Cambridge KS3 Maths: Straight-Line Graphs and Gradient | 剑桥 KS3 数学：直线图像与斜率

📚 Cambridge KS3 Maths: Straight-Line Graphs and Gradient | 剑桥 KS3 数学：直线图像与斜率

A straight-line graph is one of the most important ideas in Cambridge KS3 mathematics. It connects algebra with coordinate geometry and appears in real-world graphs such as speed-time and distance-time graphs. This article explains how to find the gradient, how to use the equation y = mx + c, and how to draw a straight line from its equation. It is ideal for learners working through Cambridge Lower Secondary Mathematics, including exercise questions like those on page 304, question 1, where you may be asked to read or calculate a gradient from a graph.

直线图像是剑桥 KS3 数学中最重要的概念之一。它将代数与坐标几何联系起来，并出现在现实世界的图像中，如速度-时间图和距离-时间图。本文将解释如何求斜率、如何使用方程 y = mx + c，以及如何根据方程画出直线。本文适合正在学习剑桥初中数学的学习者，包括类似第 304 页第 1 题的练习，题目可能要求你从图像中读取或计算斜率。

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1. What is a straight-line graph? | 什么是直线图像？

A straight-line graph is a set of points that lie on a straight line when plotted on a coordinate grid. Each point has an x-coordinate and a y-coordinate, and all the points satisfy a linear equation such as y = 2x + 1.

直线图像是坐标网格上绘制时位于同一直线上的点集。每个点有一个 x 坐标和一个 y 坐标，并且所有点都满足一个线性方程，例如 y = 2x + 1。

For any straight-line graph, the relationship between x and y can be written in the form y = mx + c, where m and c are constants.

对于任何直线图像，x 和 y 之间的关系都可以写成 y = mx + c 的形式，其中 m 和 c 是常数。

On the coordinate plane, a straight line extends forever in both directions unless we restrict it. We usually draw it by finding at least two points and joining them with a ruler.

在坐标平面上，一条直线会向两端无限延伸，除非我们加以限制。我们通常找到至少两个点，然后用直尺把它们连接起来。

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

The general equation of a straight line is y = mx + c. In this equation, x and y are variables, while m and c are constants.

直线的一般方程是 y = mx + c。在这个方程中，x 和 y 是变量，而 m 和 c 是常数。

y = mx + c

Here, m represents the gradient, also called the slope of the line. The constant c represents the y-intercept, which is the point where the line crosses the y-axis.

这里，m 表示斜率，也叫做直线的坡度。常数 c 表示 y 轴截距，也就是直线与 y 轴相交的点。

For example, in the equation y = 3x + 2, the gradient is 3 and the y-intercept is 2. In y = −2x + 5, the gradient is −2 and the y-intercept is 5.

例如，在方程 y = 3x + 2 中，斜率是 3，y 轴截距是 2。在 y = −2x + 5 中，斜率是 −2，y 轴截距是 5。

This form is powerful because it allows you to sketch a line directly without needing a table of values every time.

这种形式非常有用，因为它可以让你直接画出直线，而不必每次都列出数值表。

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3. Understanding gradient | 理解斜率

The gradient of a line measures how steep it is. A larger gradient means a steeper line. A smaller gradient means the line is less steep.

斜率衡量一条线的陡峭程度。斜率越大，线越陡；斜率越小，线越平缓。

To calculate the gradient from a graph, choose two points on the line, find how many units the line rises vertically, then divide by how many units it runs horizontally.

要从图像中计算斜率，请选择线上的两点，确定线垂直上升了多少个单位，然后除以水平移动了多少个单位。

gradient = rise ÷ run = change in y ÷ change in x

The rise is the vertical change between the two points, and the run is the horizontal change. Always divide vertical by horizontal, not the other way round.

上升量是两点之间的垂直变化量，水平量是两点之间的水平变化量。一定要用垂直量除以水平量，不要颠倒。

If the line goes up as you move from left to right, the gradient is positive. If it goes down, the gradient is negative.

如果直线从左到右向上倾斜，斜率为正。如果直线从左到右向下倾斜，斜率为负。

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4. Finding the gradient from two points | 从两点求斜率

When you know two points on a line, you can calculate the gradient using the formula below.

当你已知直线上的两个点时，可以使用下面的公式计算斜率。

m = (y₂ − y₁) ÷ (x₂ − x₁)

In this formula, (x₁, y₁) is the first point and (x₂, y₂) is the second point. It does not matter which point you call first, as long as you subtract the coordinates in the same order.

在这个公式中，(x₁, y₁) 是第一个点，(x₂, y₂) 是第二个点。哪个点称为第一个点并不重要，只要两个坐标的相减顺序一致即可。

Worked example: Find the gradient of the line passing through (1, 2) and (4, 8).

示例：求经过点 (1, 2) 和 (4, 8) 的直线的斜率。

Step 1: Label the points as x₁ = 1, y₁ = 2, x₂ = 4, y₂ = 8.

步骤 1：标记点为 x₁ = 1，y₁ = 2，x₂ = 4，y₂ = 8。

Step 2: Substitute into the formula: m = (8 − 2) ÷ (4 − 1) = 6 ÷ 3 = 2.

步骤 2：代入公式：m = (8 − 2) ÷ (4 − 1) = 6 ÷ 3 = 2。

So the gradient is 2. This means that for every 1 unit the line moves to the right, it moves 2 units up.

因此斜率为 2。这意味着直线每向右移动 1 个单位，就向上移动 2 个单位。

Another example: For the points (0, 5) and (3, 11), the gradient is (11 − 5) ÷ (3 − 0) = 6 ÷ 3 = 2.

另一个示例：对于点 (0, 5) 和 (3, 11)，斜率为 (11 − 5) ÷ (3 − 0) = 6 ÷ 3 = 2。

If the points are (2, 9) and (5, 3), then m = (3 − 9) ÷ (5 − 2) = −6 ÷ 3 = −2, giving a negative gradient.

如果点是 (2, 9) 和 (5, 3)，那么 m = (3 − 9) ÷ (5 − 2) = −6 ÷ 3 = −2，得到负斜率。

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5. Positive and negative gradients | 正斜率与负斜率

A line with a positive gradient rises from left to right. This means that as x increases, y also increases.

斜率为正的直线从左到右上升。这意味着随着 x 增大，y 也增大。

A line with a negative gradient falls from left to right. This means that as x increases, y decreases.

斜率为负的直线从左到右下降。这意味着随着 x 增大，y 减小。

A horizontal line has a gradient of zero because there is no vertical change. Its equation has the form y = c, such as y = 4.

水平线的斜率为零，因为没有垂直变化。它的方程形式为 y = c，例如 y = 4。

A vertical line does not have a gradient that is a real number, because the run is zero. It has

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