📚 IGCSE Maths: Straight Line Graphs and Coordinate Geometry | IGCSE 数学：直线图像与坐标几何
Straight line graphs are central to the IGCSE Mathematics syllabus. This unit reviews how to plot coordinates, calculate gradient, interpret y = mx + c, use midpoint and distance formulas, and solve problems involving parallel and perpendicular lines.
直线图像是 IGCSE 数学大纲的核心内容。本单元将复习如何描点、计算斜率、理解 y = mx + c、使用中点与距离公式，并解决涉及平行线和垂直线的问题。
1. Coordinates and Plotting Points | 坐标与描点
On the Cartesian plane, a point is described by an ordered pair (x, y). The first number is the x-coordinate and tells you how far to move left or right from the origin. The second number is the y-coordinate and tells you how far to move up or down.
在笛卡尔平面上，一个点用有序数对 (x, y) 表示。第一个数是 x 坐标，表示从原点向左或向右移动的距离。第二个数是 y 坐标，表示向上或向下移动的距离。
Always read the x-coordinate first. For example, the point (-3, 2) is 3 units left and 2 units up. The point (4, -1) is 4 units right and 1 unit down.
描点时总是先确定 x 坐标。例如，点 (-3, 2) 位于原点左侧 3 个单位、上方 2 个单位处。点 (4, -1) 位于原点右侧 4 个单位、下方 1 个单位处。
Plotting points accurately is the first step in drawing linear graphs. If a point does not lie exactly on grid lines, estimate its position carefully.
准确描点是绘制线性图像的第一步。如果某个点没有正好落在网格线上，需要仔细估算它的位置。
2. Gradient of a Straight Line | 直线的斜率
The gradient measures how steep a line is. It is found by dividing the vertical change by the horizontal change between two points on the line.
斜率用于衡量一条直线的倾斜程度。它等于直线上两点之间竖直变化量除以水平变化量。
Gradient m = (y₂ – y₁) / (x₂ – x₁)
A positive gradient means the line rises from left to right. A negative gradient means the line falls from left to right. A horizontal line has gradient 0, while a vertical line has undefined gradient.
斜率为正表示直线从左到右上升。斜率为负表示直线从左到右下降。水平线的斜率为 0，竖直线的斜率不存在。
For example, if a line passes through (1, 2) and (5, 10), the gradient is (10 – 2) ÷ (5 – 1) = 8 ÷ 4 = 2.
例如，如果一条直线经过 (1, 2) 和 (5, 10)，那么斜率为 (10 – 2) ÷ (5 – 1) = 8 ÷ 4 = 2。
3. The Equation y = mx + c | 方程 y = mx + c
Any non-vertical straight line can be written in the form y = mx + c. The letter m represents the gradient, and the letter c represents the y-intercept, which is the point where the line crosses the y-axis.
任何非竖直直线都可以写成 y = mx + c 的形式。字母 m 表示斜率，字母 c 表示 y 轴截距，也就是直线与 y 轴交点的纵坐标。
To identify the gradient and y-intercept from an equation, rearrange the equation until y is the subject. For y = 3x – 7, the gradient is 3 and the y-intercept is -7.
要从方程中找出斜率和 y 轴截距，需要先把方程整理成 y 为主语的形式。对于 y = 3x – 7，斜率为 3，y 轴截距为 -7。
To sketch a graph from y = mx + c, plot the y-intercept first, then use the gradient as a fraction. For y = ½x + 1, plot (0, 1) and then move 2 units right and 1 unit up to find another point.
要根据 y = mx + c 绘制图像，先标出 y 轴截距，然后把斜率看作分数来移动。对于 y = ½x + 1，先标出 (0, 1)，再向右移动 2 个单位、向上移动 1 个单位找到另一个点。
4. Finding the Equation from Two Points | 由两点求方程
If you know two points on a straight line, you can find its equation. Start by calculating the gradient using the two-point formula. Then substitute one point into y = mx + c to find c.
如果知道一条直线上的两个点，就可以求出它的方程。首先用两点公式计算斜率，然后把其中一个点代入 y = mx + c 求出 c。
For example, find the equation of the line through (2, 3) and (6, 11). First, m = (11 – 3) ÷ (6 – 2) = 8 ÷ 4 = 2. Substitute (2, 3): 3 = 2(2) + c, so c = -1. The equation is y = 2x – 1.
例如，求经过 (2, 3) 和 (6, 11) 两点的直线方程。首先，m = (11 – 3) ÷ (6 – 2) = 8 ÷ 4 = 2。代入 (2, 3)：3 = 2(2) + c，所以 c = -1。因此方程为 y = 2x – 1。
Always check your final equation by substituting both original points. If both satisfy the equation, the working is correct.
最后一定要把两个原始点都代入方程进行检查。如果两个点都满足方程，说明计算正确。
5. Midpoint of a Line Segment | 线段的中点
The midpoint is the point exactly halfway between two given points. Its coordinates are the averages of the x-coordinates and the y-coordinates.
中点是位于两个已知点正中间的点。它的坐标是两个点 x 坐标和 y 坐标的平均值。
Midpoint M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
For the points (1, 2) and (7, 8), the midpoint is ((1 + 7)/2, (2 + 8)/2) = (4, 5). This is useful in geometry problems and when finding the centre of a line segment.
对于点 (1, 2) 和 (7, 8)，中点为 ((1 + 7)/2, (2 + 8)/2) = (4, 5)。这在几何问题以及求线段中心时非常有用。
Do not subtract the coordinates when finding a midpoint. The midpoint formula uses addition, not subtraction.
求中点时不要把坐标相减。中点公式使用的是加法，而不是减法。
6. Distance Between Two Points | 两点间距离
The distance between two points is the length of the line segment joining them. It is found using Pythagoras’ theorem and the differences in the x and y coordinates.
两点间的距离就是连接这两点的线段长度。它通过毕达哥拉斯定理以及 x 坐标和 y 坐标的差来求得。
Distance d = √((x₂ – x₁)² + (y₂ – y₁)²)
For example, the distance between (1, 2) and (4, 6) is √((4 – 1)² + (6 – 2)²) = √(9 + 16) = √25 = 5.
例如，点 (1, 2) 和 (4, 6) 之间的距离为 √((4 – 1)² + (6 – 2)²) = √(9 + 16) = √25 = 5。
It does not matter which point is treated as the first point, because the differences are squared. The result is always a positive value or zero.
把哪个点作为第一个点并不重要，因为差会被平方。结果始终为正值或零。
7. Parallel and Perpendicular Lines | 平行线与垂直线
Parallel lines have the same gradient. If one line has equation y = 4x + 3, any parallel line has gradient 4, such as y = 4x – 2.
平行线具有相同的斜率。如果一条直线的方程是 y = 4x + 3，那么任何平行线的斜率都是 4，例如 y = 4x – 2。
Perpendicular lines meet at right angles. Their gradients multiply to give -1. If one gradient is m, the perpendicular gradient is -1/m.
垂直线相交成直角。它们的斜率相乘等于 -1。如果一条直线的斜率是 m，那么与它垂直的直线斜率是 -1/m。
For example, a line with gradient 2 has a perpendicular gradient of -½. A line with gradient -3 has a perpendicular gradient of ⅓.
例如，斜率为 2 的直线，其垂直线斜率为 -½。斜率为 -3 的直线，其垂直线斜率为 ⅓。
| Original gradient | Perpendicular gradient |
| 2 | -½ |
| -3 | ⅓ |
| ¾ | -4/3 |
8. Intersection of Two Lines | 两条直线的交点
The point where two straight lines cross is found by solving their equations simultaneously. The solution gives the x-coordinate and y-coordinate of the intersection.
两条直线相交的点可以通过联立方程求解得到。方程的解就是交点的 x 坐标和 y 坐标。
For example, find where y = 2x + 1 meets y = -x + 7. Set 2x + 1 = -x + 7, so 3x = 6, giving x = 2. Substitute into y = 2(2) + 1 = 5. The intersection is (2, 5).
例如，求直线 y = 2x + 1 与 y = -x + 7 的交点。令 2x + 1 = -x + 7，得到 3x = 6，因此 x = 2。代入 y = 2(2) + 1 = 5。交点为 (2, 5)。
Graphically, the intersection point should lie on both lines. You can check by plotting both equations and reading the crossing point from the grid.
从图像上看，交点应该同时位于两条直线上。可以通过绘制两条方程对应的直线并从网格中读取交点来进行检查。
9. Real-Life Linear Graphs | 实际生活中的线性图像
Linear graphs model many real-life situations, such as taxi fares, phone bills, and distance-time relationships. The gradient represents the rate of change, while the intercept often represents a fixed starting amount.
线性图像可以模拟许多实际生活中的情况，例如出租车费用、电话账单和距离-时间关系。斜率表示变化率，而截距通常表示固定的初始量。
For a distance-time graph, the gradient gives the speed. A steeper line means faster travel. The y-intercept is the starting distance from a reference point.
对于距离-时间图像，斜率表示速度。直线越陡，表示行驶越快。y 轴截距是从参考点出发的初始距离。
When interpreting real-life graphs, always state the units of the gradient. If distance is in metres and time is in seconds, the gradient has units m/s.
解释实际生活图像时，一定要说明斜率的单位。如果距离以米为单位，时间以秒为单位，那么斜率的单位就是 m/s。
10. Common Exam Mistakes | 常见考试错误
Many candidates lose marks by mixing up the midpoint and gradient formulas. The midpoint uses the average of coordinates, while the gradient uses the difference of coordinates divided by the difference of x-coordinates.
许多考生会把中点公式和斜率公式混淆而丢分。中点使用坐标的平均值，而斜率使用 y 坐标之差除以 x 坐标之差。
Another common mistake is ignoring the sign of the gradient. A line falling to the right must have a negative gradient, even if the numbers used in the formula are positive before subtraction.
另一个常见错误是忽略斜率的正负号。向右下方倾斜的直线斜率一定为负，即使代入公式的数在相减前为正。
Finally, when finding the equation of a line, do not forget to find the y-intercept. The gradient alone is not enough to determine a unique line.
最后，在求直线方程时，不要忘记求 y 轴截距。只知道斜率不足以确定一条唯一的直线。
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