Straight Line Graphs: Gradient, Intercept and Parallel Lines | 直线图像：斜率、截距与平行线

📚 Straight Line Graphs: Gradient, Intercept and Parallel Lines | 直线图像：斜率、截距与平行线

Straight line graphs are a central topic in IGCSE Mathematics. They connect algebra, coordinate geometry and real-world linear relationships. In this article, you will learn how to plot points, calculate gradient, use the equation y = mx + c, and apply these ideas to parallel and perpendicular lines.

直线图像是 IGCSE 数学的核心主题之一。它将代数、坐标几何和现实中的线性关系联系起来。本文将带你学习如何描点、计算斜率、使用直线方程 y = mx + c，并将这些方法应用于平行线和垂直线。

1. The Coordinate Plane and Plotting Points | 坐标平面与描点

The coordinate plane is formed by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). The point where they meet is the origin, written as (0, 0).

坐标平面由两条互相垂直的数轴组成：x 轴（水平）和 y 轴（垂直）。它们的交点称为原点，记作 (0, 0)。

Every point is written as an ordered pair (x, y). The first number is the horizontal position, and the second number is the vertical position. For example, the point (3, -2) means 3 units to the right and 2 units down from the origin.

每个点都用一个有序数对 (x, y) 表示。第一个数是水平位置，第二个数是竖直位置。例如，点 (3, -2) 表示从原点向右 3 个单位、向下 2 个单位。

Plotting points accurately is the first step in drawing any graph. Always label the axes with x and y, use a sensible scale, and mark points with a small cross.

准确描点是绘制任何图像的第一步。务必标注 x 轴和 y 轴，使用合理的刻度，并用小叉号标记点的位置。

2. What is a Straight Line Graph? | 什么是直线图像

A straight line graph is a visual representation of a linear equation. Every point on the line satisfies the equation, and every solution of the equation lies on the line.

直线图像是线性方程的直观表示。直线上的每个点都满足该方程，而方程的每个解都落在这条直线上。

Linear equations involve x and y to the power of 1 only. Examples include y = 2x + 5, 3x + 4y = 12, and y = -x. None of these contain x², y², xy or 1/x.

线性方程中 x 和 y 的指数均为 1。例如 y = 2x + 5、3x + 4y = 12 和 y = -x。这些方程都不含 x²、y²、xy 或 1/x。

A straight line continues forever in both directions, but in IGCSE questions we usually draw a bounded section and show key features such as intercepts.

直线在两端无限延伸，但在 IGCSE 题目中我们通常只画出有限的一段，并标出截距等关键特征。

3. Gradient: Definition and Calculation | 斜率：定义与计算

The gradient measures how steep a line is. It is the ratio of the vertical change to the horizontal change between any two points on the line.

斜率衡量直线的倾斜程度。它是直线上任意两点之间竖直变化量与水平变化量之比。

The gradient is often written as m. If a line passes through the points (x₁, y₁) and (x₂, y₂), the gradient is calculated using the formula:

斜率通常用 m 表示。如果直线经过点 (x₁, y₁) 和 (x₂, y₂)，其斜率可用以下公式计算：

m = (y₂ – y₁) ÷ (x₂ – x₁)

Always subtract the y-coordinates in the same order as the x-coordinates. For example, if A(1, 2) and B(4, 8), then m = (8 – 2) ÷ (4 – 1) = 6 ÷

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