Straight Line Graphs | 直线图

📚 Straight Line Graphs | 直线图

quation y = mx + c to finding intersections and handling parallel and perpendicular lines.

直线图是 IGCSE 数学中最基础也是最重要的内容之一。它将代数与几何紧密相连，让你能够直观地理解线性关系、通过图像解方程，并用来模拟现实世界中的各种情境。本文将从 y = mx + c 这个基本方程讲起，涵盖求交点、平行线与垂线等全部核心考点。

1. The General Equation of a Straight Line | 直线的一般方程

re, x and y are the coordinates of any point on the line, m is the gradient (slope), and c is the y-intercept, which is the point where the line crosses the y-axis.

二维平面中的任意一条直线都可以写成 y = mx + c 的形式。其中 x 和 y 是直线上任意一点的坐标，m 是梯度（斜率），c 是 y 轴截距，也就是直线与 y 轴相交的点。

l need to rearrange it into the form y = mx + c to easily read the gradient and intercept.

另一种常见形式是 ax + by = d，通常称为标准形式。你需要将它重新整理成 y = mx + c 的形式，以便快速读取梯度和截距。

y = mx + c

Example: Rearrange 2x + 3y = 6 into the form y = mx + c.

示例：将 2x + 3y = 6 改写成 y = mx + c 的形式。

ract 2x from both sides: 3y = -2x + 6. Then divide by 3: y = -⅔x + 2. So m = -⅔ and c = 2.

两边减去 2x：3y = -2x + 6，再除以 3：y = -⅔x + 2。因此 m = -⅔，c = 2。

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2. Gradient and Intercept | 梯度与截距

The gradient measures how steep a line is. It is defined as the change in y divided by the change in x, often written as "rise over run".

梯度用来衡量直线的陡峭程度，定义为 y 的变化量除以 x 的变化量，常称为“纵增量除以横增量”。

m = (y₂ - y₁) / (x₂ - x₁)

Given two points (x₁, y₁) and (x₂, y₂), you can calculate the gradient using this formula. A positive gradient means the line slopes upward from left to right; a negative gradient means it slopes downward.

已知两点 (x₁, y₁) 和 (x₂, y₂)，就可以用这个公式计算梯度。梯度为正表示直线从左到右上升；梯度为负则表示直线从左到右下降。

• A horizontal line has gradient 0, equation y = c.

  水平线的梯度为 0，方程为 y = c。

• A vertical line has an undefined gradient, equation x = k.

  垂直线的梯度无定义，方程为 x = k。

The y-intercept c is the y-value when x = 0. It is the point (0, c) on the graph.

y 轴截距 c 是当 x = 0 时的 y 值，即图像上的点 (0, c)。

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3. Drawing a Line from its Equation | 从方程绘制直线

To draw a straight line from its equation, follow these steps:

要根据方程绘制直线，请按以下步骤操作：

1. Find the y-intercept by setting x = 0. Plot this point.

   令 x = 0 找出 y 轴截距，先标出这个点。

2. nt m to find another point. If m = 2, from the intercept move 1 unit right and 2 units up.

   利用梯度 m 找出另一个点。若 m = 2，则从截距点向右移 1 个单位，再向上移 2 个单位。

3. Draw a straight line through both points, extending across the grid.

   通过这两个点画一条直线，并向两端延伸。

Alternatively, you can substitute two different x-values into the equation to find two points, then join them.

另一种方法是将两个不同的 x 值代入方程，求出两个点，然后连线。

Example: Draw the line y = 2x - 1.

示例：绘制直线 y = 2x - 1。

= 0, y = -1. When x = 2, y = 3. Plot (0, -1) and (2, 3), then draw the line through them.

当 x = 0 时，y = -1。当 x = 2 时，y = 3。标出点 (0, -1) 和 (2, 3)，然后过这两点画直线。

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4. Finding the Equation of a Line | 求直线方程

To find the equation of a straight line, you need two pieces of information: the gradient and one point on the line.

要求一条直线的方程，你需要两个条件：梯度和直线上一个点的坐标。

y - y₁ = m(x - x₁)

point (x₁, y₁), substitute these into the formula and rearrange into the form y = mx + c.

这就是点斜式。如果已知梯度 m 和一点 (x₁, y₁)，将它们代入公式，再整理成 y = mx + c 的形式即可。

Example: Find the equation of the line through (2, 5) with gradient 3.

示例：求过点 (2, 5) 且梯度为 3 的直线方程。

Using y - 5 = 3(x - 2): y - 5 = 3x - 6, so y = 3x - 1.

代入 y - 5 = 3(x - 2)：y - 5 = 3x - 6，因此 y = 3x - 1。

If you are given two points but no gradient, first calculate m using the formula, then use the point-gradient form with either point.

如果已知两点但没有给出梯度，先用公式算出 m，再用点斜式并代入其中任意一点。

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5. Parallel and Perpendicular Lines | 平行线与垂线

Parallel lines have the same gradient but different y-intercepts. If two lines have the same m, they never intersect.

平行线的梯度相同，但 y 截距不同。如果两条直线的 m 相同，它们永远不会相交。

Perpendicular lines meet at a right angle (90°). If a line has gradient m, a perpendicular line has gradient -1/m.

垂线以直角（90°）相交。如果一条直线的梯度为 m，垂直于它的直线梯度为 -1/m。

m₁ × m₂ = -1

This product rule is true for two perpendicular lines, provided neither is vertical.

这个乘积关系对两条垂线成立，前提是两者都不是竖直直线。

Example: A has equation y = ½x + 3. Line B is perpendicular to A and passes through (0, 1). Find B's equation.

示例：直线 A 的方程为 y = ½x + 3。直线 B 垂直于 A，且过点 (0, 1)。求 B 的方程。

Gradient of A is ½, so gradient of B is -2. Using y - 1 = -2(x - 0), we get y = -2x + 1.

A 的梯度为 ½，因此 B 的梯度为 -2。代入 y - 1 = -2(x - 0)，得 y = -2x + 1。

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6. Intersection of Two Lines | 两直线的交点

The intersection point of two lines is the solution to the system of their equations. Since both equations equal y, you can set the right-hand sides equal to each other and solve for x, then substitute back to find y.

两条直线的交点就是这两个方程组成的方程组的解。由于两个方程都等于 y，可以将两个等号右边部分相等，先解出 x，再代回求出 y。

m₁x + c₁ = m₂x + c₂

Example: Find the intersection of y = 2x + 1 and y = -x + 4.

示例：求直线 y = 2x + 1 和 y = -x + 4 的交点。

+ 4. Then 3x = 3, so x = 1. Substitute into y = 2(1) + 1 = 3. The intersection is (1, 3).

令 2x + 1 = -x + 4，得到 3x = 3，所以 x = 1。代入 y = 2(1) + 1 = 3。交点为 (1, 3)。

If the lines are parallel (same gradient), there is no intersection. If they are the same line, there are infinitely many intersections.

如果两直线平行（梯度相同），则没有交点。如果两直线重合，则有无穷多个交点。

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7. Real-Life Applications | 实际应用

Straight line graphs appear frequently in real-world contexts. A taxi fare might be modelled as a fixed charge plus a rate per kilometre. A phone plan might have a monthly fee plus a per-minute cost.

直线图在现实生活中非常常见。出租车费用可以建模为起步价加上每公里单价。手机套餐则可能是月租费加上每分钟通话费用。

• Distance vs. time: constant speed gives a straight line.

  距离与时间的关系：匀速运动对应一条直线。

• Currency conversion: exchange rate creates a linear relationship.

  货币兑换：汇率形成线性关系。

• Temperature conversion between Celsius and Fahrenheit is linear.

  摄氏温度与华氏温度之间的换算关系是线性的。

When interpreting graphs, the gradient often represents a rate, such as speed (£ per hour, km per litre), and the intercept represents a starting value or fixed cost.

在解读图像时，梯度通常代表某种速率，例如速度（每小时公里数、每升公里数），而截距则代表初始值或固定成本。

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8. Common Mistakes and Tips | 常见错误与技巧

Students often make errors when dealing with negative gradients, rearranging equations, or identifying the correct gradient from a graph. Be cautious with the following points:

学生在处理负梯度、方程变形以及从图像中读取梯度时经常出错。请注意以下几点：

• Always rearrange equations into y = mx + c before reading m and c.

  在读取 m 和 c 之前，务必先将方程整理成 y = mx + c 的形式。

• When calculating the gradient, keep consistent: (y₂ - y₁) over (x₂ - x₁). Do not swap the order.

  计算梯度时保持一致性：分子为 (y₂ - y₁)，分母为 (x₂ - x₁)，不要颠倒顺序。

• A negative gradient means the line goes down from left to right.

  梯度的符号为负时，直线从左到右下降。

• Remember that vertical lines have the form x = k, not y = mx + c.

  记住竖直直线的形式是 x = k，而不是 y = mx + c。

• When finding perpendicular lines, instead of the negative reciprocal, students often just take the reciprocal. Keep the minus sign!

  求垂线梯度时，学生常只取倒数而忘记变号。一定要保留负号！

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9. Practice Questions | 练习题

Try the following questions to test your understanding. Solutions are provided below.

请尝试以下题目来检验你的理解。答案附在题目之后。

1. Find the gradient of the line passing through (1, 2) and (4, 8).

   求过点 (1, 2) 和 (4, 8) 的直线的梯度。

2. Find the equation of the line with gradient -2 that passes through (3, 1).

   求梯度为 -2 且过点 (3, 1) 的直线方程。

3. Are the lines y = 3x + 2 and y = -⅓x + 5 perpendicular?

   直线 y = 3x + 2 与 y = -⅓x + 5 是否垂直？

4. Solve for the intersection of y = x + 2 and y = -2x + 8.

   求直线 y = x + 2 和 y = -2x + 8 的交点。

5. Find the equation of the line parallel to y = 4x - 1 passing through (0, 6).

   求与 y = 4x - 1 平行且过点 (0, 6) 的直线方程。

Solutions:

答案：

1. m = (8 - 2) / (4 - 1) = 6/3 = 2.

   m = (8 - 2) / (4 - 1) = 6/3 = 2。

2. y - 1 = -2(x - 3) ⇒ y = -2x + 7.

   y - 1 = -2(x - 3) ⇒ y = -2x + 7。

3. 3 × (-⅓) = -1. Yes, they are perpendicular.

   3 × (-⅓) = -1。是的，它们互相垂直。

4. x + 2 = -2x + 8 ⇒ 3x = 6 ⇒ x = 2, y = 4. Intersection at (2, 4).

   x + 2 = -2x + 8 ⇒ 3x = 6 ⇒ x = 2，y = 4。交点为 (2, 4)。

5. m = 4, c = 6, so y = 4x + 6.

   m = 4，c = 6，因此 y = 4x + 6。

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