Coordinates and Linear Graphs | 坐标与线性图像

📚 Coordinates and Linear Graphs | 坐标与线性图像

Understanding how to plot points and draw straight-line graphs is a fundamental skill in Key Stage 3 Mathematics. This topic introduces the Cartesian coordinate system and the relationship between algebraic equations and their visual representations. Mastery of these concepts lays the groundwork for more advanced algebra and geometry in later years.

理解如何描点并绘制直线图像是KS3数学中的基础技能。本主题将介绍笛卡尔坐标系，并讲解代数方程与其可视化表示之间的关系。掌握这些概念将为后续更深入的代数和几何学习奠定基础。

1. The Cartesian Plane | 笛卡尔平面

The Cartesian plane consists of two perpendicular number lines called the x-axis (horizontal) and the y-axis (vertical). The point where they intersect is the origin, labelled (0, 0). The plane is divided into four quadrants, numbered I to IV in an anti-clockwise direction, allowing both positive and negative coordinates.

笛卡尔平面由两条互相垂直的数轴组成，分别称为x轴（水平）和y轴（垂直）。它们的交点称为原点，标记为(0, 0)。该平面被分为四个象限，按逆时针方向从I到IV编号，可容纳正负坐标值。

2. Plotting Points | 描点

A coordinate pair is written as (x, y), where x gives the horizontal position and y gives the vertical position. To plot the point (3, -2), start at the origin, move 3 units to the right along the x-axis, then move 2 units down parallel to the y-axis because y is negative.

一个坐标对写作(x, y)，其中x表示水平位置，y表示垂直位置。要描画出点(3, -2)，从原点出发，沿x轴向右移动3个单位，然后由于y为负，平行于y轴向下移动2个单位。

Always read the x-coordinate first. The phrase ‘along the corridor, up the stairs’ helps beginners remember the correct order: move horizontally first, then vertically.

永远先读x坐标。短语“先走走廊，再上楼梯”可以帮助初学者记住正确的顺序：先水平移动，再垂直移动。

Point	x-coordinate	y-coordinate
A(4, 3)	4	3
B(-2, 5)	-2	5

上表列出了几个坐标点示例，明确展示了x和y分量的正负值。

3. Understanding Straight-Line Equations | 理解直线方程

A linear equation in two variables can be written in the form y = mx + c. The letter m represents the gradient (steepness) of the line, and c represents the y-intercept — the point where the line crosses the y-axis. For example, in y = 2x + 1, the gradient is 2 and the y-intercept is 1.

一个二元一次方程可以写成y = mx + c的形式。字母m表示直线的斜率（陡峭程度），c表示y轴截距——即直线与y轴相交的点。例如，在y = 2x + 1中，斜率为2，y轴截距为1。

4. Generating a Table of Values | 生成数值表

To draw a straight-line graph, first create a table of values by choosing several x-values and substituting them into the equation to find the corresponding y-values. Typically, you choose x = -2, -1, 0, 1, 2, or a similar range, to ensure a good spread of points.

要绘制直线图像，首先通过选择若干x值并将其代入方程来求出对应的y值，从而生成一个数值表。通常，你选择x = -2、-1、0、1、2或类似范围，以确保点的分布足够均匀。

For the equation y = x + 2:

If x = -2, y = -2 + 2 = 0 → point (-2, 0)

If x = -1, y = -1 + 2 = 1 → point (-1, 1)

If x = 0, y = 0 + 2 = 2 → point (0, 2)

If x = 1, y = 1 + 2 = 3 → point (1, 3)

If x = 2, y = 2 + 2 = 4 → point (2, 4)

对于方程y = x + 2：

若x = -2，y = -2 + 2 = 0 → 点(-2, 0)

若x = -1，y = -1 + 2 = 1 → 点(-1, 1)

若x = 0，y = 0 + 2 = 2 → 点(0, 2)

若x = 1，y = 1 + 2 = 3 → 点(1, 3)

若x = 2，y = 2 + 2 = 4 → 点(2, 4)

5. Drawing the Graph | 绘制图像

Plot each coordinate pair from your table onto graph paper or a coordinate grid. Use a sharp pencil and mark each point with a small cross. Once all points are plotted, use a ruler to draw a single straight line that passes through all of them. Label the line with its equation, such as y = x + 2.

将表中的每个坐标对描绘在坐标纸或坐标网格上。使用削尖的铅笔，每个点用小十字标记。所有点都描好后，用直尺画一条经过所有点的直线。用方程标记这条线，例如y = x + 2。

If the points do not line up perfectly, double-check your calculations. Linear equations always produce points that lie on a straight line. Any deviation indicates an arithmetic error or a plotting mistake.

如果点没有完美地排成直线，请重新检查计算。线性方程产生的点总是在一条直线上。任何偏差都表明存在算术错误或描点失误。

6. Gradient: Measure of Steepness | 斜率：陡峭程度的度量

Gradient (m) measures how steep a line is. It is calculated as the change in y (vertical rise) divided by the change in x (horizontal run) between any two points on the line:

m = (y₂ − y₁) ÷ (x₂ − x₁)

斜率(m)衡量直线的陡峭程度。它通过线上任意两点之间y的变化量（垂直上升高度）除以x的变化量（水平前进距离）来计算：

m = (y₂ − y₁) ÷ (x₂ − x₁)

A positive gradient means the line slopes upwards from left to right. A negative gradient means it slopes downwards from left to right. A gradient of zero gives a horizontal line that is flat.

正斜率意味着直线从左向右上升。负斜率意味着直线从左向右下降。零斜率是一条平坦的水平线。

7. The y-intercept (c) | y轴截距 (c)

The y-intercept is the y-coordinate of the point where the line crosses the y-axis. This occurs when x = 0. In the equation y = mx + c, the constant term c is the y-intercept. For y = -3x + 5, the line crosses the y-axis at (0, 5).

y轴截距是直线与y轴相交点的y坐标。这发生在x = 0时。在方程y = mx + c中，常数项c就是y轴截距。对于y = -3x + 5，直线在(0, 5)处与y轴相交。

The x-intercept is where the line crosses the x-axis (y = 0). This can be found by setting y = 0 in the equation and solving for x. For y = -3x + 5, setting 0 = -3x + 5 gives x = 5/3.

x轴截距是直线与x轴相交的位置（y = 0）。这可以通过在方程中设y = 0来求解x得到。对于y = -3x + 5，设0 = -3x + 5可得x = 5/3。

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

Horizontal lines have equations of the form y = a, where a is a constant. Every point on the line y = 4 has a y-coordinate of 4, regardless of x. Its gradient is 0 and it runs parallel to the x-axis.

水平线的方程形式为y = a，其中a为常数。直线y = 4上的每一个点的y坐标都是4，与x无关。其斜率为0，与x轴平行。

Vertical lines have equations of the form x = b, where b is a constant. Every point on the line x = -2 has an x-coordinate of -2. Vertical lines have an undefined gradient and run parallel to the y-axis.

垂直线的方程形式为x = b，其中b为常数。直线x = -2上的每一个点的x坐标都是-2。垂直线具有未定义的斜率，并与y轴平行。

9. Sketching Graphs Using Intercepts | 利用截距绘制草图

A quick method to sketch a linear graph is to find its x-intercept and y-intercept. To find the y-intercept, substitute x = 0. To find the x-intercept, substitute y = 0. Plot these two intercepts, then draw a straight line through them. This eliminates the need for a full table of values.

绘制线性图像草图的快速方法是求出其x轴截距和y轴截距。要求y轴截距，代入x = 0；要求x轴截距，代入y = 0。描出这两个截距点，然后画一条穿过它们的直线。这免去了列出完整数值表的需要。

For the equation 2x + 3y = 6:

x-intercept: set y = 0 → 2x = 6 → x = 3 → point (3, 0)

y-intercept: set x = 0 → 3y = 6 → y = 2 → point (0, 2)

Plot (3, 0) and (0, 2) and draw the straight line.

对于方程2x + 3y = 6：

x截距：设y = 0 → 2x = 6 → x = 3 → 点(3, 0)

y截距：设x = 0 → 3y = 6 → y = 2 → 点(0, 2)

描出(3, 0)和(0, 2)并画出直线。

10. Graphs from Real-Life Situations | 现实情境中的图像

Linear graphs are not limited to abstract algebra; they model many real-world contexts such as conversion between currencies, mobile phone tariffs, and distance-time relationships. When interpreting a graph, pay close attention to the labels on each axis and the units being used.

线性图像不仅限于抽象代数，它们可以为许多现实情境建模，如货币兑换、手机资费和距离-时间关系。在解读图像时，要密切注意每个坐标轴上的标签以及所用的单位。

A distance-time graph shows how distance changes over time. A straight, sloping line indicates constant speed. The gradient of the line represents the speed. A horizontal segment means the object is stationary (not moving) because distance is not changing as time increases.

距离-时间图像展示了距离随时间如何变化。一条倾斜的直线表示匀速运动。直线的斜率代表速度。水平线段表示物体静止（没有移动），因为距离不随时间的增加而改变。

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

To find the equation of a straight line given its graph, first identify the y-intercept c by observing where the line cuts the y-axis. Then, calculate the gradient m by selecting two clear points on the line and using the formula for gradient. Finally, write the equation in the form y = mx + c.

要从给定的直线图像求方程，首先观察直线与y轴的交点来确定y轴截距c。然后，在线上选取两个清晰可辨的点，利用斜率公式计算斜率m。最后，将方程写成y = mx + c的形式。

For example, if a line crosses the y-axis at (0, -3) and passes through (2, 1), the gradient is m = (1 – (-3)) ÷ (2 – 0) = 4 ÷ 2 = 2. Therefore, the equation is y = 2x – 3.

例如，如果一条直线在(0, -3)处与y轴相交，并经过(2, 1)，则斜率为m = (1 – (-3)) ÷ (2 – 0) = 4 ÷ 2 = 2。因此，方程为y = 2x – 3。

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

Two lines are parallel if they never intersect and remain the same distance apart. In the coordinate plane, parallel lines have exactly the same gradient. For instance, the lines y = 4x + 2 and y = 4x – 5 both have a gradient of 4, so they are perfectly parallel.

如果两条直线永不相交且始终保持相同距离，它们就是平行的。在坐标平面中，平行线具有完全相同的斜率。例如，直线y = 4x + 2和y = 4x – 5的斜率都为4，所以它们完全平行。

If one line has equation y = 3x and another line is parallel to it, the second line must also have gradient 3. Its specific equation depends on where it crosses the y-axis. This rule helps in constructing or identifying parallel lines without drawing them.

如果一条直线的方程是y = 3x，而另一条直线与之平行，则第二条直线的斜率也必定是3。它的具体方程取决于它与y轴的交点位置。这一规则有助于在不绘制图像的情况下构造或识别平行线。

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Coordinates and Linear Graphs | 坐标与线性图像