x, as x approaches 0 from the positive side, y approaches positive infinity; as x approaches 0 from the negative side, y approaches negative infinity. The graph consists of two branches, located in the first and third quadrants respectively.

指数函数 | Exponential Functions

指数函数的形式为 f(x) = a × b^x，其中 b > 0 且 b ≠ 1。当 b > 1 时，函数表示指数增长（exponential growth）；当 0 < b < 1 时，函数表示指数衰减（exponential decay）。

Exponential functions are in the form f(x) = a × b^x, where b > 0 and b ≠ 1. When b > 1, the function represents exponential growth; when 0 < b < 1, the function represents exponential decay.

指数函数的一个重要特征是它有一条水平渐近线 y = 0（x 轴）。无论 x 取何值，b^x 始终为正，因此指数函数的图像始终位于 x 轴上方。函数值可以无限增大，但永远不会降到零或负数。

An important characteristic of exponential functions is that they have a horizontal asymptote at y = 0 (the x-axis). No matter what value x takes, b^x is always positive, so the graph of an exponential function always lies above the x-axis. The function values can grow infinitely large but can never drop to zero or become negative.

在 IGCSE 考试中，你可能需要解指数方程，如 2^x = 8（得到 x = 3）或 3^(x+1) = 27（得到 x = 2）。这类题目通常要求你识别出两边可以写成相同底数的幂。

In the IGCSE exam, you may need to solve exponential equations such as 2^x = 8 (giving x = 3) or 3^(x+1) = 27 (giving x = 2). These problems typically require you to recognise that both sides can be written as powers of the same base.

三角函数 | Trigonometric Functions

IGCSE 数学要求你掌握三个基本三角函数：正弦函数 f(x) = sin x、余弦函数 f(x) = cos x 和正切函数 f(x) = tan x。这些函数都是周期函数（periodic functions），它们的图像有规律地重复。

IGCSE Mathematics requires you to master three basic trigonometric functions: the sine function f(x) = sin x, the cosine function f(x) = cos x, and the tangent function f(x) = tan x. These are all periodic functions, meaning their graphs repeat at regular intervals.

sin x 和 cos x 的周期为 360°（或 2π 弧度），值域为 [-1, 1]。sin x 的图像从原点开始，先上升至最大值 1（在 x = 90°），然后下降至最小值 -1（在 x = 270°），最后回到 0（在 x = 360°）。cos x 的图像与 sin x 形状相同，但向右平移了 90°：它从最大值 1 开始（在 x = 0°）。

sin x and cos x have a period of 360° (or 2π radians) and a range of [-1, 1]. The graph of sin x starts at the origin, rises to a maximum of 1 (at x = 90°), falls to a minimum of -1 (at x = 270°), and returns to 0 (at x = 360°). The graph of cos x has the same shape as sin x but is shifted 90° to the right: it starts at the maximum value of 1 (at x = 0°).

tan x 的周期为 180°（或 π 弧度）。它的图像有一系列垂直渐近线，位于 x = 90°、270°、450° 等处，在这些点函数无定义。tan x 的值域是所有实数 R，图像在渐近线之间从负无穷上升到正无穷。

tan x has a period of 180° (or π radians). Its graph has a series of vertical asymptotes at x = 90°, 270°, 450°, etc., where the function is undefined. The range of tan x is all real numbers R, and the graph rises from negative infinity to positive infinity between asymptotes.

图像变换 | Graph Transformations

理解图像变换是 IGCSE 函数章节中最实用的技能之一。给定一个基本函数 f(x) 的图像，你可以通过应用变换来绘制相关函数的图像。四种基本变换是：平移（translation）、拉伸（stretch）、反射（reflection）和压缩（compression）。

Understanding graph transformations is one of the most practical skills in the IGCSE functions chapter. Given the graph of a basic function f(x), you can sketch the graphs of related functions by applying transformations. The four basic transformations are: translation, stretch, reflection, and compression.

平移变换 | Translation

f(x) + a 将图像向上平移 a 个单位（a > 0 向上，a < 0 向下）。f(x + a) 将图像向左平移 a 个单位（a > 0 向左，a < 0 向右）。注意水平平移的方向与直觉相反：f(x - 2) 是向右平移 2 个单位，而不是向左。

f(x) + a translates the graph upward by a units (a > 0 moves up, a < 0 moves down). f(x + a) translates the graph left by a units (a > 0 moves left, a < 0 moves right). Note that the direction of horizontal translation is counterintuitive: f(x - 2) shifts the graph 2 units to the right, not left.

拉伸与反射 | Stretches and Reflections

a f(x) 在垂直方向上拉伸图像，拉伸因子为 a。如果 |a| > 1，图像被拉长；如果 0 < |a| < 1，图像被压缩。如果 a 为负，图像还会关于 x 轴反射。f(ax) 在水平方向上拉伸图像，拉伸因子为 1/a。如果 |a| > 1，图像被压缩（沿 x 轴方向缩小）；如果 0 < |a| < 1，图像被拉长（沿 x 轴方向变宽）。

a f(x) stretches the graph vertically by a factor of a. If |a| > 1, the graph is elongated; if 0 < |a| < 1, the graph is compressed. If a is negative, the graph is also reflected across the x-axis. f(ax) stretches the graph horizontally by a factor of 1/a. If |a| > 1, the graph is compressed (narrower along the x-axis); if 0 < |a| < 1, the graph is stretched (wider along the x-axis).

-f(x) 将图像关于 x 轴反射，f(-x) 将图像关于 y 轴反射。例如，y = sin x 的图像和 y = -sin x 的图像关于 x 轴对称。y = 2^x 和 y = 2^(-x) 的图像关于 y 轴对称，因为 2^(-x) = (1/2)^x。

-f(x) reflects the graph across the x-axis, and f(-x) reflects the graph across the y-axis. For example, the graphs of y = sin x and y = -sin x are symmetric about the x-axis. The graphs of y = 2^x and y = 2^(-x) are symmetric about the y-axis, because 2^(-x) = (1/2)^x.

复合函数 | Composite Functions

复合函数是将一个函数的输出作为另一个函数的输入。表示为 fg(x) 或 f(g(x))，意思是先将 x 代入 g，再将 g(x) 的结果代入 f。顺序很重要：fg(x) 表示先做 g 再做 f，这与 fg(x) = f(g(x)) 一致。

A composite function is formed when the output of one function is used as the input of another. It is written as fg(x) or f(g(x)), which means first substitute x into g, then substitute the result g(x) into f. Order matters: fg(x) means do g first then f, which matches fg(x) = f(g(x)).

例如，设 f(x) = 2x + 1，g(x) = x^2。那么 fg(x) = f(g(x)) = f(x^2) = 2(x^2) + 1 = 2x^2 + 1。反过来，gf(x) = g(f(x)) = g(2x + 1) = (2x + 1)^2 = 4x^2 + 4x + 1。请注意 fg(x) 不等于 gf(x)，因此复合函数的顺序至关重要。

For example, let f(x) = 2x + 1 and g(x) = x^2. Then fg(x) = f(g(x)) = f(x^2) = 2(x^2) + 1 = 2x^2 + 1. Conversely, gf(x) = g(f(x)) = g(2x + 1) = (2x + 1)^2 = 4x^2 + 4x + 1. Note that fg(x) is not equal to gf(x), so the order of composition is crucial.

反函数 | Inverse Functions

反函数 f^(-1)(x) 是一个”撤销”原函数 f(x) 操作的函数：如果 f(a) = b，那么 f^(-1)(b) = a。并非所有函数都有反函数 – 只有一一对应（one-to-one）的函数才有反函数。

An inverse function f^(-1)(x) is a function that “undoes” the operation of the original function f(x): if f(a) = b, then f^(-1)(b) = a. Not all functions have inverses – only one-to-one functions have inverse functions.

如何求反函数 | How to Find an Inverse Function

求反函数的标准步骤是：（1）将 f(x) 替换为 y，得到方程 y = f(x)；（2）交换 x 和 y 的位置；（3）解出 y；（4）将 y 替换为 f^(-1)(x)。例如，对于 f(x) = 2x + 3：写为 y = 2x + 3，交换得 x = 2y + 3，解出 y = (x – 3)/2，因此 f^(-1)(x) = (x – 3)/2。

The standard steps for finding an inverse function are: (1) Replace f(x) with y, obtaining the equation y = f(x); (2) Swap the positions of x and y; (3) Solve for y; (4) Replace y with f^(-1)(x). For example, for f(x) = 2x + 3: write y = 2x + 3, swap to get x = 2y + 3, solve for y = (x – 3)/2, so f^(-1)(x) = (x – 3)/2.

反函数的图像是原函数图像关于直线 y = x 的反射。这意味着原函数上的点 (a, b) 在反函数上变为点 (b, a)。f(x) 的定义域成为 f^(-1)(x) 的值域，f(x) 的值域成为 f^(-1)(x) 的定义域。

The graph of an inverse function is the reflection of the original function’s graph across the line y = x. This means that a point (a, b) on the original function becomes (b, a) on the inverse function. The domain of f(x) becomes the range of f^(-1)(x), and the range of f(x) becomes the domain of f^(-1)(x).

考试技巧与常见错误 | Exam Tips and Common Mistakes

在 IGCSE 数学考试中，函数与图像题目通常占据试卷的重要位置。以下是一些关键技巧和常见错误的总结，帮助你在考试中取得好成绩。

In IGCSE Mathematics exams, questions on functions and graphs typically occupy a significant portion of the paper. Here is a summary of key tips and common mistakes to help you perform well in the exam.

常见错误一：混淆 f(x + 2) 和 f(x) + 2。前者是水平方向向左平移 2 个单位，后者是垂直方向向上平移 2 个单位。两者产生完全不同的图像。

Common mistake 1: confusing f(x + 2) with f(x) + 2. The former is a horizontal translation 2 units to the left, while the latter is a vertical translation 2 units upward. The two produce completely different graphs.

常见错误二：忘记检查定义域约束。在解涉及分母或平方根的函数题目时，始终检查 x 是否会使分母为零或平方根内部为负数。

Common mistake 2: forgetting to check domain restrictions. When solving problems involving functions with denominators or square roots, always check whether any x-values would make a denominator zero or a radicand negative.

常见错误三：在求反函数时忘记表示定义域。如果原函数的定义域被限制（例如 x ≥ 0），那么反函数的定义域和值域也会受到约束。

Common mistake 3: forgetting to state the domain when finding inverse functions. If the original function’s domain is restricted (e.g., x ≥ 0), then the inverse function’s domain and range will also be constrained.

考试技巧一：画草图！即使题目没有明确要求画图，快速勾勒函数的草图可以帮助你直观理解问题，并验证代数计算的合理性。

Exam tip 1: sketch graphs! Even if the question does not explicitly ask for a graph, a quick sketch of the function can help you understand the problem visually and verify the reasonableness of your algebraic calculations.

考试技巧二：使用计算器验证。在 IGCSE 考试中，你可以使用计算器绘制函数图像，检查定义域和值域，计算特定点的函数值。

Exam tip 2: verify with your calculator. In the IGCSE exam, you can use your calculator to plot function graphs, check domains and ranges, and evaluate function values at specific points.

Summary | 总结

函数与图像是 IGCSE Edexcel 数学的核心内容，涵盖了从基础概念到高级变换的广泛主题。掌握函数符号 f(x)、定义域与值域、各类函数的图像特征（线性、二次、三次、倒数、指数、三角）、图像变换（平移、拉伸、反射）、复合函数以及反函数，是取得高分的关键。通过大量练习和画图辅助理解，你将能够在 IGCSE 考试中自信地应对函数相关的任何题目。

Functions and graphs form a core component of IGCSE Edexcel Mathematics, spanning a wide range of topics from fundamental concepts to advanced transformations. Mastering function notation f(x), domain and range, graph characteristics of various function types (linear, quadratic, cubic, reciprocal, exponential, trigonometric), graph transformations (translation, stretch, reflection), composite functions, and inverse functions is key to achieving a high score. Through extensive practice and sketching graphs to aid understanding, you will be able to tackle any function-related question in the IGCSE exam with confidence.

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