📚 Coordinate Geometry: Straight Line Graphs | 坐标几何：直线图像
Straight-line graphs are one of the most important topics in IGCSE Mathematics. They connect algebra and geometry, allowing us to describe relationships, predict values, and solve real-world problems. This revision guide covers every key skill you need: coordinates, distance, midpoint, gradient, equations, parallel and perpendicular lines, intercepts, and drawing graphs.
直线图像是 IGCSE 数学中最重要的内容之一。它将代数与几何联系起来，帮助我们描述关系、预测数值并解决实际问题。本复习指南涵盖你需要的全部关键技能：坐标、距离、中点、斜率、方程、平行与垂直直线、截距以及绘图。
1. The Coordinate Plane | 坐标平面
The Cartesian coordinate plane is formed by a horizontal x-axis and a vertical y-axis. Their intersection is called the origin, written as O(0, 0). The plane is divided into four quadrants, which are numbered anticlockwise starting from the top-right region.
笛卡尔坐标平面由水平 x 轴和垂直 y 轴构成。它们的交点称为原点，记作 O(0, 0)。平面被分成四个象限，从右上角开始按逆时针方向编号。
Every point is written as (x, y), with the x-coordinate first and the y-coordinate second. For example, (−3, 2) means move 3 units left and 2 units up from the origin. Always read the x-coordinate before the y-coordinate; this order is essential.
每个点写作 (x, y)，先写 x 坐标，再写 y 坐标。例如，(−3, 2) 表示从原点向左移动 3 个单位，再向上移动 2 个单位。务必先读 x 坐标，再读 y 坐标；这个顺序至关重要。
2. Plotting Points and Distance | 标点与距离
To plot a point, start at the origin and move horizontally according to the x-coordinate, then vertically according to the y-coordinate. For example, point A(2, 3) is found by moving 2 units right and 3 units up.
标点时，从原点出发，按照 x 坐标水平移动，再按照 y 坐标垂直移动。例如，点 A(2, 3) 就是向右移动 2 个单位，再向上移动 3 个单位。
The distance between two points A(x₁, y₁) and B(x₂, y₂) is found using Pythagoras' theorem. This formula is often called the distance formula:
两点 A(x₁, y₁) 和 B(x₂, y₂) 之间的距离可以用勾股定理求出。这个公式通常称为距离公式：
AB = √((x₂ − x₁)² + (y₂ − y₁)²)
Example: For A(1, 2) and B(4, 6), substitute the values: AB = √((4 − 1)² + (6 − 2)²) = √9 + 16 = √25 = 5. The distance is exactly 5 units.
例如：对于 A(1, 2) 和 B(4, 6)，代入数值：AB = √((4 − 1)² + (6 − 2)²) = √9 + 16 = √25 = 5。距离正好为 5 个单位。
3. Midpoint of a Line Segment | 线段中点
The midpoint of a line segment is exactly halfway between its two endpoints. To find it, we average the x-coordinates and average the y-coordinates.
线段的中点正好位于两个端点之间的一半位置。求中点时，我们取 x 坐标的平均值和 y 坐标的平均值。
M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
= (2, 5) and B = (8, 9), then M = ((2 + 8)/2, (5 + 9)/2) = (5, 7). The midpoint is (5, 7).
例如，若 A = (2, 5)，B = (8, 9)，则 M = ((2 + 8)/2, (5 + 9)/2) = (5, 7)。中点坐标为 (5, 7)。
oint is (−4, 0) and the other is (6, 6), the midpoint is ((−4 + 6)/2, (0 + 6)/2) = (1, 3).
该公式对于负数坐标同样适用。若一个端点为 (−4, 0)，另一个为 (6, 6)，则中点为 ((−4 + 6)/2, (0 + 6)/2) = (1, 3)。
4. Gradient of a Straight Line | 直线斜率
The gradient of a line measures how steep it is. It is the ratio of the vertical change to the horizontal change between two points on the line. It is often described as 'rise over run'.
直线的斜率衡量直线的倾斜程度。它是直线上两点之间纵坐标变化量除以横坐标变化量，通常称为“垂直变化比水平变化”。
m = (y₂ − y₁)/(x₂ − x₁)
Using two points A(x₁, y₁) and B(x₂, y₂), the gradient m is found by subtracting the y-coordinates and dividing by the difference of the x-coordinates. The denominator cannot be zero, which is why a vertical line has no defined gradient.
对于两点 A(x₁, y₁) 和 B(x₂, y₂)，斜率 m 等于两个 y 坐标之差除以两个 x 坐标之差。分母不能为零，因此垂直线没有定义的斜率。
A positive gradient means the line slopes upwards from left to right. A negative gradient means the line slopes downwards from left to right. A horizontal line has gradient 0, and a vertical line has an undefined gradient.
正斜率表示直线从左到右上升，负斜率表示直线从左到右下降。水平线的斜率为 0，垂直线的斜率无定义。
(6 − 2)/(4 − 1) = 4/3. This means the line rises 4 units for every 3 units it moves right.
例如：经过 (1, 2) 和 (4, 6) 的直线斜率 m = (6 − 2)/(4 − 1) = 4/3。这意味着直线向右移动 3 个单位，就会上升 4 个单位。
5. Equation of a Straight Line: y = mx + c | 直线方程：y = mx + c
e equation of a straight line is y = mx + c. This is known as the gradient-intercept form.
直线方程最有用的形式是 y = mx + c，称为“斜率-截距式”。
y = mx + c
In this equation, m represents the gradient of the line, and c represents the y-intercept. The y-intercept is the y-coordinate where the line crosses the y-axis, so the point is (0, c).
在这个方程中，m 表示直线的斜率，c 表示 y 轴截距。y 轴截距是直线与 y 轴交点的 y 坐标，因此该点为 (0, c)。
mple, 3x + 2y = 6 becomes y = −1.5x + 3, so the gradient is −1.5 and the y-intercept is 3.
例如，y = 2x − 3 的斜率为 2，并且直线在 (0, −3) 处穿过 y 轴。如果方程以其他形式给出，应先将其整理为 y = mx + c。例如，3x + 2y = 6 可化为 y = −1.5x + 3，因此斜率为 −1.5，y 轴截距为 3。
tical lines have the form x = k. A vertical line cannot be written in the form y = mx + c.
请记住，水平线的方程为 y = k，垂直线的方程为 x = k。垂直线不能写成 y = mx + c 的形式。
6. Finding the Equation from Two Points | 由两点求直线方程
Given two points, we can find the equation of the line that passes through them by first calculating the gradient and then using the point-gradient formula.
已知两点时，可以先计算斜率，然后使用“点斜式”公式求出经过这两点的直线方程。
y − y₁ = m(x − x₁)
gradient. After substituting the values, rearrange the equation into the form y = mx + c.
其中 (x₁, y₁) 是直线上任意一个已知点，m 为斜率。代入数值后，把方程整理成 y = mx + c 的形式。
− 2) = 4/3. Using A(2, 3), we get y − 3 = 4/3(x − 2). Simplifying gives y = (4/3)x + 1/3.
例题：求经过 A(2, 3) 和 B(5, 7) 的直线方程。首先，m = (7 − 3)/(5 − 2) = 4/3。使用 A(2, 3)，得 y − 3 = 4/3(x − 2)。整理后得到 y = (4/3)x + 1/3。
e second point. For B(5, 7), y = (4/3)(5) + 1/3 = 20/3 + 1/3 = 21/3 = 7, which is correct.
一定要代入第二个点检查答案。对于 B(5, 7)，y = (4/3)(5) + 1/3 = 20/3 + 1/3 = 21/3 = 7，结果正确。
7. Parallel and Perpendicular Lines | 平行与垂直直线
ercepts. For example, y = 3x + 2 and y = 3x − 5 are parallel because both have gradient 3.
如果两条直线斜率相同但 y 轴截距不同，则它们互相平行。例如，y = 3x + 2 和 y = 3x − 5 平行，因为两条直线的斜率都为 3。
−1. In symbols, if m₁ and m₂ are the gradients of perpendicular lines, then m₁ × m₂ = −1.
如果两条直线互相垂直，则它们斜率的乘积为 −1。用符号表示：若 m₁ 和 m₂ 是两条垂直直线的斜率，则 m₁ × m₂ = −1。
m₁ × m₂ = −1
endicular because 2 × (−1/2) = −1. A line perpendicular to gradient 4/3 has gradient −3/4.
例如，y = 2x + 1 和 y = −(1/2)x + 4 互相垂直，因为 2 × (−1/2) = −1。若一条直线斜率为 4/3，则与它垂直的直线斜率为 −3/4。
It is also important to remember that horizontal and vertical lines are perpendicular to each other. The gradient of a horizontal line is 0, while the gradient of a vertical line is undefined.
还需记住，水平线和垂直线互相垂直。水平线的斜率为 0，而垂直线的斜率无定义。
8. x- and y-Intercepts | 截距
the equation. The x-intercept is where the line crosses the x-axis; to find it, set y = 0.
直线的 y 轴截距是它与 y 轴的交点。求 y 轴截距时，令 x = 0。x 轴截距是直线与 x 轴的交点；求 x 轴截距时，令 y = 0。
y-intercept is (0, −4). Set y = 0 to get 0 = 2x − 4, so x = 2. The x-intercept is (2, 0).
例如：对于 y = 2x − 4，令 x = 0，得 y = −4，所以 y 轴截距为 (0, −4)。令 y = 0，得 0 = 2x − 4，解得 x = 2。因此 x 轴截距为 (2, 0)。
= c, the x-intercept is c/a (provided a ≠ 0) and the y-intercept is c/b (provided b ≠ 0).
若直线方程写成 ax + by = c，则 x 轴截距为 c/a（a ≠ 0），y 轴截距为 c/b（b ≠ 0）。
9. Drawing Straight Line Graphs | 绘制直线图像
To draw a straight-line graph accurately, create a table of values. Choose at least three x-values, substitute each into the equation, and calculate the corresponding y-values.
要准确绘制直线图像，可以制作一个数值表。至少选择三个 x 值，把每个 x 值代入方程，计算出对应的 y 值。
For example, to draw y = 2x + 1, choose x = −1, 0, 1, 2:
例如，为了绘制 y = 2x + 1，选择 x = −1, 0, 1, 2：
| x | −1 | 0 | 1 | 2 |
| y = 2x + 1 | −1 | 1 | 3 | 5 |
Plot the points (−1, −1), (0, 1), (1, 3), and (2, 5
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