Straight Line Graphs | 直线图

📚 Straight Line Graphs | 直线图

Straight line graphs are a fundamental topic in A-Level Mathematics, providing the basis for coordinate geometry and modelling linear relationships. In Edexcel’s specification, you are expected to master the equation of a straight line in various forms, calculate gradients and intercepts, understand parallel and perpendicular lines, and apply these concepts to solve problems in both pure mathematics and real‑world contexts.

直线图是A-Level数学中的一个基础主题，为坐标几何和线性关系建模奠定了基础。根据Edexcel考试大纲，你需要掌握各种形式的直线方程，计算梯度和截距，理解平行线和垂直线，并应用这些概念解决纯数学和现实情境中的问题。

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1. Standard Form of a Straight Line | 直线的标准形式

The most common equation of a straight line is the slope‑intercept form y = mx + c, where m represents the gradient (steepness) and c is the y‑intercept (the value where the line crosses the y‑axis).

最常见的直线方程是斜截式 y = mx + c，其中 m 表示梯度（陡峭程度），c 是 y 轴截距（直线与 y 轴相交时的取值）。

Another essential representation is the general form ax + by + c = 0, which is particularly useful for algebraic elimination and when working with integer coefficients. You must be able to convert between y = mx + c and ax + by + c = 0 efficiently, for example by rearranging y = ½x − 3 into x − 2y − 6 = 0.

另一种重要的表示是一般式 ax + by + c = 0，它在代数消元和整数系数运算中尤为实用。你必须能够熟练地在 y = mx + c 和 ax + by + c = 0 之间进行转换，比如将 y = ½x − 3 改写为 x − 2y − 6 = 0。

The point–slope form y − y₁ = m(x − x₁) is the starting point when you know a point on the line and the gradient. All these forms describe the same geometric object, so choosing the right one depends on the information available.

已知直线上一点和梯度时，点斜式 y − y₁ = m(x − x₁) 是建立方程的起点。所有这些形式描述的都是同一个几何对象，因此选择哪一种取决于已知条件。

Table: Common forms of a straight line equation

Form	Equation	Key Features
Slope‑intercept	y = mx + c	m = gradient, c = y‑intercept
Point‑slope	y − y₁ = m(x − x₁)	m = gradient, (x₁, y₁) = point on line
General	ax + by + c = 0	a, b, c are integers; gradient = −a/b

This table summarises the three primary forms you will encounter. In exams, you will often need to switch from general form to slope‑intercept form to read off the gradient and y‑intercept directly, or to show that two lines are parallel.

这张表格总结了你将遇到的三种主要形式。考试中，你经常需要从一般式转换为斜截式，以便直接读出梯度和 y 轴截距，或证明两直线平行。

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2. Calculating Gradient | 计算梯度

The gradient m quantifies the rate of change of y with respect to x. For two distinct points (x₁, y₁) and (x₂, y₂) on a straight line, the gradient is given by

梯度 m 量化了 y 相对于 x 的变化率。对于直线上的两个不同点 (x₁, y₁) 和 (x₂, y₂)，梯度公式为

m = (y₂ − y₁) / (x₂ − x₁)

It is essential to keep the order consistent: subtract the coordinates of the first point from those of the second, both in the numerator and the denominator.

保持减法顺序一致至关重要：分子和分母都必须用第二点的坐标减去第一点的坐标。

If m > 0, the line slopes upwards from left to right; if m < 0, the line slopes downwards. A gradient of zero indicates a horizontal line (y = constant), while a vertical line has an undefined gradient because the denominator x₂ − x₁ becomes zero.

若 m > 0，直线从左到右向上倾斜；若 m < 0，直线向下倾斜。梯度为零表示水平线（y = 常数），而垂直线因分母 x₂ − x₁ 为零导致梯度未定义。

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3. Intercepts and Their Meaning | 截距及其含义

The y‑intercept is the point where the line meets the y‑axis (x = 0). In y = mx + c, it is simply (0, c). The x‑intercept is the point where the line crosses the x‑axis (y = 0). You can find it by substituting y = 0 into the line’s equation and solving for x.

y 轴截距是直线与 y 轴（x = 0）的交点。在 y = mx + c 中，它就是 (0, c)。x 轴截距是直线与 x 轴（y = 0）的交点，可以通过将 y = 0 代入直线方程并解出 x 来求得。

For the general form ax + by + c = 0, the y‑intercept is −c/b (provided b ≠ 0), and the x‑intercept is −c/a (a ≠ 0). These intercepts provide two quick points for sketching the graph.

对于一般式 ax + by + c = 0，y 轴截距为 −c/b（假设 b ≠ 0），x 轴截距为 −c/a（a ≠ 0）。这两个截距为绘制图形提供了两个快捷点。

In many modelling questions, the y‑intercept represents an initial value or fixed cost, while the x‑intercept might indicate a break‑even point or root of the linear function. Always interpret intercepts in the context of the problem.

在许多建模问题中，y 轴截距代表初始值或固定成本，而 x 轴截距可能表示盈亏平衡点或线性函数的零点。请始终结合题目背景解释截距。

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4. Parallel Lines | 平行线

Two distinct lines are parallel if and only if their gradients are equal: m₁ = m₂. This condition works for any non‑vertical lines. Vertical lines are parallel to each other as well, but their gradients are undefined, so the rule is usually stated for lines expressed in slope‑intercept or comparable form.

两条不同的直线平行当且仅当它们的梯度相等：m₁ = m₂。这一条件适用于任何非垂直线。垂直线彼此平行，但它们的梯度未定义，因此该规则通常针对用斜截式或可比形式表示的直线阐述。

For example, the line through (2, 5) with gradient 3 has equation y − 5 = 3(x − 2). Any line parallel to it will also have gradient 3, such as y = 3x − 1 or 3x − y + 4 = 0.

例如，过点 (2, 5) 且梯度为 3 的直线方程为 y − 5 = 3(x − 2)。任何与之平行的直线梯度也为 3，例如 y = 3x − 1 或 3x − y + 4 = 0。

To verify parallelism, rearrange both equations into the form y = mx + c and compare their m values. If a question asks you to find a line parallel to a given line passing through a specific point, simply use the same gradient and apply the point–slope form.

要验证平行，可将两个方程都化为 y = mx + c 并比较它们的 m 值。如果题目要求找一条与给定直线平行且经过某点的直线，只需使用相同的梯度并应用点斜式即可。

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5. Perpendicular Lines | 垂直线

For two perpendicular lines with non‑zero gradients, the product of their gradients equals −1: m₁ × m₂ = −1. Equivalently, the gradient of one is the negative reciprocal of the other: m₂ = −1/m₁.

对于两条梯度均不为零的垂直线，它们的梯度乘积等于 −1：m₁ × m₂ = −1。等价地，一条线的梯度是另一条线梯度的负倒数：m₂ = −1/m₁。

Special cases: a horizontal line (m = 0) is perpendicular to a vertical line (undefined gradient), but this is best remembered conceptually rather than using the product rule.

特殊情况：水平线（m = 0）与垂直线（梯度未定义）垂直，最好从概念上加以记忆，而非套用乘积规则。

Exam questions frequently require you to find the equation of a line perpendicular to a given line passing through a stated point. The process is: first identify the given gradient m₁, then compute the perpendicular gradient m₂ = −1/m₁, and finally use y − y₁ = m₂(x − x₁).

考试题目经常要求你求出一条与给定直线垂直且经过某点的直线方程。步骤是：先确定给定梯度 m₁，计算垂直梯度 m₂ = −1/m₁，最后使用 y − y₁ = m₂(x − x₁)。

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6. Finding the Equation from Two Points | 由两点求方程

When you are given two points (x₁, y₁) and (x₂, y₂), the first step is to calculate the gradient using m = (y₂ − y₁)/(x₂ − x₁). Once m is known, substitute one of the points into y − y₁ = m(x − x₁) to obtain the equation.

当给定两点 (x₁, y₁) 和 (x₂, y₂)，第一步是用 m = (y₂ − y₁)/(x₂ − x₁) 计算梯度。求出 m 后，将其中一个点代入 y − y₁ = m(x − x₁) 即可得到方程。

Example: Points A(1, 4) and B(3, 10). Gradient m = (10 − 4)/(3 − 1) = 6/2 = 3. Using point A, the line is y − 4 = 3(x − 1), which simplifies to y = 3x + 1. You can check that B satisfies this equation, confirming the result.

示例：点 A(1, 4) 和 B(3, 10)。梯度 m = (10 − 4)/(3 − 1) = 6/2 = 3。利用点 A，直线为 y − 4 = 3(x − 1)，化简得 y = 3x + 1。你可以检验点 B 是否满足该方程，以确认结果。

Sometimes the two points have the same x‑coordinate, giving a vertical line x = k. If they have the same y‑coordinate, the line is horizontal y = k. Recognising these special cases saves time and avoids division by zero.

有时两点的 x 坐标相同，此时为垂直线 x = k。如果 y 坐标相同，则为水平线 y = k。识别这些特殊情况可以节省时间并避免除零错误。

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7. Drawing Straight Line Graphs | 绘制直线图

To sketch a straight line accurately, you typically need two points. The simplest approach is to plot the y‑intercept and then use the gradient to find a second point: from the intercept, move ‘rise’ vertically and ‘run’ horizontally according to the gradient.

要准确绘制一条直线，通常需要两个点。最简单的方法是标出 y 轴截距，然后利用梯度找到第二个点：从截距出发，按照梯度进行垂直“上升”和水平“平移”。

For y = 2x − 3, the y‑intercept is (0, −3). The gradient 2 means rise 2, run 1, so moving right 1 unit and up 2 units gives (1, −1). Drawing a straight edge through (0, −3) and (1, −1) produces the required line.

对于 y = 2x − 3，y 轴截距为 (0, −3)。梯度 2 表示上升 2、平移 1，因此右移 1 单位、上移 2 单位得到点 (1, −1)。用直尺通过 (0, −3) 和 (1, −1) 画线即可。

An alternative method uses the two intercepts: set x = 0 to find the y‑intercept and y = 0 to find the x‑intercept, as long as the line is not passing through the origin. This dual‑intercept method is especially handy for lines in general form.

另一种方法是利用两个截距：设 x = 0 求 y 截距，设 y = 0 求 x 截距，只要直线不经过原点即可。这种双截距法对于一般式直线尤为方便。

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8. Intersection of Two Lines | 两直线的交点

The intersection point of two straight lines is the solution to their simultaneous equations. You can solve by substitution, elimination, or equating the y‑expressions when both equations are in the form y = mx + c.

两条直线的交点就是它们联立方程的解。你可以通过代入法、消元法求解，或者当两个方程都为 y = mx + c 形式时直接将表达式等量代换。

Example: Find the intersection of y = 2x + 1 and y = −x + 7. Set 2x + 1 = −x + 7 → 3x = 6 → x = 2, then y = 5. Therefore the lines intersect at (2, 5).

示例：求直线 y = 2x + 1 与 y = −x + 7 的交点。令 2x + 1 = −x + 7 → 3x = 6 → x = 2，然后 y = 5。因此交点为 (2, 5)。

If the lines are parallel, there is no intersection (or infinitely many if they are the same line). In coordinate geometry problems, you may also need to find the area of a triangle formed by intersections of lines with axes – understanding intersections is crucial for such multi‑step tasks.

如果直线平行，则没有交点（若为同一条直线则有无穷多交点）。在坐标几何问题中，你可能还需要求由直线与坐标轴围成的三角形面积——理解交点是完成此类多步任务的关键。

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9. Midpoint and Distance | 中点与距离

Many straight‑line graph problems involve the segment joining two points. The midpoint M of the segment with endpoints (x₁, y₁) and (x₂, y₂) is found by averaging the x‑coordinates and y‑coordinates:

许多直线图问题会涉及连接两点的线段。以 (x₁, y₁) 和 (x₂, y₂) 为端点的线段中点 M 可通过分别对 x 坐标和 y 坐标求平均得到：

M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

The distance d between the same two points is given by Pythagoras’ theorem:

同两点间的距离 d 由勾股定理给出：

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

These formulas are directly relevant to topics such as finding the equation of a perpendicular bisector, the centre of a circle from the endpoints of a diameter, or the side length of a polygon. Always simplify surds where possible and retain exact answers unless directed otherwise.

这些公式可直接应用于求垂直平分线方程、由直径端点求圆心、或多边形的边长等问题。请尽可能化简根式并保留精确值，除非题目另有要求。

Note that the distance formula works for any pair of points, regardless of the orientation of the segment. The order of subtraction does not affect the result because of squaring.

注意，距离公式适用于任意两点，与线段方向无关。由于存在平方运算，减法的顺序不影响结果。

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10. Linear Modelling and Real‑life Applications | 线性建模与实际应用

Straight line graphs are often used to model linear relationships in economics, physics, and everyday life. The gradient represents the rate of change (

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