📚 Straight-line Graphs: Gradient, y-intercept and Solving Graphically | 直线图像：斜率、截距与图解方程
A straight-line graph is one of the most important building blocks in Cambridge KS3 Mathematics. Understanding how to plot, interpret and find the equation of a straight line will help you with algebra, geometry and real-life problems such as mobile phone tariffs, speed and conversion graphs.
直线图像是剑桥 KS3 数学中最重要的基础内容之一。理解如何绘制、解释和求出直线方程，将帮助你在代数、几何以及现实问题中得心应手，例如手机话费、速度和换算图等。
1. What Is a Straight-Line Graph? | 什么是直线图像？
A straight-line graph shows a linear relationship between two variables. On a coordinate grid, every point on the line satisfies the same linear equation. The line continues forever in both directions unless a context limits it.
直线图像表示两个变量之间的线性关系。在坐标网格上，直线上的每一个点都满足同一个线性方程。除非受到实际背景限制，直线会向两个方向无限延伸。
y = mx + c
This is the standard form of a linear equation, where m is the gradient and c is the y-intercept. In KS3 Cambridge Maths, you will meet this form again and again.
这是线性方程的标准形式，其中 m 是斜率，c 是 y 轴截距。在剑桥 KS3 数学中，你会反复遇到这种形式。
2. Plotting Points from a Table of Values | 根据数值表描点
To plot a straight line, first choose several x-values, usually −2, −1, 0, 1 and 2. Substitute each x-value into the equation to find the matching y-value. List these in a table and plot the coordinate pairs. Join the points with a ruler to make a straight line.
要绘制一条直线，通常先取几个 x 值，例如 −2、−1、0、1 和 2。将每个 x 值代入方程，求出对应的 y 值。将这些数值列入表格，并描出坐标点。用直尺连接各点，即可得到一条直线。
| x | −2 | −1 | 0 | 1 | 2 |
| y = 2x + 1 | −3 | −1 | 1 | 3 | 5 |
Plot the points (−2, −3), (−1, −1), (0, 1), (1, 3) and (2, 5). They should all lie on the same straight line. If one point is off the line, check your substitution or your plotting.
描出点 (−2, −3)、(−1, −1)、(0, 1)、(1, 3) 和 (2, 5)。它们应该都在同一条直线上。如果有一个点偏离直线，请检查代入或描点是否正确。
3. The Equation y = mx + c | 方程 y = mx + c
The letters m and c tell you everything about the line. The coefficient m controls the steepness, and the constant c tells you where the line crosses the y-axis. Always rewrite an equation into this form before describing the line.
字母 m 和 c 告诉你关于这条直线的一切。系数 m 控制直线的倾斜程度，常数 c 告诉你直线与 y 轴相交的位置。在描述直线之前，总是先把方程改写成这种形式。
- y = 3x + 2 has gradient 3 and y-intercept 2.
- y = 3x + 2 的斜率为 3，y 轴截距为 2。
- y = −x + 5 has gradient −1 and y-intercept 5.
- y = −x + 5 的斜率为 −1，y 轴截距为 5。
If b is added to mx, the line shifts up or down. For example, y = 2x and y = 2x + 4 are parallel because they have the same gradient but different y-intercepts.
如果在 mx 后面加上 b，直线会向上或向下平移。例如，y = 2x 和 y = 2x + 4 是平行的，因为它们的斜率相同，但 y 轴截距不同。
4. Understanding Gradient m | 理解斜率 m
The gradient measures how steep the line is. It is the amount the y-value increases or decreases when x increases by 1. A positive gradient slopes upwards from left to right; a negative gradient slopes downwards.
斜率表示直线的倾斜程度。它是当 x 增加 1 时，y 值增加或减少的量。正斜率表示直线从左到右向上倾斜；负斜率表示直线从左到右向下倾斜。
m = rise ÷ run = Δy ÷ Δx
To calculate gradient, choose two points on the line and divide the change in y by the change in x. Always work from left to right so that the sign is correct.
计算斜率时，在直线上选取两个点，用 y 的变化量除以 x 的变化量。总是从左到右进行计算，这样正负号才正确。
Example: if a line passes through (1, 2) and (3, 8), then Δy = 8 − 2 = 6 and Δx = 3 − 1 = 2. Therefore m = 6 ÷ 2 = 3.
例如：如果一条直线经过 (1, 2) 和 (3, 8)，那么 Δy = 8 − 2 = 6，Δx = 3 − 1 = 2。因此 m = 6 ÷ 2 = 3。
5. Understanding y-intercept c | 理解 y 轴截距 c
The y-intercept c is the point where the line crosses the y-axis. At this point, x = 0. In the equation y = 2x + 3, the y-intercept is 3, so the line passes through (0, 3).
y 轴截距 c 是直线与 y 轴相交的点。在这个点上，x = 0。在方程 y = 2x + 3 中，y 轴截距为 3，因此直线经过点 (0, 3)。
To find c from a table, look for the y-value when x = 0. If the table does not have x = 0, extend the pattern backwards until you reach x = 0.
要从表格中找 c，可以查看当 x = 0 时的 y 值。如果表格中没有 x = 0，就按规律向前延伸，直到找到 x = 0。
Example: for the points (−1, −1), (0, 1), (1, 3), the y-intercept is 1 because the point (0, 1) lies on the y-axis.
例如：对于点 (−1, −1)、(0, 1)、(1, 3)，y 轴截距为 1，因为点 (0, 1) 在 y 轴上。
6. Drawing Straight Lines Quickly | 快速绘制直线
When the equation is in the form y = mx + c, you can draw the line without a full table. First mark the y-intercept c on the y-axis. Then use the gradient m as a fraction rise/run. From the intercept, move up or down and right to find a second point, and draw the line through both points.
当方程写成 y = mx + c 的形式时，你可以不用完整表格快速画线。首先在 y 轴上标出截距 c。然后把斜率 m 写成 rise/run 的分数形式。从截距出发，向上或向下移动，再向右移动，找到第二个点，最后经过两点画出直线。
Example: draw y = (2/3)x + 1. The y-intercept is 1, so mark (0, 1). From there, rise 2 units and run 3 units right to reach (3, 3). Join (0, 1) and (3, 3).
例如：绘制 y = (2/3)x + 1。y 轴截距为 1，所以标出 (0, 1)。从这里向上移动 2 个单位，再向右移动 3 个单位，到达 (3, 3)。连接 (0, 1) 和 (3, 3)。
If the gradient is negative, move down instead of up. For y = −2x + 3, mark (0, 3), then move down 2 units and right 1 unit to reach (1, 1).
如果斜率为负，则向下移动而不是向上。对于 y = −2x + 3，标出 (0, 3)，然后向下移动 2 个单位，再向右移动 1 个单位，到达 (1, 1)。
7. Finding the Equation from a Graph | 从图像求方程
To find the equation from a graph, first read the y-intercept c directly from the y-axis. Then choose two clear points on the line and calculate the gradient using m = Δy ÷ Δx. Substitute m and c into y = mx + c.
要从图像求方程，首先直接从 y 轴读出截距 c。然后在直线上选取两个清晰的点，用 m = Δy ÷ Δx 计算斜率。将 m 和 c 代入 y = mx + c。
Example: a line crosses the y-axis at (0, −2) and passes through (3, 4). So c = −2. The gradient is m = (4 − −2) ÷ (3 − 0) = 6 ÷ 3 = 2. The equation is y = 2x − 2.
例如：一条直线与 y 轴相交于 (0, −2) 并经过 (3, 4)。所以 c = −2。斜率为 m = (4 − −2) ÷ (3 − 0) = 6 ÷ 3 = 2。方程为 y = 2x − 2。
When reading a graph, choose grid intersections so that the coordinates are exact. Avoid estimating between grid lines, because small errors can change the gradient significantly.
读取图像时，选择网格交点作为坐标，这样数值才准确。避免在网格线之间估算，因为小误差会显著改变斜率。
8. Parallel and Perpendicular Lines | 平行线与垂直线
Parallel lines have the same gradient but different y-intercepts. For example, y = 3x + 1 and y = 3x − 4 are parallel because m = 3 in both.
平行线具有相同的斜率，但 y 轴截距不同。例如，y = 3x + 1 和 y = 3x − 4 是平行的，因为两者的斜率都是 m = 3。
m₁ × m
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