Straight Line Graphs: Gradient, Intercept and y = mx + c | 直线图像：斜率、截距与 y = mx + c

📚 Straight Line Graphs: Gradient, Intercept and y = mx + c | 直线图像：斜率、截距与 y = mx + c

In IGCSE Mathematics, straight line graphs are a central topic that links algebra and geometry. You need to interpret and write equations in the form y = mx + c, calculate gradient, identify intercepts, and use these skills to solve coordinate problems. This article reviews the key ideas, worked methods and common pitfalls.

在 IGCSE 数学中，直线图像是连接代数与几何的核心主题。你需要理解并写出 y = mx + c 形式的方程，计算斜率，识别截距，并运用这些技能解决坐标问题。本文回顾关键概念、解题方法和常见错误。

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1. The Coordinate Plane and Linear Graphs | 坐标平面与线性图像

The coordinate plane is formed by a horizontal x-axis and a vertical y-axis crossing at the origin (0,0). A straight line graph is the set of points whose coordinates satisfy a linear equation in x and y.

坐标平面由水平的 x 轴和竖直的 y 轴组成，它们在原点 (0,0) 相交。直线图像是所有满足 x 与 y 线性方程的点组成的集合。

Every linear equation in two variables, such as y = 2x + 1, produces a straight line when plotted. This is why the topic is called straight line graphs.

每一个含有两个变量的线性方程，例如 y = 2x + 1，在描点后都会形成一条直线。因此该主题称为直线图像。

y = mx + c

The form y = mx + c is the most useful way to describe a straight line because the numbers m and c immediately give the steepness and vertical position.

y = mx + c 形式是描述直线最常用的方法，因为 m 和 c 直接给出了直线的倾斜程度和竖直位置。

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

In the equation y = mx + c, the coefficient m is called the gradient or slope. It tells you how much y changes when x increases by 1.

在方程 y = mx + c 中，系数 m 称为斜率或坡度。它告诉你当 x 增加 1 时，y 变化多少。

The constant c is called the y-intercept. It gives the value of y at the point where the line crosses the y-axis.

常数 c 称为 y 截距，它表示直线与 y 轴交点的 y 值。

m = Δy / Δx

A vertical line cannot be written in this form because its x-coordinate is constant. Instead, it is written as x = a, where a is the fixed x-value.

竖直直线不能写成这种形式，因为它的 x 坐标保持不变。它应写为 x = a，其中 a 是固定的 x 值。

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3. Gradient: Steepness and Direction | 斜率：陡峭程度与方向

Gradient measures steepness and direction. For two points (x₁, y₁) and (x₂, y₂), the gradient is:

斜率衡量陡峭程度和方向。对于两点 (x₁, y₁) 和 (x₂, y₂)，斜率为：

m = (y₂ − y₁) / (x₂ − x₁)

The order of subtraction must be the same in the numerator and denominator. Swapping both points gives the same result.

分子和分母的相减顺序必须一致。两个点交换后结果不变。

Example: For (1,2) and (3,8), m = (8 − 2) / (3 − 1) = 6 / 2 = 3. A gradient of 3 means y increases by 3 for each 1-unit increase in x.

示例：对于 (1,2) 和 (3,8)，m = (8 − 2) / (3 − 1) = 6 / 2 = 3。斜率为 3 表示 x 每增加 1，y 增加 3。

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4. Positive, Negative, Zero and Undefined Gradients | 正、负、零与未定义斜率

The sign of the gradient tells you the direction of the line. A positive gradient slopes upward from left to right, while a negative gradient slopes downward.

斜率的符号告诉你直线的方向。正斜率从左到右向上倾斜，负斜率从左到右向下倾斜。

A horizontal line has gradient 0 because there is no vertical change. Its equation is y = c, where c is a constant.

水平线的斜率为 0，因为没有竖直变化。其方程为 y = c，其中 c 为常数。

A vertical line has undefined gradient because the x-change is 0. Its equation is x = a.

竖直线的斜率未定义，因为 x 的变化量为 0。其方程为 x = a。

Type | 类型	Gradient | 斜率	Equation | 方程
Horizontal | 水平线	0	y = c
Vertical | 竖直线	Undefined | 未定义	x = a
Increasing | 上升直线	Positive | 正数	y = mx + c, m > 0
Decreasing | 下降直线	Negative | 负数	y = mx + c, m < 0

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5. Y-intercept and X-intercept | y 截距与 x 截距

The y-intercept c is found by substituting x = 0 into the equation. It is the starting value of y on the graph.

y 截距 c 通过将 x = 0 代入方程求得。它是图像上 y 的起始值。

The x-intercept is found by substituting y = 0 and solving for x. It is where the line crosses the x-axis.

x 截距通过将 y = 0 代入并解出 x 求得。它是直线与 x 轴的交点。

Example: For y = 2x + 6, the y-intercept is 6. Setting y = 0 gives 2x + 6 = 0, so x = −3. The intercepts are (0,6) and (−3,0).

示例：对于 y = 2x + 6，y 截距为 6。令 y = 0 得到 2x + 6 = 0，因此 x = −3。截距为 (0,6) 和 (−3,0)。

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6. Plotting a Straight Line from Its Equation | 根据方程绘制直线

A table of values is a reliable way to plot a straight line. Choose at least three x-values, calculate the corresponding y-values, and plot the coordinate pairs.

数值表是绘制直线的可靠方法。至少选择三个 x 值，计算相应的 y 值，并在坐标系中描出对应点。

For y = 2x + 1, choose x = −1, 0, 1 and 2. The y-values are −1, 1, 3 and 5, giving points (−1,−1), (0,1), (1,3) and (2,5).

对于 y = 2x + 1，选择 x = −1、0、1 和 2，对应的 y 值为 −1、1、3 和 5，得到点 (−1,−1)、(0,1)、(1,3) 和 (2,5

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