📚 Mastering Straight Line Graphs | 精通直线图像
Straight line graphs appear throughout the IGCSE Mathematics syllabus, from coordinate geometry to real-life linear models. This article revises the essential skills of plotting, finding gradients and intercepts, writing equations, and using the properties of parallel and perpendicular lines.
直线图像贯穿 IGCSE 数学课程，从坐标几何到现实生活中的线性模型。本文复习绘制直线、求斜率与截距、写方程以及运用平行线和垂直线性质的核心技能。
1. The Cartesian Plane and Coordinates | 笛卡尔平面与坐标
A Cartesian plane is formed by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). Their intersection is the origin, written as (0, 0). A point is located by an ordered pair (x, y), where x is the horizontal distance from the origin and y is the vertical distance.
笛卡尔平面由两条互相垂直的数轴组成：x 轴（水平）和 y 轴（垂直）。它们的交点称为原点，记作 (0, 0)。一个点由有序数对 (x, y) 定位，其中 x 是离原点的水平距离，y 是垂直距离。
For example, the point (3, 4) means move 3 units right from the origin and then 4 units up. The point (-2, 5) means move 2 units left and then 5 units up. Coordinates can be positive or negative depending on the quadrant.
例如，点 (3, 4) 表示从原点向右移动 3 个单位，再向上移动 4 个单位。点 (-2, 5) 表示向左移动 2 个单位，再向上移动 5 个单位。坐标的正负取决于所在象限。
2. What is a Gradient? | 什么是斜率？
The gradient, also called the slope, measures how steep a straight line is. It is calculated as the change in y divided by the change in x between two points on the line.
斜率，也称为坡度，衡量直线的倾斜程度。它等于直线上两点之间 y 的变化量除以 x 的变化量。
m = Δy/Δx = (y₂ - y₁) / (x₂ - x₁)
A positive gradient means the line rises from left to right, while a negative gradient means it falls from left to right. A horizontal line has gradient 0, and a vertical line has an undefined gradient because the change in x is zero.
正斜率表示直线从左到右上升，负斜率表示直线从左到右下降。水平线的斜率为 0，竖直线的斜率未定义，因为 x 的变化量为零。
3. The General Equation y = mx + c | 一般方程 y = mx + c
gradient and c is the y-intercept. The y-intercept is the point where the line crosses the y-axis, so its coordinates are (0, c).
直线方程通常写作 y = mx + c，其中 m 是斜率，c 是 y 轴截距。y 轴截距是直线与 y 轴交点的纵坐标，因此交点坐标为 (0, c)。
y = mx + c
s 2 and the y-intercept is 3. This means the line crosses the y-axis at (0, 3), and for every 1 unit moved to the right, the line rises by 2 units.
例如，在 y = 2x + 3 中，斜率为 2，y 轴截距为 3。这意味着直线与 y 轴交于 (0, 3)，每向右移动 1 个单位，直线上升 2 个单位。
4. Finding the Gradient from Two Points | 由两点求斜率
If two points (x₁, y₁) and (x₂, y₂) lie on a straight line, the gradient m can be found by dividing the difference in y-values by the difference in x-values.
如果直线上有两点 (x₁, y₁) 和 (x₂, y₂)，则斜率 m 可以通过 y 值之差除以 x 值之差来求得。
m = (y₂ - y₁) / (x₂ - x₁)
, 2) and (3, 8). Using the formula, m = (8 - 2) / (3 - 1) = 6/2 = 3. So the gradient is 3.
例题：求经过 (1, 2) 和 (3, 8) 的直线斜率。使用公式，m = (8 - 2) / (3 - 1) = 6/2 = 3。所以斜率为 3。
Always subtract the coordinates in the same order for both numerator and denominator. For instance, do not use y₂ - y₁ with x₁ - x₂.
计算时分子和分母必须使用相同的坐标顺序。例如，不要用 y₂ - y₁ 搭配 x₁ - x₂。
5. Finding the Equation from a Graph | 由图求方程
oints on the line to calculate the gradient m. Finally substitute m and c into y = mx + c.
要从图像求直线方程，首先观察直线与 y 轴的交点，确定 y 轴截距 c。然后选择直线上两个清晰的点计算斜率 m。最后将 m 和 c 代入 y = mx + c。
graph, you can find m first and then substitute any point into y = mx + c to solve for c.
如果一条直线经过 (0, -1) 且斜率为 2，则其方程为 y = 2x - 1。如果 y 轴截距在图像上不可见，可以先求出 m，再将任意一点代入 y = mx + c 来解出 c。
6. Drawing Straight Line Graphs | 绘制直线图像
he gradient is 1/2, move 2 units right and 1 unit up to reach (2, 2). Draw a line through these two points.
绘制直线时，首先标出 y 轴截距。然后用斜率找到另一个点。例如，方程 y = (1/2)x + 1，从 (0, 1) 开始。因为斜率为 1/2，向右移动 2 个单位，向上移动 1 个单位，到达 (2, 2)。过这两点画直线。
You can also create a table of values by choosing x-values, finding y-values, and plotting the ordered pairs. This method is especially useful when the gradient is not a simple fraction or when the line is to be drawn accurately over a larger range.
你也可以通过选择 x 值、求出 y 值并绘制有序数对来列值表。当斜率不是简单分数或需要在较大范围内精确绘制直线时，这种方法尤其有用。
7. Parallel Lines | 平行线
re parallel because both have m = 3. Their y-intercepts are different, so they never meet.
平行线具有相同的斜率。如果直线 1 的方程为 y = 3x + 2，直线 2 的方程为 y = 3x - 5，那么它们平行，因为两者的斜率都为 m = 3。它们的 y 轴截距不同，因此永不相交。
through (2, 5) has gradient 4, so 5 = 4(2) + c, giving c = -3. The equation is y = 4x - 3.
要求经过某点且与已知直线平行的直线方程，保持斜率不变，代入该点求出 c。例如，一条直线平行于 y = 4x - 1 且经过点 (2, 5)，其斜率为 4，因此 5 = 4(2) + c，解得 c = -3。该直线方程为 y = 4x - 3。
8. Perpendicular Lines | 垂直线
Perpendicular lines meet at a right angle. Their gradients are negative reciprocals of each other, so the product of their gradients is -1.
垂直线成直角相交。它们的斜率互为负倒数，因此它们的斜率乘积为 -1。
m₁ × m₂ = -1
-1/2. Given line y = -3x + 4, a perpendicular line has gradient 1/3 because -3 × 1/3 = -1.
例如，如果一条直线的斜率为 2，与它垂直的直线斜率为 -1/2。已知直线 y = -3x + 4，与它垂直的直线斜率为 1/3，因为 -3 × 1/3 = -1。
To find the equation of a perpendicular line through a given point, first find the negative reciprocal of the original gradient, then substitute the point to solve for c.
要求经过给定点且与已知直线垂直的直线方程，先求出原斜率的负倒数，再代入该点解出 c。
9. Midpoint and Distance Between Two Points | 中点与两点间距离
The midpoint of
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