📚 IGCSE Maths G-1: Straight Line Graphs | IGCSE 数学 G-1：直线图像
In IGCSE Maths, straight line graphs connect algebra with geometry. They help you understand how an equation describes a line, how to calculate its steepness or gradient, and how to find where two lines cross. This unit covers the key ideas needed for both Foundation and Higher tier questions.
在 IGCSE 数学中，直线图像把代数与几何联系起来。它们帮助你理解方程如何描述一条直线、如何计算直线的倾斜程度或斜率，以及如何找到两条直线的交点。本单元涵盖基础层级和高级层级题目所需的核心概念。
1. The Coordinate Plane | 坐标平面
A coordinate plane is formed by a horizontal x-axis and a vertical y-axis. The axes intersect at the origin, written as (0, 0).
坐标平面由水平的 x 轴和竖直的 y 轴组成。两轴在原点相交，原点记作 (0, 0)。
A point is described by an ordered pair (x, y). The first number tells you how far to move horizontally from the origin, and the second number tells you how far to move vertically.
一个点用有序数对 (x, y) 表示。第一个数表示从原点沿水平方向移动的距离，第二个数表示沿竖直方向移动的距离。
- English: The point (3, −2) means 3 units right and 2 units down.
- 中文：点 (3, −2) 表示向右 3 个单位、向下 2 个单位。
2. Plotting Points | 描点
To plot a point, start at the origin. Move along the x-axis first, then move parallel to the y-axis. A common mistake is reversing the coordinates.
描点时，从原点开始。先沿 x 轴移动，再沿与 y 轴平行的方向移动。常见错误是把两个坐标的顺序颠倒。
When you plot several points that satisfy a linear equation, they always lie on a straight line. This is why the equation is called linear.
当你描出满足某个线性方程的若干个点时，它们总是落在同一条直线上。这就是这类方程被称为线性方程的原因。
3. What is a Linear Equation? | 什么是线性方程
where m and c are constants. The highest power of x is 1, so the graph is a straight line.
含有两个变量的线性方程通常可以写成 y = mx + c，其中 m 和 c 是常数。x 的最高次数是 1，因此它的图像是一条直线。
+ 5, and y = 1/2x − 4. Equations such as x² + y = 1 are not linear because of the x² term.
例如 y = 2x + 3、y = −x + 5 和 y = 1/2x − 4。像 x² + y = 1 这样的方程不是线性方程，因为它含有 x² 项。
4. y = mx + c: Gradient and Intercept | y = mx + c：斜率与截距
gradient or slope, and c is the y-intercept. The y-intercept is the value of y when x = 0.
在 y = mx + c 中，m 是斜率或坡度，c 是 y 轴截距。y 轴截距就是当 x = 0 时 y 的值。
The gradient tells you how steep the line is and whether it goes up or down as you move to the right. A positive m gives an upward slope; a negative m gives a downward slope.
斜率表示直线的倾斜程度，以及当 x 向右移动时直线是上升还是下降。m 为正时，直线向上倾斜；m 为负时，直线向下倾斜。
| Equation | Gradient m | y-intercept c |
|---|---|---|
| y = 3x + 1 | 3 | 1 |
| y = −2x + 7 | −2 | 7 |
| y = x − 4 | 1 | −4 |
5. Calculating Gradient | 计算斜率
two points (x₁, y₁) and (x₂, y₂) is given by m = (y₂ − y₁)/(x₂ − x₁). This is often called 'rise over run'.
两点 (x₁, y₁) 和 (x₂, y₂) 之间的斜率由 m = (y₂ − y₁)/(x₂ − x₁) 给出。这通常称为“竖直变化量除以水平变化量”。
m = (8 − 2)/(4 − 1) = 6/3 = 2. This means y increases by 2 for every 1 unit increase in x.
如果一条直线经过 (1, 2) 和 (4, 8)，则 m = (8 − 2)/(4 − 1) = 6/3 = 2。这意味着 x 每增加 1 个单位，y 增加 2。
Always keep the subtraction order the same in the numerator and denominator. Swapping the order in one part changes the sign of the gradient.
分子和分母的相减顺序必须始终保持一致。如果只改变其中一个部分的顺序，就会改变斜率的正负号。
6. Finding the y-intercept | 求 y 轴截距
tuting a known point and the gradient into y = mx + c. Solve the resulting equation for c.
你可以把一个已知点和斜率代入 y = mx + c 来求 c，然后解方程得到 c。
2 and the line passes through (1, 2), then 2 = 2(1) + c, so c = 0. The equation is y = 2x.
例如，如果 m = 2 且直线经过 (1, 2)，则 2 = 2(1) + c，因此 c = 0。所求方程为 y = 2x。
Alternatively, read c directly from the graph where the line crosses the y-axis. This is usually quicker for a fully drawn graph.
或者，直接从图像上读取直线与 y 轴相交处的 c 值。如果图像已经完整画出，这通常更快。
7. Drawing a Straight Line Graph | 绘制直线图像
One method is to create a table of values. Choose at least three x-values, substitute them into the equation, and find the matching y-values.
一种方法是列数值表。至少选择三个 x 值，代入方程，求出对应的 y 值。
Plot the ordered pairs and join them with a ruler. If a point does not lie on the line, check your arithmetic because all points should be collinear.
描出这些有序数对，并用直尺连接。如果某个点不在直线上，检查你的计算，因为所有点都应在同一条直线上。
ntercept and gradient: plot c on the y-axis, then use m = rise/run to find a second point.
另一种方法利用 y 轴截距和斜率：在 y 轴上标出 c，然后利用 m = 竖直变化/水平变化 找到第二个点。
8. Finding the Equation from a Graph | 由图像求方程
To find the equation of a straight line from its graph, first identify two clear points on the line. Use them to calculate the gradient m.
要根据图像求直线方程，首先在直线上找出两个清晰点。用它们计算斜率 m。
the line crosses the y-axis; this gives c. Then write the equation in the form y = mx + c.
接着，找出直线与 y 轴的交点；这就得到 c。然后把方程写成 y = mx + c 的形式。
on the graph, substitute one known point and the gradient into y = mx + c to solve for c.
如果在图像上看不到 y 轴截距，可以把一个已知点和斜率代入 y = mx + c，解出 c。
9. Parallel and Perpendicular Lines | 平行线与垂直线
ame gradient. For example, y = 2x + 1 and y = 2x − 5 are parallel because both have m = 2.
平行线具有相同的斜率。例如，y = 2x + 1 和 y = 2x − 5 平行，因为两者的 m 都是 2。
Perpendicular lines have gradients whose product is −1. If one line has gradient m, a perpendicular line has gradient −1/m.
垂直线的斜率乘积为 −1。如果一条直线的斜率是 m，那么与它垂直的直线斜率是 −1/m。
endicular line has m = −1/3. Take care with the sign when finding the negative reciprocal.
例如，如果一条直线的 m = 3，那么垂线的 m = −1/3。求负倒数时要注意符号。
10. Intersection of Two Lines | 两直线的交点
The intersection point of two straight lines is the solution of their simultaneous equations. At that point, both equations give the same x and y values.
两条直线的交点就是它们联立方程的解。在该点，两个方程给出的 x 和 y 值相同。
You can find the intersection graphically by plotting both lines and reading the coordinates where they cross.
你可以通过绘制两条直线并读取它们相交处的坐标来从图像上找到交点。
Algebraically, set the two expressions for y equal to each other and solve for x. Then substitute x back to find y.
在代数方法中，令两个 y 的表达式相等，解出 x，再把 x 代回原式求 y。
11. Real-World Applications | 实际应用
Straight line models are used to represent constant rates, such as a mobile phone charge with a fixed monthly fee plus a cost per minute.
直线模型可用来表示恒定变化率，例如手机费用包含固定月租再加上每分钟通话费。
In such problems, the fixed amount is the y-intercept c, and the rate per unit is the gradient
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