📚 Straight Line Graphs: Gradient, Equation and Intercepts | 直线图像：斜率、方程与截距
, sequences, mensuration and, above all, in real-life modelling questions. Once you understand gradient, intercepts and the equation y = mx + c, you can plot any straight line, find equations from given conditions, and solve problems involving parallel or perpendicular lines.
在 IGCSE 数学中，直线图像是代数与几何之间的桥梁。它出现在坐标几何、函数、数列、测量以及最重要的实际情境建模题中。理解斜率、截距和方程 y = mx + c，你就能绘制任意直线、根据已知条件求方程，并解决平行线或垂直线的问题。
This revision guide follows the exam syllabus step by step, using the same vocabulary that appears in past papers: gradient, y-intercept, x-intercept, general form, negative reciprocal.
本复习指南按照考纲逐步展开，使用与历年真题一致的术语：斜率、y 轴截距、x 轴截距、一般式、负倒数。
1. The Coordinate Plane and Plotting Points | 坐标平面与描点
The coordinate plane is formed by two perpendicular number lines. The horizontal line is the x-axis, the vertical line is the y-axis, and they meet at the origin, written as (0, 0). The axes divide the plane into four regions called quadrants, numbered anticlockwise starting from the top right.
坐标平面由两条互相垂直的数轴组成。水平线叫 x 轴，垂直线叫 y 轴，它们的交点叫原点，记作 (0, 0)。两条轴把平面分成四个区域，称为象限，从右上角开始逆时针编号。
A point is located by an ordered pair (x, y). The first number is the horizontal distance from the origin, and the second number is the vertical distance. For example, the point (−3, 2) lies 3 units left and 2 units up from the origin. The order matters: (−3, 2) and (2, −3) are completely different points.
一个点用有序数对 (x, y) 表示。第一个数字是从原点出发的水平距离，第二个数字是垂直距离。例如，点 (−3, 2) 表示从原点向左 3 个单位、向上 2 个单位。顺序很重要：(−3, 2) 与 (2, −3) 是完全不同的两个点。
. Label the point clearly if a question asks for coordinates. A common exam instruction is "plot the points and draw the graph of y = 3x − 1 for −2 ≤ x ≤ 3", which means making a table of values first.
描点时，始终从原点出发，先沿 x 轴移动，再沿 y 轴方向移动。如果题目要求写出坐标，要清楚地标出该点。真题中常见的指令是"描点并画出 y = 3x − 1 在 −2 ≤ x ≤ 3 上的图像"，这意味着你要先列出数值表。
- Right and up: positive x, positive y (first quadrant). 向右向上：x、y 均为正（第一象限）。
- Left and up: negative x, positive y (second quadrant). 向左向上：x 负 y 正（第二象限）。
- Left and down: negative x, negative y (third quadrant). 向左向下：x、y 均为负（第三象限）。
- Right and down: positive x, negative y (fourth quadrant). 向右向下：x 正 y 负（第四象限）。
2. What is a Linear Equation? | 什么是线性方程？
e, y = 2x + 3 is linear, but y = x² + 1 is not, because its graph is a parabola (a curve).
线性方程是图像为一条直线的方程。它不含高于 1 次的幂，例如 x² 或 x³。比如 y = 2x + 3 是线性方程，而 y = x² + 1 不是，因为它的图像是抛物线（曲线）。
The most useful form for a straight line is the gradient-intercept form:
直线最实用的形式是斜截式：
y = mx + c
itten in this form, including lines such as 2x + 3y = 12, which rearranges to y = −⅔x + 4.
这里 m 是斜率（倾斜程度），c 是 y 轴截距（直线与 y 轴交点的纵坐标）。任何不垂直的直线都能写成这个形式，包括 2x + 3y = 12 这样的方程，移项后得到 y = −⅔x + 4。
he question gives you, but the final answer is usually expected as y = mx + c unless the question says otherwise.
考试中其他可接受的形式包括一般式 ax + by + c = 0，以及点斜式 y − y₁ = m(x − x₁)。用哪种形式取决于题目给出的条件，但除非题目另有要求，最终答案通常写成 y = mx + c。
3. Gradient: Rise Over Run | 斜率：纵向变化除以横向变化
The gradient measures how steep a line is and in which direction it slopes. For two points on the line, the gradient is the change in y divided by the change in x:
斜率衡量直线的倾斜程度及其倾斜方向。对于直线上的两点，斜率等于 y 的变化量除以 x 的变化量：
m = (y₂ − y₁) ÷ (x₂ − x₁)
the order does not matter as long as you subtract in the same direction for both coordinates.
例题：求经过 (1, 2) 和 (4, 11) 两点的直线的斜率。y 的变化量是 11 − 2 = 9，x 的变化量是 4 − 1 = 3，所以 m = 9 ÷ 3 = 3。注意只要两个坐标按同一方向相减，先减哪个点都可以。
For example, the line through (0, 5) and (4, −3) has m = (−3 − 5) ÷ (4 − 0) = −8 ÷ 4 = −2.
负斜率表示当你向右移动时直线是下降的。例如，经过 (0, 5) 和 (4, −3) 的直线，m = (−3 − 5) ÷ (4 − 0) = −8 ÷ 4 = −2。
In real-life graphs, the gradient is a rate. On a distance-time graph, the gradient is speed. On a cost-quantity graph, the gradient is the price per unit.
在实际情境图像中，斜率就是一个变化率。在路程—时间图上，斜率是速度；在费用—数量图上，斜率是单价。
4. Types of Gradient | 斜率的四种类型
Every straight line falls into one of four gradient categories. Recognising them instantly saves time in multiple-choice and sketch questions.
每条直线都属于以下四种斜率类型之一。迅速识别类型能在选择题和作图题中节省大量时间。
| Type 类型 | Gradient 斜率 | Example 例子 |
| Rising 上升 | m > 0 | y = 2x + 1 |
| Falling 下降 | m < 0 | y = −3x + 4 |
| Horizontal 水平 | m = 0 | y = 5 |
| Vertical 垂直 | undefined 不存在 | x = 3 |
A horizontal line has zero gradient because the change in y is zero. A vertical line has an undefined gradient because the change in x is zero, and division by zero is not allowed.
水平线的斜率为 0，因为 y 的变化量为 0。垂直线的斜率不存在，因为 x 的变化量为 0，而除以 0 是不允许的。
se of the axes themselves: the x-axis is the line y = 0, and the y-axis is the line x = 0.
还要注意坐标轴本身这种特殊情况：x 轴是直线 y = 0，y 轴是直线 x = 0。
5. Understanding y = mx + c | 理解 y = mx + c
For example, in y = 3x − 5, the gradient is 3 and the line crosses the y-axis at (0, −5).
在方程 y = mx + c 中，m 的值给出斜率，c 的值给出 y 轴截距。例如在 y = 3x − 5 中，斜率是 3，直线与 y 轴交于 (0, −5)。
The fastest way to sketch such a line is: plot the y-intercept on the axis, then use the gradient as a direction. A gradient of 3 means "go 1 to the right and 3 up". A gradient of −½ means "go 2 to the right and 1 down". A fraction such as ⅔ is easier to use as "right 3, up 2".
画这类直线最快的方法是：先在轴上标出 y 轴截距，再把斜率当作方向来用。斜率 3 表示"向右 1、向上 3"。斜率 −½ 表示"向右 2、向下 1"。像 ⅔ 这样的分数用"向右 3、向上 2"更方便。
Two points are enough to draw a straight line, and a third point is a good check. Always use a ruler, extend the line to the edge of the grid, and label the equation on the graph when asked.
两点即可确定一条直线，第三点则可以用来检验。务必用直尺作图，把直线延伸到网格边缘，并在题目要求时在图上标注方程。
6. Finding Intercepts | 求截距
he x-axis, so y = 0 there. Substituting these values gives you the intercepts immediately.
y 轴截距是直线与 y 轴的交点，那里 x = 0。x 轴截距是直线与 x 轴的交点，那里 y = 0。把这两个值代进去就能立刻求出截距。
12, so x = 6 and the x-intercept is (6, 0). Joining these two points draws the whole line.
例题：求 2x + 3y = 12 的两个截距。令 x = 0，得 3y = 12，所以 y = 4，y 轴截距是 (0, 4)。令 y = 0，得 2x = 12，所以 x = 6，x 轴截距是 (6, 0)。连接这两点即可画出整条直线。
an making a full table of values, especially when the equation is in the form ax + by = c.
当方程形如 ax + by = c 时，截距法通常比列完整数值表更快。
If a question asks where two lines meet, solve the two equations simultaneously. The solution is exactly the coordinates of the intersection point, which you can verify on the graph.
如果题目问两条直线在哪里相交，就联立解这两个方程。方程组的解正是交点坐标，可以在图像上验证。
7. Finding the Equation from Two Points | 由两点求直线方程
When you are given two points, follow a fixed three-step method: find the gradient, substitute one point to find c, then write the final equation.
当题目给出两点时，按固定的三步法：先求斜率，再代入一点求 c，最后写出最终方程。
itute (2, 1) into y = 3x + c, so 1 = 6 + c and c = −5. Step 3: the equation is y = 3x − 5.
例题：求经过 (2, 1) 和 (6, 13) 的直线方程。第一步，m = (13 − 1) ÷ (6 − 2) = 12 ÷ 4 = 3。第二步，把 (2, 1) 代入 y = 3x + c，得 1 = 6 + c，所以 c = −5。第三步，方程为 y = 3x − 5。
8 − 5 = 13, which matches. This check takes ten seconds and catches most arithmetic slips.
一定要用另一点检验：代入 x = 6 得 y = 18 − 5 = 13，与已知相符。这个检验只需十秒，却能抓出大多数计算失误。
both have the same x-coordinate, the line is vertical and the equation is x = that value.
如果两点的 y 坐标相同，直线是水平的，方程就是 y 等于该值。如果两点的 x 坐标相同，直线是垂直的，方程就是 x 等于该值。
8. Finding the Equation from Gradient and One Point | 由斜率和一点求直线方程
itute the gradient and the coordinates of the known point into y = mx + c and solve for c.
当斜率已知时，你只需要求 c。把斜率和已知点的坐标代入 y = mx + c，解出 c 即可。
Substituting gives 4 = −2 × 3 + c, so 4 = −6 + c and c = 10. The equation is y = −2x + 10.
例题：一条直线的斜率是 −2，且经过点 (3, 4)。代入得 4 = −2 × 3 + c，即 4 = −6 + c，所以 c = 10。方程是 y = −2x + 10。
expands to y = −2x + 10. Either method earns full marks, so use whichever you find safer.
点斜式可以更直接地得到同样答案：y − 4 = −2(x − 3) 展开得 y = −2x + 10。两种方法都能拿满分，选你更有把握的那个。
Read the question carefully for phrases such as "passes through", "cuts the y-axis at", or "has a gradient of". Each phrase tells you exactly which quantity is given.
仔细读题，注意"经过""与 y 轴交于""斜率为"这类说法，每一种说法都明确告诉你已知的是哪个量。
9. Parallel Lines | 平行线
rcepts. In symbols, if y = m₁x + c₁ is parallel to y = m₂x + c₂, then m₁ = m₂ and c₁ ≠ c₂.
平行线的斜率相同但截距不同。用符号表示，若 y = m₁x + c₁ 与 y = m₂x + c₂ 平行，则 m₁ = m₂ 且 c₁ ≠ c₂。
y = 4x + c. Substituting (2, 3) gives 3 = 8 + c, so c = −5 and the equation is y = 4x − 5.
例题：求与 y = 4x + 1 平行且经过 (2, 3) 的直线。斜率必须为 4，设 y = 4x + c。代入 (2, 3) 得 3 = 8 + c，所以 c = −5，方程为 y = 4x − 5。
Two vertical lines are always parallel to each other, and two horizontal lines are always parallel to each other. A vertical line and a horizontal line are never parallel.
两条垂直线彼此一定平行，两条水平线也彼此一定平行。垂直线与水平线永远不平行。
Never confuse parallel with perpendicular. Parallel means same gradient; perpendicular means gradients multiply to give −1.
不要把平行与垂直混淆。平行意味着斜率相同；垂直意味着两斜率之积为 −1。
10. Perpendicular Lines | 垂直线
Two lines are perpendicular if the product of their gradients is −1. Equivalently, the gradient of the perpendicular is the negative reciprocal of the original gradient.
如果两条直线的斜率之积为 −1，它们就互相垂直。等价地说，垂直线的斜率是原斜率的负倒数。
m₁ × m₂ = −1 m₂ = −1 ÷ m₁
is perpendicular to a line with gradient −½, because 2 × (−½) = −1. A line with gradient −⅗ is perpendicular to a line with gradient 5 ÷ 3.
例如，斜率为 2 的直线与斜率为 −½ 的直线垂直，因为 2 × (−½) = −1。斜率为 −⅗ 的直线与斜率为 5 ÷ 3 的直线垂直。
the perpendicular gradient is −½. Substituting (3, 6) gives 6 = −½ × 3 + c, so c = 7.5 and the equation is y = −½x + 7.5.
例题：求连接 A(1, 2) 与 B(5, 10) 的线段的垂直平分线。中点是 (3, 6)，AB 的斜率是 8 ÷ 4 = 2，所以垂直方向斜率为 −½。代入 (3, 6) 得 6 = −½ × 3 + c，所以 c = 7.5，方程为 y = −½x + 7.5。
e vertical gradient is undefined. The rule m₁ × m₂ = −1 does not apply to these two cases.
值得记住的特例：水平线与垂直线互相垂直，因为水平线斜率为 0，垂直线斜率不存在。此时 m₁ × m₂ = −1 这条规则并不适用。
11. Real-Life Applications and Modelling | 实际应用与建模
Linear models appear constantly in exam word problems. The fixed starting value becomes the y-intercept, and the constant rate of change becomes the gradient.
线性模型在考试应用题中反复出现。固定的初始值变成 y 轴截距，恒定的变化率变成斜率。
e cost in pounds and d is the distance in kilometres. The gradient 2.5 is the cost per kilometre, and the intercept 3 is the fixed fee.
例：出租车收固定起步价 3 英镑，另按每公里 2.5 英镑计费。模型为 C = 2.5d + 3，其中 C 是费用（英镑），d 是距离（公里）。斜率 2.5 是每公里费用，截距 3 是起步价。
hows that each degree Celsius is 1.8 degrees Fahrenheit, and the intercept 32 shows that 0 °C equals 32 °F.
例：温度换算用 F = 1.8C + 32。这里斜率 1.8 表示每升高 1 摄氏度等于升高 1.8 华氏度，截距 32 表示 0 °C 等于 32 °F。
On a distance-time graph, a steeper gradient means a faster speed, a horizontal section means the object is stationary, and a negative gradient means the object is returning. Always read the axes and units before interpreting the gradient.
在路程—时间图上，斜率越陡表示速度越快，水平段表示物体静止，负斜率表示物体返回。解释斜率前一定要先看清坐标轴和单位。
12. Common Mistakes and Exam Checklist | 常见错误与考试清单
Most lost marks in straight line questions come from a small number of predictable errors: reversing the subtraction in the gradient formula, losing a negative sign, or misreading c as the x-intercept.
直线题失分大多来自少数几个可预见的错误：斜率公式中减法的顺序颠倒、丢掉负号，或者把 c 误当成 x 轴截距。
- Write the formula before substituting, so the method is visible. 代入前先写出公式，让方法可见。
- Keep brackets around negative numbers, for example (−2)² or 3 − (−5). 给负数加括号，例如 (−2)² 或 3 − (−5)。
- Check the sign of the gradient by looking at the graph: down to the right means negative. 用图像检查斜率正负：向右下降为负。
- Substitute one of the original points back into your final equation. 把原始点之一代回最终方程检验。
- Give the answer in the form the question requests, usually y = mx + c. 按题目要求的形式作答，通常是 y = mx + c。
adients multiply to −1, and every non-vertical straight line can be written as y = mx + c.
最后记住核心结论：斜率等于纵向变化除以横向变化；平行线斜率相同；垂直线的斜率之积为 −1；每条不垂直的直线都可以写成 y = mx + c。
Practise by taking any two points, or any equation, and converting between the table, the graph and the equation. Fluency comes from moving between those three representations quickly.
练习方法：任取两点或任一方程，在数值表、图像与方程三种表示之间来回转换。三种表示之间快速切换，就能真正熟练。
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