IGCSE Maths: Student Book Page 675 – Straight-Line Graphs and Coordinate Geometry | IGCSE 数学：学生用书第675页——直线图像与坐标几何

📚 IGCSE Maths: Student Book Page 675 – Straight-Line Graphs and Coordinate Geometry | IGCSE 数学：学生用书第675页——直线图像与坐标几何

Welcome to this TutorHao revision guide. This article focuses on the key IGCSE Mathematics skills around straight-line graphs and coordinate geometry, which often appear in core and extended papers. We will explain the concepts clearly, give worked examples, and highlight common mistakes so you can build confidence for your exam.

欢迎阅读这篇 TutorHao 复习指南。本文重点讲解 IGCSE 数学中直线图像与坐标几何的核心技能，这些内容在基础卷和扩展卷中经常出现。我们将清晰地解释相关概念，给出详细例题，并指出常见错误，帮助你在考试中建立信心。

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1. The Coordinate Plane | 坐标平面

The coordinate plane is formed by a horizontal x-axis and a vertical y-axis. They intersect at the origin, which has coordinates (0, 0). The plane is divided into four quadrants. In the first quadrant, both x and y are positive; in the second quadrant, x is negative and y is positive; in the third quadrant, both are negative; in the fourth quadrant, x is positive and y is negative.

坐标平面由水平的 x 轴和竖直的 y 轴组成。两轴相交于原点，坐标为 (0, 0)。平面被分为四个象限。在第一象限，x 和 y 都为正；在第二象限，x 为负、y 为正；在第三象限，两者都为负；在第四象限，x 为正、y 为负。

An ordered pair is always written as (x, y). The x-coordinate tells you how far to move left or right from the origin, and the y-coordinate tells you how far to move up or down.

有序数对总是写成 (x, y)。x 坐标表示从原点向左或向右移动的距离，y 坐标表示向上或向下移动的距离。

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2. Plotting Points and Reading Coordinates | 描点与读取坐标

To plot a point such as (3, −2), start at the origin. Move 3 units to the right along the x-axis, then move 2 units down because the y-coordinate is negative. Mark the point clearly with a small cross or dot.

要描出像 (3, −2) 这样的点，从原点出发。沿 x 轴向右移动 3 个单位，因为 y 坐标为负，再向下移动 2 个单位。用小的十字或圆点清晰地标出该点。

To read coordinates from a given point, draw a dashed vertical line down to the x-axis to find the x-coordinate, then draw a dashed horizontal line across to the y-axis to find the y-coordinate. The point (−4, 5) lies in the second quadrant because its x-value is negative and its y-value is positive.

要从已知点读取坐标，先向下画一条虚线到 x 轴以找到 x 坐标，再画一条水平虚线到 y 轴以找到 y 坐标。点 (−4, 5) 位于第二象限，因为它的 x 值为负、y 值为正。

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

Any non-vertical straight line can be written in the form y = mx + c. The letter m represents the gradient, or slope, of the line. The letter c represents the y-intercept, which is the point where the line crosses the y-axis. This form is called the slope-intercept form.

任何非竖直的直线都可以写成 y = mx + c 的形式。字母 m 表示直线的斜率。字母 c 表示 y 轴截距，也就是直线与 y 轴相交的位置。这种形式称为斜截式。

y = mx + c

For example, in the equation y = 2x + 3, the gradient m is 2 and the y-intercept c is 3. This means the line rises by 2 units for every 1 unit it moves to the right, and it crosses the y-axis at (0, 3).

例如，在方程 y = 2x + 3 中，斜率 m 为 2，y 轴截距 c 为 3。这意味着直线每向右移动 1 个单位就上升 2 个单位，并且它与 y 轴相交于 (0, 3)。

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4. Gradient: Meaning and Calculation | 斜率：意义与计算

Gradient measures how steep a line is. It is calculated by dividing the change in y by the change in x. If a line rises from left to right, the gradient is positive; if it falls from left to right, the gradient is negative.

斜率衡量直线的倾斜程度。它等于 y 的变化量除以 x 的变化量。如果直线从左向右上升，斜率为正；如果从左向右下降，斜率为负。

m = (y₂ − y₁) / (x₂ − x₁)

For two points (1, 2) and (4, 8), the gradient is calculated as m = (8 − 2) / (4 − 1) = 6 / 3 = 2. A horizontal line has gradient 0, while a vertical line has undefined gradient.

对于两点 (1, 2) 和 (4, 8)，斜率计算为 m = (8 − 2) / (4 − 1) = 6 / 3 = 2。水平直线的斜率为 0，竖直直线的斜率没有定义。

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5. The y-intercept | y 轴截距

The y-intercept is the value of y when x = 0. In the equation y = mx + c, the y-intercept is simply c. On a graph, it is the point (0, c) where the line crosses the y-axis.

y 轴截距是当 x = 0 时 y 的值。在方程 y = mx + c 中，y 轴截距等于 c。在图像上，它是直线与 y 轴相交的点 (0, c)。

For the equation y = −3x + 5, the y-intercept is 5, so the line passes through (0, 5). Similarly, the x-intercept is found by setting y = 0 and solving for x. In this case, 0 = −3x + 5 gives x = 5/3, so the x-intercept is (5/3, 0).

对于方程 y = −3x + 5，y 轴截距为 5，因此直线经过 (0, 5)。类似地，求 x 轴截距时令 y = 0，解出 x。此时 0 = −3x + 5 得 x = 5/3，所以 x 轴截距为 (5/3, 0)。

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

To draw a straight line, you only need two points. One reliable method is to choose two values of x, such as x = 0 and x = 1, and substitute them into the equation to find the corresponding y-values. Then plot the two points and connect them using a ruler.

要画一条直线，你只需要两个点。一种可靠的方法是选择两个 x 值，例如 x = 0 和 x = 1，代入方程求出对应的 y 值。然后描出这两个点，并用直尺连接。

For y = 2x − 1, when x = 0, y = −1; when x = 2, y = 3. Plot (0, −1) and (2, 3), then draw the line through both points. Always label the line with its equation and use a sharp pencil for accuracy.

对于 y = 2x − 1，当 x = 0 时 y = −1；当 x = 2 时 y = 3。描出 (0, −1) 和 (2, 3)，然后过这两点画直线。始终标出直线对应的方程，并使用

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