📚 Graphs and Properties of Linear Functions | 线性函数的图像与性质
A linear function is one of the most fundamental concepts in A-Level mathematics. It produces a straight-line graph and serves as the building block for calculus, coordinate geometry, and mathematical modelling. In this article, we will explore the definition, graphical properties, and the key techniques you need to master linear functions for your exams.
线性函数是 A-Level 数学中最基本的概念之一。它对应一条直线图像，是微积分、坐标几何和数学建模的基石。本文将系统讲解线性函数的定义、图像性质以及考试中必备的解题技巧。
1. Definition and Standard Form | 定义与标准形式
A linear function is a polynomial of degree one. Its general form is f(x) = ax + b, where a and b are real constants and a ≠ 0. In two variables, the equation of a straight line is usually written as y = mx + c, where m represents the gradient (slope) and c represents the y-intercept.
线性函数是一次多项式。其一般形式为 f(x) = ax + b，其中 a 和 b 为实数常数，且 a ≠ 0。在二维坐标中，直线方程通常写作 y = mx + c，其中 m 表示斜率（梯度），c 表示 y 轴截距。
Another common form is the general equation Ax + By + C = 0, which is frequently used when dealing with vertical lines or standard-form questions. If B ≠ 0, this equation can be rearranged into the slope-intercept form y = -(A/B)x – C/B.
另一种常见形式是直线的一般方程 Ax + By + C = 0，在处理竖直线或标准形式题目时经常用到。若 B ≠ 0，可以将其整理为斜截式 y = -(A/B)x – C/B。
In terms of domain and range, a linear function f(x) = mx + c has the set of all real numbers as its domain. Unless m = 0, the range is also all real numbers; if m = 0, the range is the single value {c}. The graph of any linear function is always a straight line, and the rate of change is constant everywhere along that line.
就定义域和值域而言，线性函数 f(x) = mx + c 的定义域为全体实数。当 m ≠ 0 时，值域也为全体实数；当 m = 0 时，值域为单元素集合 {c}。线性函数的图像始终是一条直线，且直线上任意一点的变化率都恒定不变。
2. Gradient and Intercept | 斜率与截距
The gradient (or slope) of a line measures its steepness. Given two points (x₁, y₁) and (x₂, y₂) on the line, the gradient is calculated as:
直线的斜率衡量其陡峭程度。已知直线上两点 (x₁, y₁) 和 (x₂, y₂)，斜率的计算公式为：
m = (y₂ – y₁) / (x₂ – x₁)
If m > 0, the line slopes upwards from left to right; if m < 0, it slopes downwards from left to right; if m = 0, the line is horizontal. A vertical line has an undefined gradient because x₂ - x₁ = 0, and its equation is of the form x = k. The greater the absolute value of m, the steeper the line.
当 m > 0 时，直线从左向右上升；当 m < 0 时，直线从左向右下降；当 m = 0 时，直线水平。竖直线的斜率不存在，因为 x₂ - x₁ = 0，其方程为 x = k。|m| 越大，直线越陡峭。
The y-intercept is the point where the line crosses the y-axis, given by (0, c). The x-intercept is found by setting y = 0, giving x = -c/m, provided that m ≠ 0.
y 轴截距是直线与 y 轴的交点，坐标为 (0, c)。求 x 轴截距时令 y = 0，解得 x = -c/m，前提是 m ≠ 0。
Worked Example 1: Find the gradient and both intercepts of the line 2y = 4x – 6.
例题 1：求直线 2y = 4x – 6 的斜率与两个截距。
Solution: Divide every term by 2: y = 2x – 3. Hence m = 2 and c = -3. The y-intercept is (0, -3). To find the x-intercept, set y = 0: 0 = 2x – 3, so x = 1.5. The line passes through (0, -3) and (1.5, 0).
解答：各项同时除以 2：y = 2x – 3。因此 m = 2，c = -3。y 截距为 (0, -3)。求 x 截距时令 y = 0：0 = 2x – 3，解得 x = 1.5。直线经过 (0, -3) 和 (1.5, 0) 两点。
3. Graphing Linear Functions | 绘制线性函数图像
To sketch the graph of a linear function, you only need two points. The fastest method is to use the y-intercept together with the gradient, or to find both intercepts.
绘制线性函数图像只需两个点。最快捷的方法是利用 y 截距配合斜率，或者直接求出两个坐标轴截距。
Step-by-step guide:
分步指南：
- Identify the gradient m and y-intercept c from y = mx + c. / 从 y = mx + c
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导