📚 Coordinate Geometry: Straight Line Graphs | 坐标几何：直线图像
In IGCSE Mathematics, coordinate geometry connects algebraic equations with geometric shapes on the Cartesian plane. Straight line graphs are examined repeatedly because they test gradient, intercepts, midpoint, distance, and the ability to move between a graph and its equation. This guide covers the core methods and common pitfalls.
在 IGCSE 数学中，坐标几何将代数方程与笛卡尔平面上的几何图形联系起来。直线图像反复出现在考试中，因为它考查斜率、截距、中点、距离以及在图像与方程之间转换的能力。本指南涵盖核心方法和常见易错点。
1. Coordinates and Plotting Points | 坐标与描点
A Cartesian plane has a horizontal x-axis and a vertical y-axis. Every point is written as an ordered pair (x, y). The first number tells you how far to move horizontally from the origin (0, 0); the second tells you how far to move vertically. For example, the point (-3, 2) lies 3 units to the left and 2 units up.
笛卡尔平面有一条水平的 x 轴和一条垂直的 y 轴。每个点都写成一个有序对 (x, y)。第一个数字表示从原点 (0, 0) 水平移动的距离；第二个数字表示垂直移动的距离。例如，点 (-3, 2) 位于向左 3 个单位、向上 2 个单位的位置。
2. Gradient of a Straight Line | 直线的斜率
The gradient m describes the direction and steepness of a line. It is defined as the change in y divided by the change in x. Given two points (x₁, y₁) and (x₂, y₂), use the formula:
斜率 m 描述直线的方向和倾斜程度。它的定义是 y 的变化量除以 x 的变化量。给定两点 (x₁, y₁) 和 (x₂, y₂)，使用公式：
m = (y₂ – y₁) / (x₂ – x₁)
A positive gradient slopes upwards from left to right; a negative gradient slopes downwards. A zero gradient gives a horizontal line, while a vertical line has undefined gradient.
斜率为正时，直线从左到右向上倾斜；斜率为负时，直线从左到右向下倾斜。斜率为零表示水平直线，而垂直直线的斜率未定义。
| Gradient m | Direction of line | 中文含义 |
|---|---|---|
| m > 0 | Slopes up to the right | 向右上方倾斜 |
| m < 0 | Slopes down to the right | 向右下方倾斜 |
| m = 0 | Horizontal line | 水平线 |
| m undefined | Vertical line | 垂直线 |
3. Midpoint of a Line Segment | 线段的中点
The midpoint of a line segment is the point exactly halfway between the two endpoints. To find it, average the x-coordinates and average the y-coordinates:
线段的中点是恰好位于两个端点中间的点。求中点时，取 x 坐标的平均值和 y 坐标的平均值：
M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)
This point is the centre of symmetry of the segment and is useful in geometry and coordinate proof questions.
这个点是线段对称中心，常用于几何题和坐标证明题。
4. Length of a Line Segment | 线段的长度
The length of a line segment is found using Pythagoras’ theorem. The horizontal difference and vertical difference form the two shorter sides of a right-angled triangle. The distance d between (x₁, y₁) and (x₂, y₂) is:
线段的长度通过勾股定理求出。水平差和垂直差构成直角三角形的两条直角边。两点 (x₁, y₁) 和 (x₂, y₂) 之间的距离 d 为：
d = √((x₂ – x₁)² + (y₂ – y₁)²)
Always write the final length as a positive value. If you get a surd such as √20, simplify it to 2√5 in exact form.
最终长度始终写成正值。如果得到根式如 √20，应在精确形式下化简为 2√5。
5. Equation of a Straight Line: y = mx + c | 直线方程：y = mx + c
The equation of a straight line is usually written as y = mx + c. The coefficient m is the gradient, and the constant c is the y-intercept, where the line crosses the y-axis. For example, y = 2x – 5 has gradient 2 and crosses the y-axis at (0, -5).
直线方程通常写作 y = mx + c。系数 m 是斜率，常数 c 是 y 轴截距，即直线与 y 轴的交点纵坐标。例如，y = 2x – 5 的斜率为 2，与 y 轴交于点 (0, -5)。
6. Finding the Equation from Two Points | 由两点求直线方程
To find the equation of a line from two points, first calculate m using the gradient formula. Then substitute one of the points into y – y₁ = m(x – x₁) and rearrange to the form y = mx + c. For example, points A(1, 3) and B(5, 11) give gradient (11 – 3) / (5 – 1) = 2, so y – 3 = 2(x – 1), which becomes y = 2x + 1.
要根据两点求直线方程，首先用斜率公式计算 m。然后将其中一个点代入 y – y₁ = m(x – x₁)，并整理成 y = mx + c 的形式。例如，点 A(1, 3) 和 B(5, 11) 的斜率为 (11 – 3) / (5 – 1) = 2，因此 y – 3 = 2(x – 1)，整理得 y = 2x + 1。
7. Parallel and Perpendicular Lines | 平行线与垂直线
Parallel lines have the same gradient. If line L₁ has gradient m₁ and line L₂ has gradient m₂, then L₁ ∥ L₂ means m₁ = m₂. Perpendicular lines meet at right angles, and their gradients multiply to -1: m₁ × m₂ = -1. Therefore the gradient of a perpendicular line is the negative reciprocal.
平行直线具有相同的斜率。如果直线 L₁ 的斜率为 m₁，直线 L₂ 的斜率为 m₂，那么 L₁ ∥ L₂ 意味着 m₁ = m₂。垂直直线成直角相交，它们的斜率乘积为 -1：m₁ × m₂ = -1。因此，垂直直线的斜率是原斜率的负倒数。
8. Intercepts and Sketching | 截距与画图
To find the x-intercept, set y = 0 and solve for x. To find the y-intercept, set x = 0 and solve for y. Once you have both intercepts, plot them and draw a straight line through them. This is the fastest way to sketch a linear graph accurately.
求 x 轴截距时，令 y = 0 并解出 x。求 y 轴截距时，令 x = 0 并解出 y。得到两个截距后，描出它们并画一条穿过两点的直线。这是快速且准确地绘制线性图像的方法。
For example, the line 2x + 3y = 12 has x-intercept (6, 0) and y-intercept (0, 4). Plot both and connect them to sketch the graph.
例如，直线 2x + 3y = 12 的 x 轴截距为 (6, 0)，y 轴截距为 (0, 4)。描出这两点并连接即可画出图像。
9. Intersection of Two Lines | 两条直线的交点
The intersection of two straight lines is the point that satisfies both equations. You can solve the two linear equations simultaneously by substitution or elimination. The solution x and y gives the coordinates of the intersection point. In a graph, this point is where the two lines cross.
两条直线的交点是同时满足两个方程的点。可以通过代入法或消元法解这两个线性方程。解出的 x 和 y 就是交点坐标。在图像中，这个点就是两条直线相交的位置。
For example, y = 2x + 1 and y = -x + 4 give 2x + 1 = -x + 4, so x = 1 and y = 3. The intersection is (1, 3).
例如，y = 2x + 1 与 y = -x + 4 联立得 2x + 1 = -x + 4，解得 x = 1，y = 3。交点坐标为 (1, 3)。
10. Real-Life Applications and Exam Tips | 实际应用与考试技巧
Straight line graphs appear in real-life contexts such as travel graphs, conversion graphs, and cost models. Always label axes, choose a sensible scale, and check that your gradient and intercept match the equation. In exams, show your working clearly because method marks are awarded for correct gradient or substitution even if the final answer is wrong.
直线图像出现在实际情境中，如行程图像、单位换算图像和成本模型。始终标注坐标轴，选择合理的比例，并检查斜率和截距是否与方程一致。考试中要清晰展示解题过程，因为即使最终答案有误，正确的斜率计算或代入步骤也能获得方法分。
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