Coordinates and Straight-line Graphs | 坐标与直线图

📚 Coordinates and Straight-line Graphs | 坐标与直线图

In KS3 mathematics, working with coordinates and straight-line graphs is a fundamental skill. It involves plotting points on a Cartesian grid, understanding the equation of a line in the form y = mx + c, and interpreting the meaning of gradient and intercept. These concepts form the basis for linear relationships in algebra and real-world data analysis.

在 KS3 数学中，学习坐标和直线图是一项基本技能。它包括在笛卡尔坐标格上描点、理解形如 y = mx + c 的直线方程，以及解释斜率和截距的意义。这些概念构成了代数中线性关系以及现实世界数据分析的基础。

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1. The Cartesian Coordinate System | 笛卡尔坐标系

The Cartesian plane consists of two perpendicular number lines, the x-axis (horizontal) and the y-axis (vertical), which intersect at the origin (0,0). Every point is defined by an ordered pair (x, y), where x is the distance along the horizontal axis and y is the distance along the vertical axis. The plane is divided into four quadrants: QI (+,+), QII (-,+), QIII (-,-), and QIV (+,-).

笛卡尔平面由两条垂直的数轴构成：x 轴（水平）和 y 轴（垂直），它们相交于原点 (0,0)。每个点由一个有序对 (x, y) 定义，其中 x 是沿水平方向的距离，y 是沿垂直方向的距离。该平面分为四个象限：第一象限 (+,+)、第二象限 (-,+)、第三象限 (-,-) 和第四象限 (+,-)。

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2. Plotting Points Accurately | 准确描点

To plot a point such as (3, -2), start at the origin, move 3 units to the right along the x-axis, then 2 units down parallel to the y-axis. Using graph paper or a digital grid helps maintain precision. Always label the axes and use a consistent scale. Plotting several points that satisfy a rule is the first step toward drawing a graph of a function.

要描出 (3, -2) 这样的点，从原点开始，沿 x 轴向右移动 3 个单位，再沿 y 轴平行方向向下移动 2 个单位。使用坐标纸或数字网格有助于保持精度。始终给坐标轴添加标签并使用一致的刻度。描出满足某一规则的多个点是绘制函数图像的第一步。

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3. Coordinates in All Four Quadrants | 四个象限中的坐标

Negative coordinates appear in QII, QIII, and QIV. For example, (-4, 5) lies in QII, (-3, -1) in QIII, and (6, -2) in QIV. Pupils should be comfortable reading and plotting points in any quadrant. This skill is essential when dealing with graphs that extend beyond the first quadrant, including horizontal and vertical lines with negative intercepts.

负坐标出现在第二、第三和第四象限。例如，(-4, 5) 位于第二象限，(-3, -1) 位于第三象限，(6, -2) 位于第四象限。学生应当能够熟练地读取和描出任意象限中的点。当处理超出第一象限的图形时（包括具有负截距的水平和垂直直线），这项技能至关重要。

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4. Horizontal and Vertical Lines | 水平线与垂直线

A horizontal line has a constant y-value; its equation is y = k, where k is a number. For instance, y = 3 is a horizontal line passing through all points with y-coordinate 3. A vertical line has a constant x-value; its equation is x = h, such as x = -2. These lines are parallel to the axes and have gradients of 0 and undefined, respectively.

水平线具有恒定的 y 值；其方程为 y = k，其中 k 是一个数。例如，y = 3 是一条通过所有 y 坐标为 3 的点的水平线。垂直线具有恒定的 x 值；其方程为 x = h，例如 x = -2。这些线平行于坐标轴，斜率分别为 0 和未定义。

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5. Introducing the Equation y = mx + c | 引入方程 y = mx + c

The standard form of a linear equation is y = mx + c, where m represents the gradient (steepness) and c represents the y-intercept (where the line crosses the y-axis). The gradient is the ratio of vertical change to horizontal change. A positive gradient slopes upward, a negative gradient slopes downward. The y-intercept is the value of y when x = 0.

线性方程的标准形式是 y = mx + c，其中 m 表示斜率（坡度），c 表示 y 轴截距（直线与 y 轴相交的点）。斜率是垂直变化与水平变化之比。正斜率向上倾斜，负斜率向下倾斜。y 轴截距是当 x = 0 时的 y 值。

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6. Finding the Gradient from Two Points | 由两点求斜率

The gradient m is calculated using the formula (change in y) / (change in x), which is commonly written as m = (y₂ – y₁) / (x₂ – x₁). If A = (1, 2) and B = (4, 8), then m = (8 – 2) / (4 – 1) = 6/3 = 2. This means for every 1 unit moved to the right, the line rises by 2 units. A negative gradient indicates a fall in y as x increases.

斜率 m 使用公式 (y 的变化量) / (x 的变化量) 来计算，通常写作 m = (y₂ – y₁) / (x₂ – x₁)。如果 A = (1, 2) 且 B = (4, 8)，则 m = (8 – 2) / (4 – 1) = 6/3 = 2。这意味着每向右移动 1 个单位，直线就上升 2 个单位。负斜率表示随着 x 的增加，y 下降。

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7. Interpreting Gradient and Real-life Meaning | 解释斜率及现实意义

In real-world contexts, gradient represents rate of change, such as speed (distance/time) or cost per item. A line with equation y = 5x + 10 might represent a total cost y given the number of items x, where the gradient (5) is the unit cost and the y-intercept (10) is a fixed charge. Understanding the meaning of gradient and intercept helps in modelling practical situations.

在现实情境中，斜率代表变化率，例如速度（距离/时间）或每件物品的成本。方程为 y = 5x + 10 的直线可以表示给定物品数量 x 时的总成本 y，其中斜率 (5) 是单位成本，y 轴截距 (10) 是固定费用。理解斜率和截距的含义有助于对实际问题进行建模。

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8. Determining the Y-intercept from a Graph or Equation | 通过图形或方程确定 y 轴截距

The y-intercept c is simply the y-value when x = 0. On a graph, it is the point where the line crosses the y-axis. In an equation like y = 3x – 4, c = -4, so the line crosses at (0, -4). If only given a graph, you can read this value directly. Knowing the intercept helps write the equation of a line when combined with the gradient.

y 轴截距 c 就是当 x = 0 时的 y 值。在图形上，它是直线与 y 轴相交的点。在方程 y = 3x – 4 中，c = -4，因此直线在 (0, -4) 处相交。如果只给出了图形，你可以直接读取这个值。知道截距并结合斜率有助于写出直线的方程。

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9. Drawing a Straight-line Graph from its Equation | 根据方程绘制直线图

To draw the graph of y = 2x + 1, first note the y-intercept (0, 1). Then use the gradient m = 2 to find a second point: from (0, 1), move 1 unit right and 2 units up to reach (1, 3). Plot both points and draw a straight line through them. Extend the line across the grid and label it with its equation. A table of values can also be used as an alternative method.

要绘制 y = 2x + 1 的图形，首先注意 y 轴截距为 (0, 1)。然后使用斜率 m = 2 找到第二个点：从 (0, 1) 出发，向右移动 1 个单位，再向上移动 2 个单位，到达 (1, 3)。描出这两点，并通过它们画出一条直线。将直线延伸穿过整个网格，并标注其方程。也可以通过制表求值的方法来绘图。

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10. Checking if a Point Lies on a Line | 验证一个点是否在直线上

Substitute the x-coordinate of the point into the equation and see if the result matches the y-coordinate. For example, does (3, 7) lie on y = 2x + 1? Calculate 2(3) + 1 = 7, which matches. So, (3, 7) is on the line. This method is used to verify plotted points and to solve problems involving coordinate geometry.

将点的 x 坐标代入方程，看结果是否与 y 坐标一致。例如，(3, 7) 是否在直线 y = 2x + 1 上？计算 2(3) + 1 = 7，结果一致，因此 (3, 7) 在直线上。该方法用于验证所描的点以及解决涉及坐标几何的问题。

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11. Finding the Equation from a Graph | 根据图形求方程

To find the equation of a straight-line graph, first identify the y-intercept c. Then choose two clear points on the line and calculate the gradient m = (y₂ – y₁)/(x₂ – x₁). If the line passes through (0, 2) and (3, 8), c = 2, m = (8 – 2)/(3 – 0) = 6/3 = 2, so the equation is y = 2x + 2. This reverse process consolidates understanding of the linear form.

要找到一条直线图的方程，首先确定 y 轴截距 c。然后在直线上选取两个清晰的点，计算斜率 m = (y₂ – y₁)/(x₂ – x₁)。如果直线通过 (0, 2) 和 (3, 8)，则 c = 2，m = (8 – 2)/(3 – 0) = 6/3 = 2，因此方程为 y = 2x + 2。这个逆向过程巩固了对线性形式的理解。

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

Two distinct lines are parallel if and only if they have the same gradient. Lines y = 3x + 4 and y = 3x – 1 are parallel because m = 3 in both cases. Their y-intercepts are different, so they never meet. Understanding this property helps in constructing and identifying parallel lines without needing to draw them.

当且仅当两条不同的直线具有相同的斜率时，它们才是平行的。直线 y = 3x + 4 和 y = 3x – 1 是平行的，因为两者的斜率 m 都是 3。它们的 y 轴截距不同，所以永远不会相交。理解这一性质有助于在不绘图的情况下构建和识别平行线。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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