Grade 8H Maths Unit 4: Working in the Cartesian Plane – Key Concepts & Worked Answers | 8年级高阶数学第4单元：笛卡尔平面核心概念与答案解析

📚 Grade 8H Maths Unit 4: Working in the Cartesian Plane – Key Concepts & Worked Answers | 8年级高阶数学第4单元：笛卡尔平面核心概念与答案解析

This revision guide covers the essential skills in Grade 8H Maths Unit 4: Working in the Cartesian Plane. You will consolidate coordinate plotting, master the equation y = mx + c, calculate gradient and intercepts, and apply these ideas to problem-solving. Each section pairs a concise explanation with a worked example, so you can check your answers and strengthen exam technique.

本复习指南涵盖 8 年级高阶数学第 4 单元“笛卡尔平面”的核心技能。你将巩固坐标描点、掌握一次方程 y = mx + c、计算斜率与截距，并将这些知识应用于解题。每个小节都以简明讲解搭配例题，帮助你核对答案并提升考试技巧。

1. The Cartesian Plane and Coordinates | 笛卡尔平面与坐标

The Cartesian plane is formed by two perpendicular number lines: the horizontal x-axis and the vertical y-axis. The point where they meet is called the origin, written as (0, 0).

笛卡尔平面由两条互相垂直的数轴组成：水平的 x 轴和竖直的 y 轴。它们的交点称为原点，记作 (0, 0)。

A point is written as an ordered pair (x, y). The first number is the x-coordinate, and the second number is the y-coordinate. For example, the point (3, −2) means move 3 units right and 2 units down from the origin.

一个点用有序数对 (x, y) 表示。第一个数字是 x 坐标，第二个数字是 y 坐标。例如，点 (3, −2) 表示从原点向右移动 3 个单位、向下移动 2 个单位。

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2. Plotting Points and Naming Quadrants | 描点与象限命名

To plot a point, start at the origin. Move horizontally according to the x-coordinate, then vertically according to the y-coordinate. Positive x moves right; negative x moves left. Positive y moves up; negative y moves down.

描点时从原点出发。先按 x 坐标水平移动，再按 y 坐标竖直移动。x 为正向右，x 为负向左；y 为正向上，y 为负向下。

The plane is divided into four quadrants. Quadrant I has both coordinates positive, Quadrant II has x negative and y positive, Quadrant III has both negative, and Quadrant IV has x positive and y negative.

平面分为四个象限。第一象限两个坐标都为正，第二象限 x 为负、y 为正，第三象限两个坐标都为负，第四象限 x 为正、y 为负。

Quadrant	Sign of x	Sign of y	Example
I	+	+	(2, 5)
II	−	+	(−2, 5)
III	−	−	(−2, −5)
IV	+	−	(2, −5)

Remember: the signs tell you which direction to move from the origin, not the size of the coordinate.

记住：符号只表示从原点出发的移动方向，并不表示坐标的大小。

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3. Understanding Linear Equations y = mx + c | 理解一次方程 y = mx + c

y = mx + c

In the equation of a straight line, m represents the gradient or slope, and c represents the y-intercept, which is the value of y when x = 0.

在直线方程中，m 表示斜率（坡度），c 表示 y 截距，即 x = 0 时 y 的值。

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

例如，在 y = 2x + 1 中，斜率为 2，y 截距为 1。这意味着直线与 y 轴交于 (0, 1)，并且每向右移动 1 个单位就上升 2 个单位。

To find the coordinates of points on the line, substitute different x-values into the equation. If x = 3, then y = 2(3) + 1 = 7, so (3, 7) lies on the line.

要找到直线上的点的坐标，可以将不同的 x 值代入方程。若 x = 3，则 y = 2(3) + 1 = 7，所以 (3, 7) 在直线上。

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4. Gradient: Rise over Run | 斜率：竖直距离除以水平距离

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

斜率衡量直线的倾斜程度。它用直线上两点之间 y 的变化量除以 x 的变化量来计算。

m = Δy / Δx = (y₂ − y₁) / (x₂ − x₁)

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

例题：求经过点 (1, 2) 和 (4, 8) 的直线的斜率。

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

代入公式：m = (8 − 2) / (4 − 1) = 6 / 3 = 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.

斜率为正表示直线从左到右向上倾斜，斜率为负表示直线从左到右向下倾斜。水平直线的斜率为 0，竖直直线的斜率未定义。

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5. Intercepts: Where Lines Cross the Axes | 截距：直线与坐标轴的交点

The y-intercept is the point where the line crosses the y-axis. In the form y = mx + c, this is simply c, because x = 0 at that point.

y 截距是直线与 y 轴相交的点。在 y = mx + c 的形式中，y 截距就是 c，因为在该点 x = 0。

To find the x-intercept, set y = 0 and solve for x. The x-intercept is where the line crosses the x-axis.

要求 x 截距，令 y = 0 并解出 x。x 截距是直线与 x 轴相交的点。

Example: For the equation 2x + 3y = 12, find both intercepts.

例题：对于方程 2x + 3y = 12，求两个截距。

Set x = 0: 3y = 12, so y = 4. The y-intercept is (0, 4). Set y = 0: 2x = 12, so x = 6. The x-intercept is (6, 0).

令 x = 0：3y = 12，所以 y = 4。y 截距为 (0, 4)。令 y = 0：2x = 12，所以 x = 6。x 截距为 (6, 0)。

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6. Drawing Straight Lines from a Table of Values | 根据数值表绘制直线

A reliable way to draw a straight line is to create a table of x-values, substitute them into the equation to find y, and then plot the resulting coordinate pairs.

绘制直线的一种可靠方法是先列出 x 值的表格，代入方程求出 y，然后描出对应的坐标点。

Example: Draw y = x + 3 for x = −2, −1, 0, 1, 2.

例题：绘制 y = x + 3，其中 x = −2, −1, 0, 1, 2。

x	−2	−1	0	1	2
y = x + 3	1	2	3	4	5

Plot the points (−2, 1), (−1, 2), (0, 3), (1, 4), (2, 5) and join them with a straight edge. All points should lie on a single straight line.

描出点 (−2, 1)、(−1, 2)、(0, 3)、(1, 4)、(2, 5)，并用直尺连接。所有点都应落在同一条直线上。

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7. Parallel Lines and Their Gradients | 平行线及其斜率

Parallel lines never meet and always have the same gradient. If one line has gradient m, any line parallel to it must also have gradient m.

平行线永不相交，并且总是具有相同的斜率。如果一条直线的斜率为 m，那么任何与它平行的直线的斜率也一定为 m。

Example: y = 3x + 2 and y = 3x − 5 are parallel because both have gradient 3. Their y-intercepts are different, so they are distinct lines.

例题：y = 3x + 2 和 y = 3x − 5 是平行的，因为两者的斜率都为 3。它们的 y 截距不同，因此是两条不同的直线。

When asked to write the equation of a line parallel to a given line, keep the same m and choose a suitable c from the information provided.

当题目要求写出与已知直线平行的直线方程时，保持相同的 m，并根据题给信息选择合适的 c。

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8. 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. Its coordinates are found by averaging the x-coordinates and averaging the y-coordinates.

线段的中点是两端点之间恰好位于中间的点。它的坐标通过分别求两个 x 坐标的平均值和两个 y 坐标的平均值得到。

M = ( (x₁ + x₂) / 2 , (y₁ + y₂) / 2 )

Example: Find the midpoint of the segment joining (2, 5) and (−4, 9).

例题：求连接 (2, 5) 和 (−4, 9) 的线段的中点。

x-coordinate: (2 + (−4)) / 2 = −2 / 2 = −1. y-coordinate: (5 + 9) / 2 = 14 / 2 = 7. The midpoint is (−1, 7).

x 坐标：(2 + (−4)) / 2 = −2 / 2 = −1。y 坐标：(5 + 9) / 2 = 14 / 2 = 7。中点为 (−1, 7)。

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9. Coordinate Geometry in Context | 坐标几何的实际应用

Coordinate geometry skills are often combined in problem-solving. You may need to find a missing coordinate, write the equation of a line from a description, or identify the type of triangle formed by three points.

坐标几何技能经常在解决问题时综合运用。你可能需要求缺失的坐标、根据描述写出直线方程，或判断三个点构成的三角形类型。

Horizontal lines have the form y = a, where a is a constant. Vertical lines have the form x = b, where b is a constant.

水平直线的形式为 y = a，其中 a 为常数。竖直直线的形式为 x = b，其中 b 为常数。

Example: A line passes through (0, 5) and has gradient −2. Write its equation.

例题：一条直线经过 (0, 5) 且斜率为 −2。写出它的方程。

Since it passes through (0, 5), the y-intercept c = 5. The gradient m = −2. The equation is y = −2x + 5.

因为它经过 (0, 5)，所以 y 截距 c = 5。斜率 m = −2。因此方程为 y = −2x + 5。

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10. Common Mistakes and How to Avoid Them | 常见错误与避免方法

A common error is swapping the x- and y-coordinates when plotting a point. Always write the x-coordinate first and move horizontally before moving vertically.

一个常见错误是在描点时把 x

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