Coordinates and Straight-Line Graphs | 坐标与直线图像

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

In the Cambridge KS3 exercise linked to p171_2.pdf, you are asked to complete a table of values, plot points on a coordinate grid, draw a straight line, and then describe its gradient and y-intercept. This article revises each of these skills step by step.

在剑桥 KS3 练习 p171_2.pdf 中，你需要完成数值表、在坐标网格上描点、画出直线，并描述它的斜率和 y 轴截距。本文逐步复习这些技能。

1. Cartesian Coordinates | 笛卡尔坐标

A coordinate grid consists of a horizontal x-axis and a vertical y-axis. Coordinates are always written in the form (x, y), where the x-value tells you how far to move left or right from the origin, and the y-value tells you how far to move up or down.

坐标网格由水平的 x 轴和竖直的 y 轴组成。坐标总是写成 (x, y) 的形式，其中 x 值表示从原点向左或向右移动多少，y 值表示向上或向下移动多少。

For example, the point (2, 3) is 2 units right of the origin and 3 units up. The point (-1, 4) is 1 unit left and 4 units up. The origin itself is (0, 0).

例如，点 (2, 3) 在原点右侧 2 个单位、上方 3 个单位。点 (-1, 4) 在原点左侧 1 个单位、上方 4 个单位。原点本身是 (0, 0)。

The grid is divided into four quadrants. In the first quadrant both coordinates are positive. In the second quadrant x is negative and y is positive. In the third quadrant both are negative, and in the fourth quadrant x is positive and y is negative.

网格分为四个象限。第一象限中两个坐标都为正。第二象限中 x 为负、y 为正。第三象限中两个都为负，第四象限中 x 为正、y 为负。

2. Plotting Points from a Table | 根据表格描点

A table of values helps you find coordinates that lie on a straight line. For example, for the rule y = 2x + 1, you substitute each x-value into the rule to calculate y.

数值表帮助你找到位于直线上的坐标。例如，对于规则 y = 2x + 1，你将每个 x 值代入规则来计算 y。

Each pair such as (-2, -3), (0, 1) and (2, 5) can be plotted as a point. You need at least three points to draw a reliable straight line, because an error in one point is easier to spot when more points are present.

每一对如 (-2, -3)、(0, 1) 和 (2, 5) 都可以描成一个点。你至少需要三个点才能可靠地画出直线，因为当点更多时，更容易发现某个点的错误。

When you substitute negative values, be careful with directed numbers. For y = -2x + 3, if x = -1, then y = -2(-1) + 3 = 2 + 3 = 5.

当代入负值时，要小心有向数。对于 y = -2x + 3，如果 x = -1，那么 y = -2(-1) + 3 = 2 + 3 = 5。

3. Understanding y = mx + c | 理解直线方程 y = mx + c

All straight-line graphs can be written in the gradient-intercept form y = mx + c. The letter m represents the gradient or slope, and the letter c represents the y-intercept, which is where the line crosses the y-axis.

所有直线图像都可以写成斜截式 y = mx + c。字母 m 表示斜率或坡度，字母 c 表示 y 轴截距，即直线与 y 轴相交的位置。

y = mx + c

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

在方程 y = 2x + 1 中，斜率 m 是 2，y 轴截距 c 是 1。这表示直线每向右移动 1 个单位就上升 2 个单位，并且它在 (0, 1) 处穿过 y 轴。

If the equation is written in a different order, such as y = 1 + 2x, it still has gradient m = 2 and y-intercept c = 1, but it is often easier to rewrite it as y = 2x + 1 first.

如果方程以不同的顺序写出，例如 y = 1 + 2x，它的斜率仍然是 m = 2，y 轴截距仍然是 c = 1，但通常先把它改写为 y = 2x + 1

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