Straight Line Graphs | 直线图像

📚 Straight Line Graphs | 直线图像

Straight line graphs are one of the most important topics in IGCSE Mathematics. They appear in almost every exam paper, either as direct questions or as tools for solving problems in coordinate geometry, algebra and even statistics. This revision guide will help you understand the core ideas, avoid common mistakes and gain confidence in answering exam-style questions.

直线图像是 IGCSE 数学中最重要的话题之一。它们几乎出现在每一份试卷中，既可能是直接考查的题目，也可能作为坐标系几何、代数甚至统计解题的工具。本复习指南将帮助你理解核心概念、避免常见错误，并自信地应对考试风格的问题。

1. The Equation of a Straight Line | 直线方程

The general equation of a straight line is written as y = mx + c, where m is the gradient and c is the y-intercept. This is the most common form used in IGCSE Edexcel questions.

直线的一般方程写作 y = mx + c，其中 m 是斜率，c 是 y 轴截距。这是在 IGCSE Edexcel 试题中最常用的形式。

Another useful form is ax + by = d, called the standard form. You can rearrange any standard form equation into y = mx + c by making y the subject.

另一种有用形式是 ax + by = d，称为标准形式。你可以将任何标准形式的方程通过移项改写成 y = mx + c。

Remember: every straight line has a unique equation, but a given equation can be written in different but equivalent ways.

记住：每条直线都有唯一的方程，但一个给定的方程可以写成不同但等价的形式。

2. Gradient | 斜率

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

斜率衡量一条线的陡峭程度。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 between points (2, 3) and (6, 11) is m = (11 − 3)/(6 − 2) = 8/4 = 2.

例如，点 (2, 3) 和 (6, 11) 之间的斜率为 m = (11 − 3)/(6 − 2) = 8/4 = 2。

A positive gradient means the line slopes upwards from left to right. A negative gradient means the line slopes downwards. A horizontal line has gradient 0, and a vertical line has an undefined gradient.

正斜率表示直线从左到右向上倾斜。负斜率表示直线向下倾斜。水平线斜率为 0，而垂直线斜率未定义。

3. Intercepts | 截距

The y-intercept is the point where the line crosses the y-axis. It is the value of y when x = 0. In the equation y = mx + c, the y-intercept is c.

y 轴截距是直线与 y 轴相交的点。它是当 x = 0 时 y 的值。在方程 y = mx + c 中，y 轴截距是 c。

The x-intercept is the point where the line crosses the x-axis. It is the value of x when y = 0. To find it, set y = 0 and solve for x.

x 轴截距是直线与 x 轴相交的点。它是当 y = 0 时 x 的值。要求它，令 y = 0 并解出 x。

For example, for the line y = 3x − 6, the y-intercept is −6. Setting 3x − 6 = 0 gives x = 2, so the x-intercept is 2.

例如，对于直线 y = 3x − 6，y 轴截距是 −6。令 3x − 6 = 0 得到 x = 2，所以 x 轴截距是 2。

4. Drawing a Straight Line Graph | 绘制直线图像

To draw a straight-line graph, you only need two points. A third point is useful to check your work.

要绘制直线图像，只需要两个点。第三个点可以用来检查你的作图。

The most reliable method is to choose three x-values, substitute them into the equation to find the corresponding y-values, and plot the resulting coordinates.

最可靠的方法是选择三个 x 值，将它们代入方程求出对应的 y 值，然后描出所得坐标。

For y = 2x + 1, choose x = 0, 1, 2:

对于 y = 2x + 1，选择 x = 0, 1, 2：

x	0	1	2
y	1	3	5

Plot (0, 1), (1, 3) and (2, 5), then draw a straight line through them. Always extend the line across the grid.

描出 (0, 1)、(1, 3) 和 (2, 5)，然后画一条穿过它们的直线。始终将直线延伸穿过坐标格。

5. Finding the Equation from a Graph | 从图像求方程

If you are given a graph, you can find its equation by determining the gradient m and the y-intercept c.

如果给你一个图像，你可以通过确定斜率 m 和 y 轴截距 c 来求其方程。

First, read the y-intercept directly from the graph. Then choose two clear points on the line and use the gradient formula.

首先，直接从图像读取 y 轴截距。然后在直线上选择两个清晰点，并使用斜率公式。

Suppose the y-intercept is 2 and the line passes through (1, 5) and (2, 8). The gradient is (8 − 5)/(2 − 1) = 3. The equation is y = 3x + 2.

假设 y 轴截距是 2，直线经过 (1, 5) 和 (2, 8)。斜率为 (8 − 5)/(2 − 1) = 3。方程为 y = 3x + 2。

When the y-intercept is not easy to read, substitute both points into y = mx + c and solve the simultaneous equations.

当 y 轴截距不容易读出时，将两个点代入 y = mx + c，然后解联立方程。

6. Parallel and Perpendicular Lines | 平行与垂直直线

Two lines are parallel if they have the same gradient. They never meet, and their equations have the same m value but different c values.

两条直线平行当且仅当它们有相同的斜率。它们永不相交，其方程具有相同的 m 值但不同的 c 值。

For example, y = 4x + 1 and y = 4x − 3 are parallel.

例如，y = 4x + 1 和 y = 4x − 3 是平行的。

Two lines are perpendicular if the product of their gradients is −1. So if line A has gradient m, line B has gradient −1/m.

两条直线垂直当且仅当它们的斜率乘积为 −1。所以如果直线 A 的斜率为 m，则直线 B 的斜率为 −1/m。

m₁ × m₂ = −1

For instance, a line with gradient 2 is perpendicular to a line with gradient −½. Vertical and horizontal lines are also perpendicular, but the gradient rule cannot be applied directly.

例如，斜率为 2 的直线垂直于斜率为 −½ 的直线。垂直线和水平线也互相垂直，但斜率规则不能直接应用。

7. Midpoint and Length of a Segment | 线段的中点与长度

Given two points (x₁, y₁) and (x₂, y₂), the midpoint is the average of the x-coordinates and the average of the y-coordinates.

给定两点 (x₁, y₁) 和 (x₂, y₂)，中点是 x 坐标的平均值和 y 坐标的平均值。

Midpoint = ((x₁ + x₂)/2 , (y₁ + y₂)/2)

The length of the line segment is found using Pythagoras’ theorem on the difference in x and difference in y.

线段的长度使用毕达哥拉斯定理，基于 x 之差和 y 之差求得。

Length = √((x₂ − x₁)² + (y₂ − y₁)²)

For points (1, 2) and (5, 6), the midpoint is (3, 4) and the length is √(4² + 4²) = √32 = 4√2.

对于点 (1, 2) 和 (5, 6)，中点是 (3, 4)，长度是 √(4² + 4²) = √32 = 4√2。

8. Special Cases: Horizontal and Vertical Lines | 特殊情况：水平线与垂直线

A horizontal line has gradient 0 and its equation is y = k, where k is a constant. No matter what x is, y equals k.

水平线的斜率为 0，其方程为 y = k，其中 k 是常数。无论 x 是多少，y 都等于 k。

A vertical line has an undefined gradient and its equation is x = k, where k is a constant. All points on the line have the same x-coordinate.

垂直线的斜率未定义，其方程为 x = k，其中 k 是常数。直线上所有点的 x 坐标都相同。

For example, y = 2 is a horizontal line through (0, 2), and x = −3 is a vertical line through (−3, 0).

例如，y = 2 是穿过 (0, 2) 的水平线，x = −3 是穿过 (−3, 0) 的垂直线。

9. Solving Simultaneous Equations Graphically | 用图像解联立方程

Two straight-line equations can be solved together. Graphically, the solution is the point where the two lines intersect.

两个直线方程可以联立求解。图像上，解就是两条直线相交的点。

For example, draw y = x + 1 and y = −2x + 4. The lines intersect at the point (1, 2), so the solution is x = 1, y = 2.

例如，画出 y = x + 1 和 y = −2x + 4。两条线交于点 (1, 2)，所以解为 x = 1，y = 2。

If the lines are parallel, there is no solution. If the equations represent the same line, there are infinitely many solutions.

如果两条线平行，则无解。如果两个方程表示同一条直线，则有无穷多组解。

Always check your graphical solution by substituting the x and y values back into both equations.

始终将 x 和 y 值代入两个方程来检验你的图像解。

10. Exam Tips and Common Mistakes | 考试技巧与常见错误

Always use a ruler when drawing straight-line graphs. A freehand line may lose marks exactly.

绘制直线图像时务必使用直尺。徒手画线可能会因不精确而失分。

When finding the gradient, always subtract coordinates in the correct order. Do not change the order between the numerator and the denominator.

求斜率时，务必以正确的顺序进行坐标相减。不要在分子和分母之间交换顺序。

Check whether the question asks for an exact value or a decimal approximation. If it asks for an exact value, leave answers in surd or fractional form.

检查题目要求的是精确值还是小数近似值。如果要求精确值，请保留根式或分数形式。

For perpendicular lines, the gradient of the perpendicular line is the negative reciprocal. Do not forget the negative sign.

对于垂直直线，垂直线的斜率是负倒数。不要忘记负号。

Practice converting between y = mx + c and ax + by = d quickly. Many questions require this skill.

练习快速在 y = mx + c 与 ax + by = d 之间转换。许多问题都需要这一技能。

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Straight Line Graphs | 直线图像