📚 Linear Graphs: Plotting Straight Lines from y = mx + c | 线性图像：从 y = mx + c 绘制直线
This guide covers the essential skills needed to understand and plot linear graphs in the Cambridge KS3 Mathematics curriculum. You will learn how to identify the gradient and the y-intercept, draw straight lines from their equations, read information off a graph, and use linear models to solve real problems.
本指南涵盖 Cambridge KS3 数学课程中理解和绘制线性图像所需的核心技能。你将学习如何识别斜率与 y 轴截距，根据方程绘制直线，从图像中读取信息，并利用线性模型解决实际问题。
1. Coordinates and the Cartesian Plane | 坐标与笛卡尔平面
A Cartesian plane is formed by two perpendicular number lines called the x-axis (horizontal) and the y-axis (vertical). The point where they meet is the origin, written as (0, 0). Every point on the plane is described by an ordered pair (x, y), where x is the horizontal distance from the origin and y is the vertical distance.
笛卡尔平面由两条互相垂直的数轴组成，分别称为 x 轴（水平）和 y 轴（竖直）。它们的交点称为原点，记作 (0, 0)。平面上的每个点都用一个有序数对 (x, y) 来描述，其中 x 是到原点的水平距离，y 是垂直距离。
The two axes divide the plane into four regions called quadrants. Moving to the right is positive on the x-axis; moving up is positive on the y-axis. Quadrant I contains points with x > 0 and y > 0, Quadrant II has x < 0 and y > 0, Quadrant III has x < 0 and y < 0, and Quadrant IV has x > 0 and y < 0.
两条轴把平面分成四个区域，称为象限。沿 x 轴向右为正，沿 y 轴向上为正。第一象限的点满足 x > 0 且 y > 0；第二象限 x < 0、y > 0；第三象限 x < 0、y < 0；第四象限 x > 0、y < 0。
For example, the point (3, -2) means move 3 units to the right along the x-axis, then 2 units down parallel to the y-axis. It always matters which coordinate comes first: (3, -2) and (-2, 3) are two completely different points, sitting in different quadrants.
例如，点 (3, -2) 表示沿 x 轴向右移动 3 个单位，再沿 y 轴方向向下移动 2 个单位。坐标的顺序非常重要：(3, -2) 和 (-2, 3) 是两个完全不同的点，位于不同象限。
When you draw a graph by hand, always label both axes, mark the scale clearly at regular intervals, and use a ruler for the axes as well as for the line itself. Neat graphs make errors obvious and earn method marks in tests.
手绘图像时，务必标注两条轴、按等距标明刻度，并用直尺画轴和直线。作图整洁既能让错误一目了然，也能在考试中拿到过程分。
2. What Makes a Graph Linear? | 什么样的图像是线性的？
mbers. The key feature is that x appears to the first power only: there are no x² terms, no 1 ÷ x terms, and no square-root terms.
线性图像就是一条直线。它的方程总能整理成 y = mx + c 的形式，其中 m 和 c 是常数。其关键特征是 x 只出现一次幂：不含 x² 项、不含 1 ÷ x 项，也不含根号项。
Because a straight line has a constant rate of change, equal steps along the x-axis always produce equal steps along the y-axis. In a table of values this shows up as a constant difference between successive y-values, which is a very quick way to check whether a set of points is really linear.
因为直线的变化率恒定，所以 x 每增加相同的一步，y 的变化也相同。在数值表中，这表现为相邻 y 值之差恒定，这是判断一组点是否真正共线的最快方法。
y = m x + c
ast, y = x² + 1 is not linear, because its graph is a curve (a parabola) rather than a straight line.
任何能整理成该形式的方程都是线性的。例如 2y = 6x + 8 是线性的，因为两边同时除以 2 得到 y = 3x + 4。相反，y = x² + 1 不是线性的，因为它的图像是曲线（抛物线）而不是直线。
Two points are enough to fix a straight line, but examiners usually expect at least three points plotted and joined with a ruler, so that a plotting mistake is immediately visible.
两点即可确定一条直线，但考试通常要求至少 plotting 三个点并用直尺连成直线，这样一旦描点出错就能立刻发现。
3. The Gradient m: Steepness and Direction | 斜率 m：陡峭程度与方向
The gradient m measures how steep the line is and in which direction it slopes. It is defined as the vertical change divided by the horizontal change between any two points on the line. You will often hear this described as "rise over run".
斜率 m 衡量直线的陡峭程度以及倾斜方向。它定义为直线上任意两点之间的竖直变化量除以水平变化量，常被形象地称为"纵向变化除以横向变化"。
m = (y₂ - y₁) ÷ (x₂ - x₁)
Choose any two points (x₁, y₁) and (x₂, y₂) that lie on the line and substitute their coordinates into the formula. Because the line is straight, it does not matter which pair of points you pick: the answer is always the same.
在直线上任取两点 (x₁, y₁) 和 (x₂, y₂)，把坐标代入公式即可。由于直线是笔直的，取哪一对点都无所谓，结果永远相同。
- If m > 0, the line rises from left to right (positive gradient). 若 m > 0，直线从左向右上升（正斜率）。
- If m < 0, the line falls from left to right (negative gradient). 若 m < 0，直线从左向右下降（负斜率）。
- If m = 0, the line is horizontal. 若 m = 0，直线水平。
- A vertical line has an undefined gradient, because the horizontal change is zero and you cannot divide by zero. 竖直线的斜率不存在，因为水平变化量为零，不能除以零。
a line with m = ½. A small gradient such as m = 0.1 describes a line that is almost flat.
陡峭程度随 |m| 的增大而增大。m = 3 的直线比 m = ½ 的直线陡得多；m = 0.1 这样的斜率则描绘出一条几乎水平的直线。
4. The y-intercept c | y 轴截距 c
uting x = 0 into the equation gives y = c straight away. The intercept is therefore the point (0, c).
y = mx + c 中的 c 就是 y 轴截距：即直线与 y 轴交点处的 y 坐标。交点处恰好 x = 0，因此把 x = 0 代入方程立刻得到 y = c。所以截距点就是 (0, c)。
thing. What matters is that the term written alone, without an x attached to it, is the intercept.
有些教材把方程写成 y = c + mx，含义完全相同。关键是：那个没有与 x 相乘、单独出现的项就是截距。
- y = 2x + 3 crosses the y-axis at (0, 3). y = 2x + 3 与 y 轴交于 (0, 3)。
- y = -x + 4 crosses the y-axis at (0, 4). y = -x + 4 与 y 轴交于 (0, 4)。
- y = ½x - 1 crosses the y-axis at (0, -1). y = ½x - 1 与 y 轴交于 (0, -1)。
- y = 5x crosses the y-axis at the origin, (0, 0), because c = 0. y = 5x 经过原点 (0, 0)，因为 c = 0。
The y-intercept tells you where to start. Once you have marked (0, c) on the grid, the gradient tells you where to go next. Together they let you draw a line accurately without making a table at all.
y 轴截距告诉你从哪里开始。在网格上标出 (0, c) 之后，斜率就告诉你下一步往哪里走。二者结合，你完全可以不列数值表就准确画出直线。
5. Reading y = mx + c: Worked Examples | 解读 y = mx + c：例题精讲
The most common exam task is simply to state the gradient and the y-intercept of a given equation. The skill is to identify m and c correctly, especially when the equation is not yet in the standard form.
最常见的考题就是直接说出给定方程的斜率与 y 轴截距。关键技能是正确辨认 m 和 c，尤其是在方程尚未写成标准形式的时候。
| Equation 方程 | Gradient m 斜率 | y-intercept 截距 |
|---|---|---|
| y = 4x + 7 | 4 | (0, 7) |
| y = -3x + 2 | -3 | (0, 2) |
| y = ⅔x - 5 | ⅔ | (0, -5) |
| y = 8 - 2x | -2 | (0, 8) |
| 3x + y = 5 | -3 | (0, 5) |
| 2y = 6x + 8 | 3 | (0, 4) |
not 8. Always put the x-term in front of the constant term before reading off the values, because a careless reading here loses marks very quickly.
注意第四行：y = 8 - 2x 等同于 y = -2x + 8，所以斜率是 -2，而不是 8。读值前一定要先把含 x 的项写在常数项前面，因为这里一粗心就很容易丢分。
y = 3x + 4. Rearranging into y = mx + c is a skill worth practising until it is automatic.
最后两行必须先变形。由 3x + y = 5 两边同减 3x 得 y = -3x + 5；由 2y = 6x + 8 各项同除以 2 得 y = 3x + 4。把方程整理成 y = mx + c 是一项值得反复练习到自动化的技能。
6. Plotting a Line Using a Table of Values | 用数值表绘制直线
The table-of-values method is the safest approach when the equation is unfamiliar or when the gradient is a fraction. Choose a small set of x-values, usually -2, -1, 0, 1 and 2, substitute each one into the equation, and record the matching y-value.
当方程不熟悉或斜率是分数时，数值表法是最稳妥的方法。选取一小组 x 值，通常取 -2、-1、0、1、2，逐个代入方程，记录对应的 y 值。
tuting each x-value gives the following table, and the constant difference of 2 between successive y-values confirms the gradient of 2.
例题：绘制 y = 2x - 1。把每个 x 值代入可得下表，相邻 y 值之差恒为 2，正好印证了斜率为 2。
| x | -2 | -1 | 0 | 1 | 2 |
|---|---|---|---|---|---|
| y = 2x - 1 | -5 | -3 | -1 | 1 | 3 |
Now plot each ordered pair (x, y) as a small cross on the grid and join them with a single straightedge line that extends beyond the first and last points. The line should pass through (0, -1), confirming the intercept, and rise 2 units for every 1 unit to the right, confirming the gradient.
然后在网格上把每个有序数对 (x, y) 描成小十字，用直尺连成一条直线，并让直线向两端延伸超出首末两点。这条线应经过 (0, -1) 以印证截距，并且每向右 1 格上升 2 格，以印证斜率。
If your plotted points do not sit on a straight line, do not force them onto one. Check the arithmetic for the point that is out of place, since a single substitution error is by far the most likely explanation.
如果描出的点不在同一条直线上，不要强行把它们连成直线。应检查那个明显偏离的点所对应的计算，因为单点代入错误是最常见的原因。
7. Plotting from the Intercept and Gradient | 用截距和斜率作图
le. Mark the intercept (0, c) first, then use the gradient as a movement instruction: go 1 unit right and m units up if m is positive, or m units down if m is negative.
当方程已经是 y = mx + c 的形式时，可以使用更快的作图方法。先标出截距点 (0, c)，再把斜率当作移动指令：向右走 1 格，若 m 为正则向上 m 格，若 m 为负则向下 m 格。
ercept is (0, 1); from there go 3 units right and 2 units up to reach (3, 3). A second such step gives (6, 5), and the three points (0, 1), (3, 3) and (6, 5) lie on a single straight line.
对于写成 a ÷ b 的分数斜率，向右移动 b 格、向上移动 a 格。例如 y = ⅔x + 1 的截距为 (0, 1)；从该点向右 3 格、向上 2 格到达 (3, 3)。再重复一次得到 (6, 5)，三点 (0, 1)、(3, 3)、(6, 5) 恰好共线。
- y = 3x - 2: start at (0, -2), then right 1, up 3, giving (1, 1). y = 3x - 2：从 (0, -2) 出发，右 1、上 3，得到 (1, 1)。
- y = -x + 5: start at (0, 5), then right 1, down 1, giving (1, 4). y = -x + 5：从 (0, 5) 出发，右 1、下 1，得到 (1, 4)。
- y = ½x: start at (0, 0), then right 2, up 1, giving (2, 1). y = ½x：从 (0, 0) 出发，右 2、上 1，得到 (2, 1)。
2y = 10, rearrange it to y = -2x + 5 first, then use the intercept-and-gradient approach.
这种方法比列数值表快得多，但只有在方程形式正确时才能使用。如果题目给出 4x + 2y = 10 之类，先整理成 y = -2x + 5，再使用截距加斜率的方法。
8. Finding the Equation of a Line from a Graph | 从图像求直线方程
Sometimes you are shown a straight-line graph and asked to write down its equation. Work in two steps: find c first, then find m.
有时题目给出一条直线图像，要求写出它的方程。分两步进行：先求 c，再求 m。
nvenient points on the line and apply the gradient formula, counting squares if necessary.
第一步：读出直线与 y 轴的交点。若交于 (0, 3)，则 c = 3。第二步：在直线上取两个方便的点，套用斜率公式，必要时直接数格子。
is y = 2x + 3. Always test your answer with a third point if one is visible on the graph.
例题：一条直线经过 (0, 3) 和 (2, 7)。则 c = 3，m = (7 - 3) ÷ (2 - 0) = 4 ÷ 2 = 2，所以方程为 y = 2x + 3。如果图上还能看到第三个点，务必用它检验答案。
If the line does not pass through a convenient grid intersection, choose two points that do, or read the coordinates as carefully as you can and expect the gradient to be a simple fraction such as ½ or ¾ rather than a long decimal.
如果直线没有经过方便的格点交点，就选两个确实落在格点上的点，或者尽可能仔细地读出坐标，并且预期斜率会是 ½、¾ 之类的简单分数，而不是冗长的小数。
9. Special Cases: Horizontal, Vertical and Through the Origin | 特殊情况：水平线、垂直线与过原点直线
ferent: it is not a graph of y in terms of x at all, and its equation is written as x = a.
水平线的斜率为零，因此方程就是 y = c，线上每个点的 y 坐标都相同。垂直线则不同：它根本不是 y 关于 x 的函数图像，其方程写成 x = a。
- y = 4 is the horizontal line through (0, 4). y = 4 是经过 (0, 4) 的水平线。
- y = -2 is the horizontal line through (0, -2). y = -2 是经过 (0, -2) 的水平线。
- x = 3 is the vertical line through (3, 0). x = 3 是经过 (3, 0) 的垂直线。
- x = -1 is the vertical line through (-1, 0). x = -1 是经过 (-1, 0) 的垂直线。
he coordinates are equal, and y = -x, which has gradient -1 and passes through points such as (2, -2) and (-3, 3).
当 c = 0 时，直线经过原点，方程为 y = mx。最常用的两个例子是 y = x（斜率为 1，经过所有横纵坐标相等的点）和 y = -x（斜率为 -1，经过 (2, -2)、(-3, 3) 等点）。
igin still has a y-intercept; it is simply (0, 0), so c = 0 and you must write it as such.
常见的考试陷阱是以为每条直线都必须在 y 轴上有"看得见"的截距。经过原点的直线同样有 y 轴截距，它就是 (0, 0)，即 c = 0，作答时必须明确写出这一点。
10. Parallel and Perpendicular Lines | 平行线与垂直线
would be the same line. So y = 3x + 1 and y = 3x - 4 are parallel, both having gradient 3.
两条直线平行的充要条件是斜率相同。它们的截距必须不同，否则就是同一条直线。因此 y = 3x + 1 与 y = 3x - 4 平行，二者斜率都是 3。
nt is the negative reciprocal of the other. If m₁ = 2 then m₂ = -½, because 2 × (-½) = -1.
两条直线垂直的充要条件是斜率之积为 -1，也就是说一个斜率是另一个的负倒数。若 m₁ = 2，则 m₂ = -½，因为 2 × (-½) = -1。
m₁ × m₂ = -1
- Parallel to y = 4x - 7 through (0, 2): y = 4x + 2. 与 y = 4x - 7 平行且过 (0, 2)：y = 4x + 2。
- Perpendicular to y = 4x - 7 through (0, 2): y = -¼x + 2. 与 y = 4x - 7 垂直且过 (0, 2)：y = -¼x + 2。
- rizontal lines are perpendicular to vertical lines; y = 5 and x = -3 meet at right angles. 水平线与垂直线互相垂直；y = 5 与 x = -3 成直角相交。
Note that a horizontal line has gradient 0 and the reciprocal of 0 is undefined, which is exactly why the negative-reciprocal rule breaks down for horizontal and vertical pairs and they must be handled as a special case.
注意水平线的斜率为 0，而 0 的倒数不存在，这正解释了为什么负倒数规则对水平线与垂直线这一对失效，必须作为特殊情况单独处理。
11. Using Linear Graphs to Solve Problems | 利用线性图像解决问题
me journey can all be represented by y = mx + c, and each part of the equation has a real meaning.
只要某个量以恒定速率变化，就可以用线性图像
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