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Straight-line Graphs and Coordinates | 直线图与坐标

2026-08-21 11:45 · 本文部分内容由AI辅助创作

📚 Straight-line Graphs and Coordinates | 直线图与坐标

In Cambridge KS3 mathematics, straight-line graphs connect algebra with geometry. When you plot a linear equation, the points always lie along a straight line. This topic helps you understand coordinates, gradient, intercepts, and how equations describe visual patterns. These skills are foundational for further study at IGCSE and beyond.

在剑桥 KS3 数学中，直线图将代数与几何联系起来。当你绘制线性方程的图像时，所有点总是落在一条直线上。这一主题帮助你理解坐标、斜率、截距，以及方程如何描述视觉规律。这些技能为 IGCSE 及更高阶段的学习打下基础。

1. The Coordinate Plane | 坐标系

The coordinate plane is formed by a horizontal x-axis and a vertical y-axis. The point where they cross is called the origin, written as (0, 0). Coordinates are always written in the form (x, y). The x-value tells you how far to move left or right; the y-value tells you how far to move up or down.

坐标系由水平的 x 轴和垂直的 y 轴构成。两轴相交的点称为原点，记作 (0, 0)。坐标始终写成 (x, y) 的形式。x 值表示向左或向右移动多少；y 值表示向上或向下移动多少。

2. Plotting Points | 描点

To plot a point such as (3, 2), start at the origin. Move 3 units right along the x-axis, then 2 units up parallel to the y-axis. Mark the point with a clear dot. Common mistakes include swapping x and y or moving in the wrong direction for negative coordinates.

要绘制例如 (3, 2) 这样的点，从原点开始。沿 x 轴向右移动 3 个单位，再平行于 y 轴向上移动 2 个单位。用清晰的圆点标出该点。常见错误包括把 x 和 y 颠倒，或负坐标方向移动错误。

3. Using a Table of Values | 使用数值表

n, and calculate the matching y-value. For y = 2x + 1, if x = -1, then y = 2(-1) + 1 = -1.

在绘制直线之前，我们通常先制作一个数值表。至少选择三个 x 值，将每个值代入方程，并计算出对应的 y 值。对于 y = 2x + 1，如果 x = -1，则 y = 2(-1) + 1 = -1。

y = 2x + 1

x -2 -1 0 1 2
y = 2x + 1 -3 -1 1 3 5

Plot the ordered pairs (-2, -3), (-1, -1), (0, 1), (1, 3) and (2, 5). They should all lie on the same straight line.

描出有序数对 (-2, -3)、(-1, -1)、(0, 1)、(1, 3) 和 (2, 5)。它们都应落在同一条直线上。

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

osses the y-axis. For example, in y = 3x - 2, the gradient is 3 and the y-intercept is -2.

任何直线都可以写成 y = mx + c 的形式。字母 m 表示斜率或坡度。字母 c 表示 y 轴截距，也就是直线与 y 轴相交的位置。例如，在 y = 3x - 2 中，斜率为 3，y 轴截距为 -2。

y = mx + c

5. Gradient: Slope of a Line | 斜率：直线的坡度

Gradient measures how steep a line is. It is the ratio of vertical change to horizontal change between two points. If a line goes up from left to right, the gradient is positive. If it goes down from left to right, the gradient is negative.

斜率衡量直线有多陡。它是两点之间垂直变化与水平变化的比值。如果直线从左到右向上倾斜，斜率为正；如果从左到右向下倾斜，斜率为负。

m = (y₂ - y₁) ÷ (x₂ - x₁)

, 3) and (4, 9), the rise is 9 - 3 = 6 and the run is 4 - 1 = 3. Therefore, m = 6 ÷ 3 = 2.

例如，使用点 (1, 3) 和 (4, 9)，垂直变化为 9 - 3 = 6，水平变化为 4 - 1 = 3。因此，m = 6 ÷ 3 = 2。

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6. Intercept on the y-axis | y 轴截距

ph, look where the line crosses the y-axis. To find it from an equation, substitute x = 0.

y 轴截距是当 x = 0 时 y 的值。它是点 (0, c)。要从图像中找到它，观察直线与 y 轴相交的位置。要从方程中找到它，将 x = 0 代入。

r y = -x + 4, substitute x = 0 to get y = -0 + 4 = 4. The y-intercept is therefore (0, 4).

对于 y = -x + 4，代入 x = 0 得到 y = -0 + 4 = 4。因此 y 轴截距为 (0, 4)。

7. Drawing a Straight-line Graph | 绘制直线图

To draw a straight-line graph, follow these steps. First, make a table of values. Second, plot each ordered pair on the coordinate grid. Third, draw a straight line through the points using a ruler. Finally, label the line with its equation. The line should extend across the whole grid.

要绘制直线图，请遵循以下步骤。首先，制作一个数值表。其次，在坐标网格上描出每个有序数对。第三，用直尺通过各点画一条直线。最后，用方程标注这条直线。直线应延伸至整个网格。

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

ations of the form x = h. They are parallel to the y-axis and their gradient is undefined.

水平线的方程形式为 y = k。它们平行于 x 轴，斜率为 0。垂直线的方程形式为 x = h。它们平行于 y 轴，斜率未定义。

y = k and x = h

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

lel because both have m = 2. They have different y-intercepts, so they are distinct lines.

平行线永不相交，且斜率完全相同。例如，y = 2x + 1 和 y = 2x - 5 是平行的，因为两者的 m 都等于 2。它们的 y 轴截距不同，因此它们是两条不同的直线。

y = 2x + 1 and y = 2x - 5

10. Reading Values from a Graph | 从图像读取数值

Once a straight-line graph is drawn, you can use it to estimate unknown values. Find a given x on the x-axis, draw a vertical line up to the graph, then read across to the y-axis. This gives the corresponding y-value. This is called interpolation when the value lies within the plotted range.

直线图绘制完成后，你可以用它来估算未知数值。在 x 轴上找到给定的 x，向上画一条垂直线到达图像，然后平移到 y 轴读数。这样就得到对应的 y 值。当数值位于已绘制范围内时，这称为内插。

2, if x = 3, then y = 0.5(3) + 2 = 3.5. You can check this by reading the graph at x = 3.

对于 y = 0.5x + 2，如果 x = 3，则 y = 0.5(3) + 2 = 3.5。你可以通过在 x = 3 处读取图像来检验。

11. Common Misconceptions | 常见误解

Students often mix up the x-axis and y-axis. Always write coordinates as (x, y), not (y, x). Another error is misreading negative coordinates, especially in all four quadrants. A third mistake is calculating gradient as horizontal change divided by vertical change; it must be vertical over horizontal.

学生经常混淆 x 轴和 y 轴。坐标始终写成 (x, y)，而不是 (y, x)。另一个错误是误读负坐标，尤其是在四个象限中。第三个错误是把斜率计算成水平变化除以垂直变化；斜率必须是垂直变化除以水平变化。