📚 Straight Line Graphs: y = mx + c | 直线图像：y = mx + c
llel lines and perpendicular lines will help you solve many questions confidently. This revision guide explains the key ideas, formulas and common pitfalls step by step.
直线图像是 IGCSE 数学中最常考查的主题之一。它将代数、坐标几何和现实建模联系在一起。扎实理解 y = mx + c、斜率、截距、平行线和垂直线，将帮助你自信地解决许多题目。本复习指南逐步讲解关键概念、公式和常见易错点。
1. What is a Straight Line Graph? | 什么是直线图像？
A straight line graph is a collection of points that all lie on the same straight line in the coordinate plane. In IGCSE Mathematics, we normally work on the Cartesian plane, which has a horizontal x-axis and a vertical y-axis. The position of any point is given by an ordered pair (x, y). A straight line graph can always be described by a linear equation, and its graph never curves or bends.
直线图像是坐标平面内所有位于同一条直线上的点的集合。在 IGCSE 数学中，我们通常在包含水平 x 轴和垂直 y 轴的笛卡尔平面上研究它。任意点的位置用一个有序数对 (x, y) 表示。直线图像始终可以用一个线性方程来描述，其图像永远不会弯曲。
2. The General Equation y = mx + c | 一般方程 y = mx + c
nd c, you can describe the exact position and steepness of the line without needing to plot many points.
IGCSE 中直线方程的标准形式是 y = mx + c。字母 m 表示直线的斜率或坡度，字母 c 表示 y 轴截距。y 轴截距是当 x = 0 时 y 的值。如果知道 m 和 c，就可以描述直线的确切位置和倾斜程度，而无需绘制很多点。
y = mx + c
+ 2, the gradient is 3 and the y-intercept is 2, so the line crosses the y-axis at (0, 2).
例如，在 y = 3x + 2 中，斜率为 3，y 轴截距为 2，因此直线在 (0, 2) 处与 y 轴相交。
3. Understanding Gradient m | 理解斜率 m
ard. If m = 0, the line is horizontal; if the line is vertical, its gradient is undefined.
斜率 m 衡量直线的倾斜程度。它等于两点之间的垂直变化量除以水平变化量。斜率为正的直线从左到右向上倾斜，斜率为负的直线从左到右向下倾斜。如果 m = 0，直线是水平的；如果直线是垂直的，则斜率没有定义。
m = rise ÷ run = (y₂ − y₁) ÷ (x₂ − x₁)
- m > 0: the line slopes upward.
- m > 0：直线向上倾斜。
- m < 0: the line slopes downward.
- m < 0：直线向下倾斜。
- m = 0: the line is horizontal.
- m = 0：直线是水平的。
4. Understanding y-intercept c | 理解 y 轴截距 c
always passes through (0, c). Changing the value of c moves the whole line up or down parallel to itself, but it does not change the gradient.
y 轴截距是直线与 y 轴相交的点。在这个点上，x = 0。由方程 y = mx + c 可知，当 x = 0 时，y = c。因此直线总是经过点 (0, c)。改变 c 的值会使整条直线向上或向下平移，但不改变斜率。
le y = 2x − 3 crosses at (0, −3). Both lines have the same gradient, so they are parallel.
例如，y = 2x + 5 在 (0, 5) 处与 y 轴相交，而 y = 2x − 3 在 (0, −3) 处与 y 轴相交。两条直线的斜率相同，因此它们平行。
5. Plotting a Straight Line from a Table of Values | 通过数值表绘制直线
To draw a straight line graph, choose at least three x-values, substitute each one into the equation, and calculate the corresponding y-values. Then plot the ordered pairs (x, y) on the coordinate plane and join them with a ruler. Three points are useful because they help you spot mistakes: if the points are not collinear, you have made an error.
要绘制直线图像，至少选择三个 x 值，将它们分别代入方程，并计算出对应的 y 值。然后在坐标平面上标出有序数对 (x, y)，并用直尺连接。三个点很有用，因为如果这些点不在同一条直线上，就说明计算出错了。
| x | −1 | 0 | 1 | 2 |
|---|---|---|---|---|
| y = 2x + 1 | −1 | 1 | 3 | 5 |
These points are (−1, −1), (0, 1), (1, 3) and (2, 5). Plot them and draw one straight line through all of them.
这些点是 (−1, −1)、(0, 1)、(1, 3) 和 (2, 5)。标出这些点，并画一条经过所有点的直线。
6. Finding the Equation from a Graph | 根据图像求方程
use them to calculate the gradient m. Finally substitute m and c into the form y = mx + c.
如果给出直线图像并要求求方程，首先直接从 y 轴读出 y 轴截距 c。接着在直线上选取两个明确的点，用它们计算斜率 m。最后将 m 和 c 代入 y = mx + c 的形式。
m = (y₂ − y₁) ÷ (x₂ − x₁)
es through (2, 10), then c = 4 and m = (10 − 4) ÷ (2 − 0) = 3. The equation is y = 3x + 4.
例如，如果一条直线在 (0, 4) 处与 y 轴相交，并经过点 (2, 10)，则 c = 4，m = (10 − 4) ÷ (2 − 0) = 3。因此方程为 y = 3x + 4。
7. Gradient Between Two Points | 两点间的斜率
Given two points A(x₁, y₁) and B(x₂, y₂), the gradient of the line segment AB is found by subtracting the y-coordinates and dividing by the difference in x-coordinates. Be consistent with the order of subtraction in the numerator and denominator, because switching the order in only one part will change the sign of the gradient.
给定两点 A(x₁, y₁) 和 B(x₂, y₂)，线段 AB 的斜率等于 y 坐标之差除以 x 坐标之差。分子和分母的减法顺序必须保持一致，因为只改变其中一个部分的顺序会改变斜率的符号。
m = (y₂ − y₁) ÷ (x₂ − x₁)
) ÷ (4 − 1) = 9 ÷ 3 = 3. This means y increases by 3 units for every 1 unit increase in x.
对于 A(1, 2) 和 B(4, 11)，m = (11 − 2) ÷ (4 − 1) = 9 ÷ 3 = 3。这意味着 x 每增加 1 个单位，y 增加 3 个单位。
8. Parallel and Perpendicular Lines | 平行线与垂直线
Parallel lines always have the same gradient. If line 1 has gradient m₁ and line
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