📚 Graphs of Functions | 函数图像
A graph of a function gives a visual way to understand how one quantity changes with another. For IGCSE Edexcel Mathematics, mastering different types of function graphs, their shapes, key features and how to use them to solve equations is essential. This article covers the main function types you will encounter, from straight lines to trigonometric curves, and explains how to interpret and sketch them without relying on graph-plotting software.
函数图像提供了一种直观的方式来理解一个量如何随另一个量变化。对于 Edexcel IGCSE 数学，掌握不同类型函数图像的形状、关键特征以及如何利用它们解方程至关重要。本文涵盖你将遇到的主要函数类型，从直线到三角函数曲线，并解释如何在不依赖绘图软件的情况下解读和绘制它们的草图。
1. What is a Function? | 什么是函数？
A function is a rule that assigns exactly one output to each input. We often write f(x) = … or y = … . For example, f(x) = 2x + 3 means that if you input x = 1, the output is f(1) = 2(1) + 3 = 5. Every x-value in the domain produces a single y-value.
函数是一种对每个输入只分配一个输出的规则。我们通常写作 f(x) = … 或 y = … 。例如，f(x) = 2x + 3 表示如果输入 x = 1，输出就是 f(1) = 2(1) + 3 = 5。定义域中的每个 x 值都会产生一个唯一的 y 值。
You can test whether a graph represents a function using the vertical line test: if any vertical line cuts the graph more than once, then the relation is not a function. All the graphs we study in IGCSE (linear, quadratic, cubic, reciprocal, exponential, trigonometric) are functions on their given domains.
你可以用垂直线检验判断一个图像是否表示函数：如果任何一条垂直线与图像相交超过一次，那么该关系就不是函数。我们在 IGCSE 中学习的所有图像（一次、二次、三次、倒数、指数、三角函数）在给定定义域上都是函数。
2. Domain and Range | 定义域与值域
The domain of a function is the set of all possible input (x) values for which the function is defined. The range is the set of all possible output (f(x) or y) values the function can take. For example, for f(x) = √x, the domain is x ≥ 0 because you cannot take the square root of a negative number in real numbers; the range is also y ≥ 0.
函数的定义域是函数有定义的所有可能输入值（x）的集合。值域是函数可以取到的所有可能输出值（f(x) 或 y）的集合。例如，对于 f(x) = √x，定义域为 x ≥ 0，因为在实数范围内不能对负数开平方；值域也为 y ≥ 0。
For quadratic functions like f(x) = x², the domain is all real numbers (ℝ), but the range is y ≥ 0 because squares are never negative. When sketching graphs, always consider whether the domain is restricted, which affects the shape we draw.
对于像 f(x) = x² 这样的二次函数，定义域是所有实数（ℝ），但值域是 y ≥ 0，因为平方永远不会为负。在绘制草图时，始终要考虑定义域是否受限，这会影响我们绘制的形状。
- English: Domain – possible x-values; Range – possible y-values.
- 中文：定义域——可能的 x 值；值域——可能的 y 值。
3. Straight-Line Graphs | 直线图
A linear function has the form y = mx + c, where m is the gradient (slope) and c is the y-intercept (the point where the line crosses the y-axis). For instance, y = 2x − 3 has gradient 2 and y-intercept −3. To plot the graph, you can start at (0, −3) and use the gradient to find another point: move 1 unit right and 2 units up to reach (1, −1).
一次函数的形式为 y = mx + c，其中 m 是斜率，c 是 y 轴截距（直线与 y 轴的交点）。例如，y = 2x − 3 的斜率为 2，y 轴截距为 −3。要画出图像，你可以从 (0, −3) 开始，利用斜率找到另一个点：向右移动 1 个单位，向上移动 2 个单位到达 (1, −1)。
Straight lines can also be presented in the form ax + by + c = 0. Rearranging into y = mx + c helps identify gradient and intercept. Horizontal lines have equation y = k (gradient 0), and vertical lines have equation x = k (undefined gradient, not a function).
直线也可以表示为 ax + by + c = 0 的形式。将其转化为 y = mx + c 有助于识别斜率和截距。水平线的方程为 y = k（斜率为 0），垂直线的方程为 x = k（斜率无定义，不是函数）。
4. Quadratic Graphs | 二次函数图像
Quadratic functions have the general form y = ax² + bx + c, where a ≠ 0. Their graphs are parabolas. If a > 0, the parabola opens upwards (∪ shape) and has a minimum point. If a < 0, it opens downwards (∩ shape) and has a maximum point. The turning point can be found by completing the square or using symmetry; the y-intercept is c, and the x-intercepts (roots) are found by solving ax² + bx + c = 0, often by factorising or using the quadratic formula.
二次函数的一般形式为 y = ax² + bx + c，其中 a ≠ 0。它们的图像是抛物线。如果 a > 0，抛物线开口向上（∪ 形）且有一个最小值点；如果 a < 0，抛物线开口向下（∩ 形）且有一个最大值点。顶点可以通过配方法或利用对称性求得；y 轴截距为 c，而 x 轴截距（根）通过解 ax² + bx + c = 0 得到，常通过因式分解或二次公式求解。
For example, y = x² − 4x + 3 factorises to (x − 1)(x − 3), so the roots are x = 1 and x = 3. The y-intercept is 3. The vertex is midway between the roots at x = 2, giving y = (2)² − 4(2) + 3 = −1, so the vertex is (2, −1).
例如，y = x² − 4x + 3 因式分解为 (x − 1)(x − 3)，因此根为 x = 1 和 x = 3。y 轴截距为 3。顶点位于两根的中点 x = 2，代入得 y = 2² − 4×2 + 3 = −1，因此顶点为 (2, −1)。
5. Cubic Graphs | 三次函数图像
Cubic functions have the form y = ax³ + bx² + cx + d, with a ≠ 0. If a > 0, the graph generally comes from the bottom left and goes to the top right; if a < 0, it comes from top left and goes to bottom right. They can have up to two turning points. Simple cubics like y = x³ have a point of inflection at the origin, with no turning point.
三次函数的形式为 y = ax³ + bx² + cx + d，其中 a ≠ 0。如果 a > 0，图像通常从左下方延伸到右上方；如果 a < 0，则从左上方延伸到右下方。它们最多可以有两个驻点。像 y = x³ 这样的简单三次函数在原点有一个拐点，没有驻点。
To sketch a cubic, find the y-intercept (d) and factorise the function if possible to find x-intercepts. For instance, y = x³ − 2x² − x + 2 factorises to (x − 1)(x + 1)(x − 2), giving roots at x = −1, 1, and 2. The y-intercept is 2. Knowing the sign of a tells you the overall shape.
要画出三次函数的草图，先找到 y 轴截距（d），如果可能，对函数进行因式分解以找到 x 轴截距。例如，y = x³ − 2x² − x + 2 可以因式分解为 (x − 1)(x + 1)(x − 2)，根为 x = −1、1 和 2。y 轴截距为 2。知道 a 的正负就可以确定整体形状。
6. Reciprocal Graphs | 倒数函数图像
A basic reciprocal function is y = 1/x, where x ≠ 0. The graph consists of two separate branches in the first and third quadrants. As x gets very large positive or negative, y approaches 0, so the x-axis (y = 0) is a horizontal asymptote. As x approaches 0 from the right, y → +∞; from the left, y → −∞, so the y-axis (x = 0) is a vertical asymptote.
基本的倒数函数是 y = 1/x，其中 x ≠ 0。图像由第一象限和第三象限的两条独立分支组成。当 x 取非常大的正值或负值时，y 趋近于 0，因此 x 轴（y = 0）是一条水平渐近线。当 x 从右侧趋近于 0 时，y → +∞；从左侧趋近于 0 时，y → −∞，因此 y 轴（x = 0）是一条垂直渐近线。
More general reciprocal functions, like y = a/(x − h) + k, have vertical asymptote x = h and horizontal asymptote y = k. For example, y = 2/(x − 1) − 3 has asymptotes x = 1 and y = −3. The ‘a’ value stretches the graph and determines which quadrants the branches occupy.
更一般的倒数函数，如 y = a/(x − h) + k，有垂直渐近线 x = h 和水平渐近线 y = k。例如，y = 2/(x − 1) − 3 的渐近线为 x = 1 和 y = −3。a 值会拉伸图像并决定分支位于哪几个象限。
7. Exponential Graphs | 指数函数图像
Exponential functions have the form y = aˣ (or y = a^x) where a > 0 and a ≠ 1. If a > 1, the graph shows exponential growth: it passes through (0, 1), rises slowly for negative x, then increases rapidly for positive x. The x-axis (y = 0) is an asymptote. If 0 < a < 1, the graph decays, approaching the x-axis from above as x increases.
指数函数的形式为 y = aˣ（或 y = a^x），其中 a > 0 且 a ≠ 1。如果 a > 1，图像呈指数增长：经过点 (0, 1)，在 x 为负时缓慢上升，然后在 x 为正时快速增长。x 轴（y = 0）是一条渐近线。如果 0 < a < 1，图像呈指数衰减，随着 x 增大从上方趋近 x 轴。
At IGCSE, you often work with y = 2ˣ, y = 3ˣ and their simple transformations, such as y = 2ˣ + 3 (shifts up) or y = 2ˣ⁻¹. You may also see exponential functions related to compound growth or decay problems. Recognising the basic shape helps you sketch transformations quickly.
在 IGCSE 阶段，你通常会遇到 y = 2ˣ、y = 3ˣ 及其简单变换，例如 y = 2ˣ + 3（向上平移）或 y = 2ˣ⁻¹。你还可能见到与复利增长或衰减问题相关的指数函数。认识基本形状有助于你快速画出变换后的草图。
8. Trigonometric Graphs | 三角函数图像
The three trigonometric functions you need for IGCSE are y = sin x, y = cos x and y = tan x, usually with x in degrees from 0° to 360°. The sine graph starts at 0, rises to 1 at 90°, returns to 0 at 180°, goes to −1 at 270°, and back to 0 at 360°. Its range is −1 ≤ y ≤ 1, and it has a period of 360°.
IGCSE 需要掌握的三个三角函数是 y = sin x、y = cos x 和 y = tan x，x 通常以度为单位，范围为 0° 到 360°。正弦图像从 0 开始，在 90° 处上升到 1，在 180° 处回到 0，在 270° 处下降到 −1，在 360° 处回到 0。其值域为 −1 ≤ y ≤ 1，周期为 360°。
The cosine graph starts at 1 at 0°, decreases to 0 at 90°, to −1 at 180°, returns to 0 at 270° and back to 1 at 360°. It also has range −1 ≤ y ≤ 1 and period 360°. The tangent graph has asymptotes at 90° and 270°, repeats every 180°, and its range is all real numbers. You must be able to sketch these graphs and read off solutions to equations like sin x = 0.5.
余弦图像在 0° 处从 1 开始，在 90° 处下降至 0，在 180° 处降至 −1，在 270° 处回到 0，在 360° 处回到 1。它的值域也是 −1 ≤ y ≤ 1，周期为 360°。正切图像在 90° 和 270° 处有渐近线，每 180° 重复一次，值域为所有实数。你必须能够画出这些图像的草图并读出像 sin x = 0.5 这样的方程的解。
9. Using Graphs to Solve Equations | 利用图像解方程
One powerful application of function graphs is solving equations. To solve f(x) = 0, you find the x-coordinates where the graph y = f(x) crosses the x-axis. For example, the roots of x² − 4x + 3 = 0 are the x-intercepts of the parabola, x = 1 and x = 3. Similarly, to solve f(x) = g(x), you can draw both graphs on the same axes and read the x-values where they intersect.
函数图像的一个重要应用是解方程。要解 f(x) = 0，你需要在图像 y = f(x) 与 x 轴相交处找到 x 坐标。例如，x² − 4x + 3 = 0 的根就是抛物线与 x 轴的交点，即 x = 1 和 x = 3。类似地，要解 f(x) = g(x)，可以在同一坐标系中画出两个图像，然后读出它们交点的 x 值。
You can also solve equations by rearranging them into a form that matches a known graph. For instance, to solve x³ − 3x = 1, you could draw y = x³ − 3x and y = 1, or draw y = x³ − 3x − 1 and find where it crosses the x-axis. This method gives approximate solutions if exact algebraic solutions are not required.
你还可以通过将方程重新整理为与已知图像匹配的形式来求解。例如，要解 x³ − 3x = 1，你可以画出 y = x³ − 3x 和 y = 1，或者画出 y = x³ − 3x − 1 并找到它与 x 轴的交点。如果不需要精确的代数解，这种方法可以给出近似解。
10. Sketching Graphs from Factors | 根据因式绘制图像
When a polynomial function is given in factorised form, you can identify the x-intercepts immediately. For a quadratic like y = (x − p)(x − q), the graph crosses the x-axis at p and q. The y-intercept is found by setting x = 0. The sign of the leading coefficient tells you whether the parabola opens up or down.
当多项式函数以因式分解形式给出时，你可以立即确定 x 轴截距。对于像 y = (x − p)(x − q) 这样的二次函数，图像在 p 和 q 处穿过 x 轴。y 轴截距通过令 x = 0 求得。首项系数的正负告诉你抛物线是开口向上还是向下。
For a cubic given as y = (x − a)(x − b)(x − c), the roots are a, b and c. The y-intercept is −a × b × c after expanding or by setting x = 0. If any factor is repeated, the graph touches the x-axis instead of crossing it. This principle also applies to graphed trigonometric equations after factorising expressions like 2 sin x − 1 = 0.
对于以 y = (x − a)(x − b)(x − c) 形式给出的三次函数，根为 a、b 和 c。y 轴截距为 −a × b × c（展开后或将 x = 0 代入）。如果有任何因式重复，图像将触碰但不穿过 x 轴。这一原理也适用于将像 2 sin x − 1 = 0 这样的表达式因式分解后得到的三角函数图像。
Always consider the end behaviour governed by the highest power. An even-degree polynomial with positive leading coefficient will go up on both ends; an odd-degree with positive leading coefficient goes down on the left and up on the right.
始终要考虑由最高次幂决定的末端走势。具有正首项系数的偶次多项式两端均向上；具有正首项系数的奇次多项式左端向下、右端向上。
11. Transformation of Graphs | 图像变换
Understanding how graphs shift can save you time. The graph of y = f(x) + k is a vertical translation of y = f(x) by k units up if k > 0 (down if k < 0). The graph of y = f(x − h) is a horizontal translation by h units to the right if h > 0 (left if h < 0). For example, y = (x − 2)² is the parabola y = x² shifted 2 units right.
理解图像如何平移可以为你节省时间。y = f(x) + k 的图像是 y = f(x) 的垂直平移，若 k > 0 则向上平移 k 个单位（若 k < 0 则向下）。y = f(x − h) 是水平平移，若 h > 0 则向右平移 h 个单位（若 h < 0 则向左）。例如，y = (x − 2)² 就是抛物线 y = x² 向右平移 2 个单位。
Other transformations include reflections: y = −f(x) reflects the graph in the x-axis; y = f(−x) reflects in the y-axis. Stretches, like y = af(x), multiply y-coordinates by a, affecting the graph steepness. These transformations help you sketch variations of standard functions quickly.
其他变换包括对称：y = −f(x) 将图像关于 x 轴对称；y = f(−x) 关于 y 轴对称。拉伸变换，如 y = af(x)，将 y 坐标乘以 a，会影响图像的陡峭程度。这些变换能帮助你快速绘制标准函数的变体草图。
12. Summary | 总结
Function graphs are a central topic in Edexcel IGCSE Mathematics. You should be able to recognise and sketch linear, quadratic, cubic, reciprocal, exponential and trigonometric functions, identify their key features such as intercepts, turning points and asymptotes, and use graphs to solve equations in context. Always link what you see on the graph back to the algebra, and practise sketching by hand rather than relying on calculators.
函数图像是 Edexcel IGCSE 数学的核心主题。你应该能够识别并绘制一次、二次、三次、倒数、指数和三角函数的图像，识别它们的关键特征，如截距、驻点和渐近线，并利用图像在实际情境中解方程。始终将图像上的信息与代数联系起来，多进行手绘草图的练习，而不要依赖计算器。
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