Linear Functions: Graphs and Properties | 一次函数图像与性质

📚 Linear Functions: Graphs and Properties | 一次函数图像与性质

Linear functions are one of the most fundamental topics in algebra. They appear in almost every mathematics examination, from GCSE and IGCSE to A-Level and beyond. In this article, we will explore the graph and properties of linear functions in detail, including the meaning of the gradient and intercept, how to draw the graph, and how parallel and perpendicular lines relate to each other.

一次函数是代数中最基础的内容之一，几乎出现在每一场数学考试中，无论是 GCSE、IGCSE，还是 A-Level 及更高阶课程。本文将详细讲解一次函数的图像与性质，包括斜率与截距的含义、如何绘制图像，以及平行线与垂直线之间的关系。

1. Definition of a Linear Function | 一次函数的定义

A linear function is a function whose graph is a straight line. Its standard form is:

一次函数是图像为一条直线的函数，其标准形式为：

y = mx + c

Here, x is the independent variable, y is the dependent variable, m is the gradient (slope) of the line, and c is the y-intercept — the point where the line crosses the y-axis.

其中 x 是自变量，y 是因变量，m 是直线的斜率（梯度），c 是纵截距，也就是直线与 y 轴交点的纵坐标。

Some textbooks write the equation as y = ax + b or f(x) = mx + c. Regardless of the letters used, the key point is that the highest power of x is 1. This is why it is called a first-degree (linear) function.

有些教材将方程写成 y = ax + b 或 f(x) = mx + c。无论使用哪种字母，关键在于 x 的最高次数为 1。这就是它被称为一次（线性）函数的原因。

The main features of a linear function are:

一次函数的主要特点如下：

• The graph is always a straight line.

  图像总是一条直线。

• The rate of change (gradient) is constant throughout.

  变化率（斜率）处处恒定。

• Every linear function has exactly one y-intercept.

  每个一次函数有且仅有一个纵截距。

• It has at most one x-intercept, unless it is a horizontal line y = c where c ≠ 0.

  它至多有一个横截距，除非是水平线 y = c 且 c ≠ 0 的情形。

2. The Gradient m | 斜率 m

The gradient measures how steep a line is. It tells us how much y changes when x increases by 1 unit. A larger gradient means a steeper line.

斜率衡量直线的陡峭程度，表示当 x 增加 1 个单位时 y 的变化量。斜率越大，直线越陡。

Given two points (x₁, y₁) and (x₂, y₂) on the line, the gradient is calculated as:

已知直线上两点 (x₁, y₁) 和 (x₂, y₂)，斜率计算公式为：

m = (y₂ − y₁) / (x₂ − x₁) = rise / run

This formula is sometimes called “rise over run”: rise is the vertical change (Δy) and run is the horizontal change (Δx).

该公式有时称为“纵比横”（rise over run）：rise 代表纵向变化量（Δy），run 代表横向变化量（Δx）。

Example: Find the gradient of the line passing through A(1, 2) and B(4, 11).

示例：求经过 A(1, 2) 和 B(4, 11) 两点的直线的斜率。

m = (11 − 2) / (4 − 1) = 9 / 3 = 3

So the line rises 3 units for every 1 unit it moves to the right. It is important to subtract the coordinates in the same order: y₂ − y₁ over x₂ − x₁, not y₂ − y₁ over x₁ − x₂.

因此，直线向右移动 1 个单位时上升 3 个单位。注意坐标必须按相同顺序相减：分子为 y₂ − y₁，分母为 x₂ − x₁，不能写成 x₁ − x₂。

3. The y-intercept c | 纵截距 c

The y-intercept is the value of y when x = 0. In the equation y = mx + c, it is the constant term c. Graphically, it is the point where the line crosses the y-axis, written as (0, c).

纵截距是当 x = 0 时 y 的值。在方程 y = mx + c 中，它就是常数项 c

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