📚 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。
- Move right for positive x and left for negative x. — 正 x 向右移动，负 x 向左移动。
- Move up for positive y and down for negative y. — 正 y 向上移动，负 y 向下移动。
- The plane is divided into four quadrants. — 坐标平面分为四个象限。
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, 2) means x = 3 and y = 2. — (3, 2) 表示 x = 3 且 y = 2。
- The order in a coordinate matters: (2, 3) is a different point. — 坐标中顺序很重要：(2, 3) 是不同的点。
- Use a sharp pencil and grid paper for accurate plotting. — 使用削尖的铅笔和方格纸以准确描点。
3. Using a Table of Values | 使用数值表
Before drawing a line, we usually create a table of values. Choose at least three x-values, substitute each into the equation, 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
Any straight line can be written in the form y = mx + c. The letter m represents the gradient or steepness. The letter c represents the y-intercept, which is where the line crosses 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
- m controls how steep the line is. — m 控制直线的陡峭程度。
- c controls where the line crosses the y-axis. — c 控制直线与 y 轴的交点。
- If c is not written, it is 0. — 如果 c 没有写出来，它就是 0。
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)
For example, using the points (1, 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。
- Positive gradient: line slopes upward. — 正斜率：直线向上倾斜。
- Negative gradient: line slopes downward. — 负斜率：直线向下倾斜。
- Zero gradient: horizontal line. — 零斜率：水平线。
6. Intercept on the y-axis | y 轴截距
The y-intercept is the value of y when x = 0. It is the point (0, c). To find it from a graph, 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。
For 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)。
- The y-intercept always has x-coordinate 0. — y 轴截距的 x 坐标始终为 0。
- It shows the starting value when input is zero. — 它表示输入为零时的起始值。
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。
- Step 1: Create a table of values. — 第 1 步：制作数值表。
- Step 2: Plot the points carefully. — 第 2 步：仔细描点。
- Step 3: Join the points with a ruler. — 第 3 步：用直尺连接各点。
- Step 4: Label the graph with its equation. — 第 4 步：用方程标注图像。
8. Horizontal and Vertical Lines | 水平线与垂直线
Horizontal lines have equations of the form y = k. They are parallel to the x-axis and have gradient 0. Vertical lines have equations 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
- y = -3 is a horizontal line crossing the y-axis at -3. — y = -3 是一条水平线，在 -3 处与 y 轴相交。
- x = 2 is a vertical line crossing the x-axis at 2. — x = 2 是一条垂直线，在 2 处与 x 轴相交。
- Horizontal lines do not rise or fall. — 水平线不上升也不下降。
9. Parallel Lines and Their Gradients | 平行线及其斜率
Parallel lines never meet and have exactly the same gradient. For example, y = 2x + 1 and y = 2x – 5 are parallel 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
- Same m means the lines are parallel. — m 相同表示两条直线平行。
- Different c means they are not the same line. — c 不同表示它们不是同一条直线。
- Parallel lines always maintain the same distance apart. — 平行线始终保持相同的距离。
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。
For y = 0.5x + 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。
- Use a ruler to align the reading accurately. — 使用直尺使读数更准确。
- Interpolation is reliable within the plotted range. — 在已绘制范围内，内插是可靠的。
- Extrapolation beyond the range is less reliable. — 超出范围的外推可靠性较低。
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。
- Confusing (x, y)
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