Straight Line Graphs | 直线图

📚 Straight Line Graphs | 直线图

Straight line graphs are one of the most important topics in IGCSE Mathematics. They appear in algebra, coordinate geometry, and even in solving real-world problems. Understanding how to find the equation of a line, its gradient, and its intercepts is essential for success in your Edexcel exam.

直线图是 IGCSE 数学中最重要的主题之一。它在代数、坐标几何甚至解决现实问题中都会出现。理解如何求直线的方程、斜率和截距，是你在 Edexcel 考试中取得好成绩的关键。

1. The Coordinate Plane | 坐标平面

The coordinate plane is a two-dimensional grid formed by the x-axis (horizontal) and the y-axis (vertical). Every point is written as an ordered pair (x, y), where x is the horizontal position and y is the vertical position.

坐标平面是由 x 轴（水平）和 y 轴（垂直）组成的二维网格。每个点写成有序数对 (x, y)，其中 x 是水平位置，y 是垂直位置。

For example, the point (3, 2) means move 3 units right and 2 units up from the origin (0, 0).

例如，点 (3, 2) 表示从原点 (0, 0) 向右移动 3 个单位，再向上移动 2 个单位。

The x-coordinate is always written first.

x 坐标总是写在前面。
The y-coordinate is always written second.

y 坐标总是写在后面。
The point where the axes meet is called the origin.

坐标轴相交的点称为原点。

2. The Equation y = mx + c | 方程 y = mx + c

Every straight line can be written in the form y = mx + c, where m is the gradient (steepness) of the line and c is the y-intercept (where the line crosses the y-axis).

每条直线都可以写成 y = mx + c 的形式，其中 m 是直线的斜率（陡峭程度），c 是 y 截距（直线与 y 轴相交的点）。

y = mx + c

Here, m and c are both constants. For example, in the equation y = 2x + 3, the gradient m = 2 and the y-intercept c = 3.

这里，m 和 c 都是常数。例如，在方程 y = 2x + 3 中，斜率 m = 2，y 截距 c = 3。

3. Finding the Gradient | 求斜率

The gradient of a line measures how steep it is. It is defined as the change in y divided by the change in x between two points on the line.

直线的斜率衡量它的陡峭程度。它定义为直线上两点之间 y 的变化量除以 x 的变化量。

m = (y₂ − y₁) / (x₂ − x₁)

For example, the gradient of the line passing through the points A(1, 2) and B(4, 14) is:

例如，经过点 A(1, 2) 和 B(4, 14) 的直线的斜率为：

m = (14 − 2) / (4 − 1) = 12 / 3 = 4

A positive gradient means the line slopes upwards from left to right; a negative gradient means it slopes downwards.

正斜率表示直线从左到右向上倾斜；负斜率表示它向下倾斜。

4. Finding the y-intercept | 求 y 截距

The y-intercept is the point where the line crosses the y-axis. At this point, x = 0. In the equation y = mx + c, the y-intercept is simply c.

y 截距是直线与 y 轴相交的点。在该点处，x = 0。在方程 y = mx + c 中，y 截距就是 c。

For example, the line y = −3x + 5 has a y-intercept of 5. This means the line crosses the y-axis at (0, 5).

例如，直线 y = −3x + 5 的 y 截距为 5。这意味着直线在 (0, 5) 处与 y 轴相交。

To find the x-intercept, set y = 0 and solve for x.

要求 x 截距，令 y = 0 并解出 x。

5. Drawing a Straight Line | 画直线

To draw a straight line from its equation, you need at least two points. A common method is to use a table of values.

要从方程画一条直线，你至少需要两个点。常用方法是使用数值表。

For example, to draw y = 2x − 1, choose some x-values:

例如，要画 y = 2x − 1，选择一些 x 值：

x	0	1	2
y	−1	1	3

The points (0, −1), (1, 1) and (2, 3) all lie on the line. Plot them and join them with a straight line.

点 (0, −1)、(1, 1) 和 (2, 3) 均在直线上。把它们画出来并用直线连接。

6. Finding the Equation of a Line | 求直线方程

Given two points or one point and the gradient, you can find the equation of a straight line.

已知两个点或一个点和斜率，你就可以求出直线的方程。

Method: first find the gradient m, then substitute one point into y = mx + c to find c.

方法：先求斜率 m，然后把一个点代入 y = mx + c 求出 c。

Example: A line passes through (2, 5) with gradient 3. Substitute x = 2, y = 5, m = 3:

例子：一条直线经过 (2, 5) 且斜率为 3。代入 x = 2，y = 5，m = 3：

5 = 3(2) + c → c = 5 − 6 = −1

So the equation is y = 3x − 1.

因此方程为 y = 3x − 1。

7. Parallel Lines | 平行线

Two lines are parallel if they have the same gradient. This means they never meet, no matter how far they are extended.

两条直线如果斜率相同，那么它们平行。这意味着无论延伸多远，它们都不会相交。

For example, y = 2x + 1 and y = 2x − 4 are parallel because both have gradient m = 2.

例如，y = 2x + 1 和 y = 2x − 4 是平行的，因为斜率为 m = 2。

If a problem asks for a line parallel to a given line, simply keep the same gradient and choose a different y-intercept.

如果题目要求一条与给定直线平行的直线，只需保持相同的斜率，选择不同的 y 截距即可。

8. Perpendicular Lines | 垂直线

Two lines are perpendicular if the product of their gradients is −1. In other words, if one line has gradient m, the perpendicular line has gradient −1/m.

两条直线垂直，如果它们斜率的乘积为 −1。换句话说，如果一条直线的斜率为 m，垂直线的斜率为 −1/m。

m₁ × m₂ = −1

For example, if a line has gradient 3, a perpendicular line has gradient −1/3. Using Unicode, we write −⅓.

例如，如果一条直线的斜率为 3，垂线的斜率为 −1/3。用 Unicode 写为 −⅓。

Note: a horizontal line (gradient 0) is perpendicular to a vertical line (which has no gradient).

注意：水平线（斜率 0）垂直于竖直线（没有斜率）。

9. Intersection of Two Lines | 两直线的交点

The point where two lines cross is called the intersection point. To find it, solve the two equations simultaneously.

两条直线相交的点称为交点。要求这个点，需要联立解两个方程。

Example: Solve y = 2x + 1 and y = −x + 4.

例子：解 y = 2x + 1 和 y = −x + 4。

Set the right-hand sides equal: 2x + 1 = −x + 4

令右边相等：2x + 1 = −x + 4

3x = 3 → x = 1

Then y = 2(1) + 1 = 3. The intersection point is (1, 3).

然后 y = 2(1) + 1 = 3。交点就是 (1, 3)。

10. Real-World Applications | 实际应用

Straight line graphs are used to model relationships in physics, economics, and everyday life. For example, distance-time graphs and speed-time graphs are straight lines when the speed is constant.

直线图用于模拟物理、经济和日常生活中的关系。例如，当速度恒定时，距离-时间图和速度-时间图就是直线。

In a distance-time graph, the gradient represents speed. The steeper the line, the faster the moving object.

在距离-时间图中，斜率代表速度。直线越陡，物体运动越快。

Another common application is converting between units, such as temperature. The formula C = (5/9)(F − 32) can be graphed as a straight line.

另一个常见应用是单位换算，例如温度。公式 C = (5/9)(F − 32) 可以画成一条直线。

Understanding gradients and intercepts helps you interpret these graphs quickly and accurately.

理解斜率和截距可以帮助你快速准确地解读这些图表。

11. Common Exam Tips | 考试常见提示

Here are some important tips for straight line graph questions in your Edexcel IGCSE exam.

以下是一些关于 Edexcel IGCSE 考试中直线图题目的重要提示。

Always write the equation in the form y = mx + c before reading off the gradient or intercept.

在读出斜率或截距之前，始终把方程写成 y = mx + c 的形式。
When finding the gradient, be careful with negative coordinates.

求斜率时，注意负坐标。
Use a ruler to draw straight lines — your line must pass exactly through the plotted points.

用尺子画直线——你的直线必须准确穿过画出的点。
If two lines are perpendicular, remember to use m₂ = −1/m₁, not the same gradient.

如果两条直线垂直，记得使用 m₂ = −1/m₁，而不是相同的斜率。

12. Practice Question | 练习题

Find the equation of the line that passes through the points (3, 7) and (5, 11). Give your answer in the form y = mx + c.

求经过点 (3, 7) 和 (5, 11) 的直线方程。用 y = mx + c 的形式给出答案。

Solution: Gradient m = (11 − 7) / (5 − 3) = 4 / 2 = 2. Then substitute (3, 7): 7 = 2(3) + c → c = 1. So the equation is y = 2x + 1.

解答：斜率 m = (11 − 7) / (5 − 3) = 4 / 2 = 2。然后代入 (3, 7)：7 = 2(3) + c → c = 1。所以方程为 y = 2x + 1。

Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Straight Line Graphs | 直线图