📚 Graphs of Functions (Graphs 8) | 函数图像 (第8章)
Graphs are visual representations of mathematical relationships. In IGCSE Mathematics, understanding function graphs is essential for interpreting data, solving equations, and analysing real-world situations. This article covers the key types of graphs you need to master, including linear, quadratic, cubic, reciprocal, exponential, and trigonometric functions, along with transformations and graphical solutions.
图像是数学关系的直观表示。在IGCSE数学中，理解函数图像对于解读数据、解方程以及分析现实情境至关重要。本文将涵盖你需要掌握的主要图像类型，包括线性函数、二次函数、三次函数、倒数函数、指数函数和三角函数，以及图像变换和图解法解题。
1. Introduction to Cartesian Coordinates | 笛卡尔坐标系简介
The Cartesian plane consists of a horizontal x-axis and a vertical y-axis intersecting at the origin (0,0). Each point is described by coordinates (x, y). Understanding how to plot points and sketch graphs is the first step in mastering function graphs.
笛卡尔平面由水平的x轴和垂直的y轴组成，两轴在原点(0,0)处相交。每个点用坐标(x, y)描述。掌握如何描点和绘制图像是学好函数图像的第一步。
Key features of graphs include the slope (gradient), intercepts (where the graph crosses the axes), turning points (maximum or minimum), and asymptotes (lines the graph approaches but never touches). Recognizing these features helps in sketching and interpreting graphs.
图像的主要特征包括斜率（梯度）、截距（图像与坐标轴的交点）、转折点（极大值或极小值）和渐近线（图像趋近但永不相交的直线）。识别这些特征有助于绘制和理解图像。
2. Linear Graphs | 一次函数图像（直线）
A linear function has the form y = mx + c < 0, it slopes downwards. The x-intercept can be found by setting y = 0 and solving for x.
一次函数的形式为 y = mx + c，其中 m 代表斜率，c 代表 y 轴截距。图像是一条直线。如果 m > 0，直线向上倾斜；如果 m < 0，直线向下倾斜。x 轴截距可以通过令 y = 0 解方程求得。
both intercepts. For example, y = 3x − 2 has gradient 3 and crosses the y-axis at (0, −2).
绘制直线图像只需要两个点。常用的方法包括标出 y 轴截距并利用斜率，或者直接求出两个截距。例如，y = 3x − 2 的斜率为 3，与 y 轴交于 (0, −2)。
3. Quadratic Graphs | 二次函数图像（抛物线）
The standard quadratic function is y = ax² + bx + c, where a ≠ 0. Its graph is a parabola, which is symmetric about a vertical line called the axis of symmetry. If a > 0, the parabola opens upwards and has a minimum point; if a < 0, it opens downwards and has a maximum point.
标准二次函数形式为 y = ax² + bx + c (a ≠ 0)。其图像为一条抛物线，关于一条称为对称轴的竖直线对称。如果 a > 0，抛物线开口向上，有最低点；如果 a < 0，则开口向下，有最高点。
ntercept is at (0, c), and the x-intercepts (roots) are found by solving ax² + bx + c = 0.
转折点（顶点）可通过配方法或公式 x = −b/(2a) 求得。y 轴截距为 (0, c)，x 轴截距（根）通过解方程 ax² + bx + c = 0 得出。
4. Cubic Graphs | 三次函数图像
A cubic function is written as y = ax³ + bx² + cx + d with a ≠ 0. Its graph is a smooth curve with up to two turning points. The shape depends on the sign of a: for a > 0, the graph generally rises to the right; for a < 0, it falls to the right.
三次函数的形式为 y = ax³ + bx² + cx + d (a ≠ 0)。图像是一条光滑曲线，最多有两个转折点。形状取决于 a 的符号：当 a > 0 时，图像总体上向右上升；当 a < 0 时，向右下降。
Cubic graphs can have one, two, or three real roots depending on how many times they cross the x-axis. They are not symmetric like parabolas, but some have rotational symmetry about a point of inflection.
三次函数图像与 x 轴的交点可以是一个、两个或三个实数根。它们不像抛物线那样具有轴对称性，但有些三次图像关于拐点成中心对称。
5. Reciprocal Graphs | 倒数函数图像（双曲线）
The basic reciprocal function is y = 1/x symptotes at x = 0 (vertical) and y = 0 (horizontal). The graph never touches these lines.
最基本的倒数函数是 y = 1/x（或更一般地 y = a/x）。图像由两个独立的分支组成，称为双曲线。渐近线为 x = 0（垂直）和 y = 0（水平）。图像永远不接触这些直线。
n the first and third quadrants; if a < 0, they are in the second and fourth quadrants. Reciprocal graphs are useful for modelling inverse proportion.
当 a > 0 时，y = a/x 的分支位于第一和第三象限；当 a < 0 时，位于第二和第四象限。倒数图像在反比例建模中非常有用。
6. Exponential Graphs | 指数函数图像
Exponential functions take the form y = aˣ te. For a > 1, the graph increases rapidly (exponential growth); for 0 < a < 1, it decreases (exponential decay).
指数函数的形式为 y = aˣ，其中 a > 0 且 a ≠ 1。图像总是经过 (0,1)，因为 a⁰ = 1。x 轴（y = 0）是一条水平渐近线。当 a > 1 时，图像快速增长（指数增长）；当 0 < a < 1 时，图像递减（指数衰减）。
A key exponential function in mathematics is y = eˣ ial graphs are used in compound interest, population growth, and radioactive decay models.
数学中一个重要的指数函数是 y = eˣ，其中 e ≈ 2.718。指数图像被广泛应用于复利、人口增长和放射性衰变等模型中。
7. Trigonometric Graphs | 三角函数图像
CSE, x is usually in degrees. The sine and cosine graphs are periodic with period 360° and have a maximum of 1 and minimum of −1.
三个关键的三角函数图像是 y = sin x、y = cos x 和 y = tan x。在IGCSE中，x 通常以度为单位。正弦和余弦图像是周期为 360° 的周期函数，最大值是 1，最小值是 −1。
d repeats every 180°. Understanding these shapes is essential for solving trigonometric equations graphically.
正弦图像从原点开始并向上移动；余弦图像从其最大值 (0,1) 开始。正切图像在 x = 90°, 270° 等处有渐近线，周期为 180°。理解这些形状对于用图像解三角方程至关重要。
8. Transformations of Graphs | 图像变换
transformations are translation (shift), reflection, and stretch. For a function y = f(x):
图像变换是指对基本函数进行平移、反射或拉伸。主要的变换包括平移（移动）、反射和拉伸。对于函数 y = f(x)：
- f(x) + a translates the graph vertically by a units (up if a > 0).
- f(x + a) translates horizontally by −a units (left if a > 0).
- −f(x) reflects the graph in the x-axis.
- f(−x) reflects in the y-axis.
- a f(x) stretches vertically by factor a.
f(x) + a 将图像纵向平移 a 个单位（a > 0 时向上）；f(x + a) 横向平移 −a 个单位（a > 0 时向左）；−f(x) 关于 x 轴反射；f(−x) 关于 y 轴反射；a f(x) 纵向拉伸为原来的 a 倍。
Applying these transformations correctly allows you to sketch related graphs quickly without plotting many points.
正确应用这些变换，可以快速绘制相关图像，而无需描出大量点。
9. Using Graphs to Solve Equations | 利用图像解方程
= g(x) on the same axes. The x-coordinates of their intersection points are the solutions.
图像可用于求方程的近似解。要解 f(x) = g(x)，在同一直角坐标系中画出 y = f(x) 和 y = g(x)，它们的交点横坐标就是方程的解。
d off where y = 0. Alternatively, rearrange to x² = 2x + 3 and plot y = x² and y = 2x + 3.
常见的考试题型是在已有的二次函数图像上添加一条合适的直线来解方程。例如，要解 x² − 2x − 3 = 0，你可以画出 y = x² − 2x − 3 的图像，并找出 y = 0 时的 x 值。或者，将方程变形为 x² = 2x + 3，再画出 y = x² 和 y = 2x + 3 的图像。
10. Gradients and Areas under Curves | 曲线斜率与曲线下的面积
The gradient at a point on a curve is found by drawing a tangent and calculating its slope. This gives an instantaneous rate of change. For example, on a distance–time graph, the gradient represents speed.
曲线上某一点的斜率可以通过画出该点的切线并计算切线斜率来求得。这表示瞬时变化率。例如，在距离-时间图像中，斜率代表速度。
Estimating the area under a curve between two points involves splitting the area into simple shapes (rectangles or trapeziums) and summing their areas. This is useful when the graph represents a rate (e.g., velocity–time graph gives distance).
估算曲线下两点之间的面积，可以将该区域划分成简单图形（如矩形或梯形），然后将这些面积相加。当图像表示速率时，这一方法非常有用（例如，速度-时间图像下的面积表示距离）。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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