📚 Straight Line Graphs: Gradient, Intercept and y = mx + c | 直线图像：斜率、截距与 y = mx + c
ify 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。
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。
oduces 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
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。
2. The General Equation of a Straight Line | 直线的一般方程
ent 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
s 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。
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。
(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。
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。
dient 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 |
5. Y-intercept and X-intercept | y 截距与 x 截距
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。
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。
cept 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)。
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。
, 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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