📚 Straight Line Graphs | 直线方程
In this chapter review we consolidate the key concepts of straight line graphs, including gradient, different forms of linear equations, conditions for parallel and perpendicular lines, distance between two points, midpoint coordinates, and real‑world modelling with linear functions. A solid command of these topics will underpin all further coordinate geometry and calculus work on the Edexcel AS/A‑Level Pure Mathematics course.
本章复习将系统梳理直线图像的核心知识，包括梯度、直线方程的不同形式、平行与垂直的条件、两点间的距离、中点坐标以及线性函数在实际问题中的建模。扎实掌握这些内容将为后续的坐标几何和微积分学习打下坚实基础，适用于 Edexcel AS/A‑Level 纯数学课程。
1. Gradient of a Straight Line | 直线的梯度
The gradient (m) of a straight line measures its steepness and direction. If a line passes through two points (x₁, y₁) and (x₂, y₂), the gradient is given by:
梯度 (m) 表示直线的倾斜程度和方向。如果一条直线经过两点 (x₁, y₁) 和 (x₂, y₂)，其梯度公式为：
m = (y₂ − y₁) / (x₂ − x₁)
A positive gradient means the line rises as x increases, a negative gradient means the line falls, a zero gradient gives a horizontal line, and an undefined gradient (denominator zero) corresponds to a vertical line.
正梯度表示直线随 x 增加而上升，负梯度表示下降，梯度为零时直线水平，分母为零时梯度无定义，对应竖直线。
2. Equation y = mx + c | 斜截式方程 y = mx + c
most common form for a straight line is y = mx + c, where m is the gradient and c is the y‑intercept – the coordinate where the line crosses the y‑axis (0, c). This form allows direct reading of both steepness and intercept.
直线最常用的形式是 y = mx + c，其中 m 为梯度，c 为直线在 y 轴上的截距，即直线与 y 轴的交点 (0, c)。该形式可直接读出倾斜程度和截距。
For any point on the line, substituting its coordinates satisfies the equation. To find the equation from a graph, identify the y‑intercept and calculate the gradient using two clear points.
对于直线上的任意一点，代入坐标都能满足方程。若要从图像求方程，可找出 y 截距并利用两个清晰点计算梯度。
3. Finding the Equation from Two Points | 已知两点求直线方程
When two points (x₁, y₁) and (x₂, y₂) are known, first calculate the gradient m = (y₂ − y₁) / (x₂ − x₁). Then use the point‑slope form y − y₁ = m(x − x₁) and simplify to y = mx + c or ax + by + c = 0.
已知两点 (x₁, y₁) 和 (x₂, y₂) 时，先计算梯度 m = (y₂ − y₁) / (x₂ − x₁)，再使用点斜式 y − y₁ = m(x − x₁) 并化简为 y = mx + c 或一般式 ax + by + c = 0。
It does not matter which point is used for the point‑slope formula, as both will lead to the same final equation. Always check your result by substituting both given points.
点斜式中选用哪个点都可以，最终方程一致。最后务必用给定的两个点代入检验。
4. Parallel Lines | 平行线
Two distinct straight lines are parallel if and only if their gradients are equal. That is, for lines with gradients m₁ and m₂, the condition is:
两条不重合的直线平行的充要条件是它们的梯度相等。即，若两条直线的梯度分别为 m₁ 和 m₂，则：
m₁ = m₂
us;a/b (provided b ≠ 0). Parallel lines never intersect and maintain a constant distance apart.
如果直线以一般式 ax + by + c = 0 给出，它们的梯度均为 −a/b（假设 b ≠ 0）。平行线永不相交，且保持恒定的间距。
5. Perpendicular Lines | 垂直线
Two lines are perpendicular if the product of their gradients equals −1. Symbolically, if line 1 has gradient m₁ and line 2 has gradient m₂, then:
两条直线垂直的条件是它们的梯度之积为 −1。用符号表示，若第一条线梯度为 m₁，第二条为 m₂，则：
m₁ × m₂ = −1
This is equivalent to m₂ = −1 / m&# lines: a horizontal line (m = 0) is perpendicular to a vertical line (undefined gradient).
这也等价于 m₂ = −1 / m₁。需要注意水平线与竖直线的情况：水平线梯度为 0，与梯度无定义的竖直线相互垂直。
6. Distance Between Two Points | 两点间的距离
The length of the straight line segment joining two points (x₁, y₁) and (x₂, y₂) is found using Pythagoras’ theorem. The distance formula is:
连接两点 (x₁, y₁) 和 (x₂, y₂) 的线段长度可通过勾股定理求出。距离公式为：
d = √[(x₂ − x₁)² + (y₂ − y₁)²]
This formula works in any order of the points and always gives a non‑negative result. It is the foundation for finding the radius of a circle and distances in vectors.
该公式不依赖点的顺序，结果恒为非负值。它是求圆半径及向量距离的基础。
7. Midpoint Formula | 中点公式
The midpoint of the line segment joining (x₁, y₁) and (x₂, y₂) has coordinates that are the averages of the x‑coordinates and y‑coordinates respectively:
连接 (x₁, y₁) 和 (x₂, y₂) 的线段中点坐标分别为两个横坐标和两个纵坐标的平均值：
M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
The midpoint is often used alongside gradient to find the perpendicular bisector of a line segment, a skill frequently examined in coordinate geometry problems involving circles.
中点常与梯度结合使用，以求出线段的垂直平分线，这是解决涉及圆的坐标几何题时的常见技巧。
8. Equation in the Form ax + by + c = 0 | 一般式方程 ax + by + c = 0
gers or real numbers. In this form, the gradient is −a/b (provided b ≠ 0) and the y‑intercept is −c/b. The x‑intercept is −c/a when a ≠ 0.
直线方程也可以写作 ax + by + c = 0，其中 a, b, c 为整数或实数。此形式下，梯度为 −a/b（b ≠ 0），y 截距为 −c/b，x 截距为 −c/a（a ≠ 0）。
examining conditions of parallelism or perpendicularity without converting to y = mx + c.
这一形式便于联立方程组求解，也可不经化为斜截式直接判断平行或垂直条件。
9. Drawing Straight Line Graphs | 绘制直线图像
pts (by setting x = 0 then y = 0). For vertical lines (x = k) or horizontal lines (y = c), the graph is immediate.
绘制直线图像时，至少找出满足方程的两个点并用直尺连接。最容易找的点通常是轴线截距（分别设 x = 0 和 y = 0）。对于竖直线 x = k 或水平线 y = c，可直接画出。
Always label the line with its equation and ensure the graph extends across the full coordinate grid. Checking a third point minimises plotting errors.
务必标出直线的方程，并使图像贯穿整个坐标格。多验证一个点可减少绘图误差。
10. Intersection of Two Lines | 两直线的交点
found by solving their equations simultaneously. If the lines are given in the form y = m₁x + c₁ and y = m₂x + c₂, setting m₁x + c₁ = m₂x + c₂ gives the x‑coordinate; then substitute to find y.
两直线的交点可通过联立它们的方程求得。若方程已写成 y = m₁x + c₁ 和 y = m₂x + c₂，令 m₁x + c₁ = m₂x + c₂ 解得 x 坐标，再代入求 y。
If the lines are parallel and distinct, the simultaneous equations have no solution. If they are the same line (coincident), there are infinitely many solutions. Intersection is a key method for solving geometrical constraints in modelling contexts.
若两直线平行且不重合，则联立方程无解。若为同一直线（重合），有无穷多解。求交点是建模情境中解决几何约束的重要方法。
11. Key Formulas Summary | 关键公式总结
The following table brings together the essential formulas for straight line graphs that must be recalled quickly in exams.
下表汇集了直线图像的核心公式，考试中需熟练运用。
| Concept / 概念 | Formula / 公式 |
|---|---|
| Gradient / 梯度 | m = (y₂ − y₁) / (x₂ − x₁) |
| Equation (slope‑intercept) / 斜截式 | y = mx + c |
| Equation (general) / 一般式 | ax + by + c = 0 |
| Parallel condition / 平行条件 | m₁ = m₂ |
| Perpendicular condition / 垂直条件 | m₁ × m₂ = −1 |
| Distance / 两点距离 | d = √[(x₂ − x₁)² + (y₂ − y₁)²] |
| Midpoint / 中点 | M = ((x₁ + x₂)/2, (y₁ + y₂)/2) |
| Point‑slope form / 点斜式 | y − y₁ = m(x − x₁) |
| Gradient from general form / 一般式求梯度 | m = −a/b (b ≠ 0) |
Familiarity with these formulas enables efficient problem‑solving. Practice converting between different forms, using gradients to analyse relationships, and applying algebraic methods to linear models.
熟练掌握这些公式能大幅提高解题效率。建议多练习不同形式间的转换，运用梯度分析几何关系，并将代数方法用于线性建模。
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