📚 Straight Line Graphs: Plotting, Gradient and Intercept | 直线图像：描点、斜率与截距
In KS3 Cambridge Mathematics, straight line graphs are a central algebra topic. They connect coordinates, equations, tables of values and real-life rates of change.
在 KS3 剑桥数学中，直线图像是核心代数主题。它们把坐标、方程、数值表和现实生活中的变化率联系起来。
This revision guide covers plotting points, recognising linear relationships, using the equation y = mx + c, calculating gradient and intercept, and applying these skills to exam-style questions.
本复习指南涵盖描点、识别线性关系、使用方程 y = mx + c、计算斜率和截距，以及将这些技能应用于试题型问题。
1. Coordinates and the Cartesian Plane | 坐标与笛卡尔平面
A graph is drawn on a coordinate grid with two axes. The horizontal axis is the x-axis and the vertical axis is the y-axis. A point is written as (x, y).
图像绘制在有两条坐标轴的坐标网格上。水平轴是 x 轴，竖直轴是 y 轴。一个点写作 (x, y)。
The origin is (0, 0), where the axes cross. Positive x-values go to the right and positive y-values go upwards.
原点是 (0, 0)，即两轴相交处。x 轴正方向向右，y 轴正方向向上。
For example, the point (3, 5) means moving 3 units right from the origin and then 5 units up.
例如，点 (3, 5) 表示从原点向右移动 3 个单位，再向上移动 5 个单位。
2. Plotting Points and Linear Patterns | 描点与线性规律
To plot a point, start at the origin. Move along the x-axis by the x-coordinate, then move vertically by the y-coordinate.
描点时从原点开始。先按 x 坐标沿 x 轴移动，再按 y 坐标竖直移动。
If points follow a linear pattern, they lie on a straight line. For example, the points (−2, −4), (−1, −1), (0, 2), (1, 5) and (2, 8) all lie on one line.
如果点遵循线性规律，它们就位于同一直线上。例如，点 (−2, −4)、(−1, −1)、(0, 2)、(1, 5) 和 (2, 8) 都在一条直线上。
You can check this by plotting them on squared paper or by looking at how the y-value changes each time x increases by 1.
你可以通过在方格纸上描点或观察每次 x 增加 1 时 y 值如何变化来检验这一点。
In the example above, as x increases by 1, y increases by 3. This constant rate tells us the graph will be a straight line.
在上面的例子中，当 x 每增加 1 时，y 增加 3。这个恒定的变化率告诉我们图像将是一条直线。
3. Recognising a Straight Line Graph | 识别直线图像
A linear graph is a straight line. It represents a relationship where y changes at a constant rate as x changes.
线性图像是一条直线。它表示当 x 改变时，y 以恒定速率变化的关系。
If the rate is not constant, the graph will curve. At KS3 level, we focus on straight line graphs and their equations.
如果变化率不恒定，图像会弯曲。在 KS3 阶段，我们重点关注直线图像及其方程。
For example, y = 2x + 3 is linear, but y = x² is not linear because the graph of y = x² is a curve.
例如，y = 2x + 3 是线性的，但 y = x² 不是线性的，因为 y = x² 的图像是一条曲线。
4. The Equation y = mx + c | 方程 y = mx + c
The general equation of a straight line is:
直线的一般方程是：
y = mx + c
Here m is the gradient and c is the y-intercept. The gradient measures steepness, and the y-intercept is where the line crosses the y-axis.
这里 m 是斜率，c 是 y 轴截距。斜率衡量倾斜程度，y 轴截距是直线与 y 轴的交点。
For example, the line y = 2x + 3 has gradient 2 and y-intercept 3.
例如，直线 y = 2x + 3 的斜率为 2，y 轴截距为 3。
If the equation is written in a different form, always rearrange it to the form y = mx + c first.
如果方程以其他形式书写，一定要先把它整理成 y = mx + c 的形式。
5. Understanding Gradient (m) | 理解斜率（m）
Gradient is calculated as rise over run:
斜率按“垂直增量除以水平增量”计算：
m = rise ÷ run = (y₂ − y₁) ÷ (x₂ − x₁)
A positive gradient slopes upwards from left to right. A negative gradient slopes downwards from left to right.
正斜率从左到右向上倾斜。负斜率从左到右向下倾斜。
If m = 0, the line is horizontal. If the line is vertical, the gradient is undefined.
如果 m = 0，直线是水平的。如果直线是竖直的，斜率未定义。
For example, y = 3x − 5 has positive gradient 3, so it goes uphill. y = −2x + 1 has negative gradient −2, so it goes downhill.
例如，y = 3x − 5 的斜率为正 3，因此它从左到右向上。y = −2x + 1 的斜率为负 −2，因此它从左到右向下。
6. Understanding y-intercept (c) | 理解 y 轴截距（c）
The y-intercept c is the value of y when x = 0.
y 轴截距 c 是 x = 0 时的 y 值。
To find c from a graph, read the y-coordinate where the line crosses the y-axis.
要从图像上找到 c，读取直线穿过 y 轴处的 y 坐标。
In the equation y = 4x + 7, the line crosses the y-axis at (0, 7). In y = 4x − 2, it crosses at (0, −2).
在方程 y = 4x + 7 中，直线与 y 轴交于 (0, 7)。在 y = 4x − 2 中，它交于 (0, −2)。
Do not confuse the y-intercept with the x-intercept. The x-intercept is where the line crosses the x-axis, and it is found by setting y = 0.
不要把 y 轴截距与 x 轴截距混淆。x 轴截距是直线与 x 轴的交点，通过令 y = 0 求出。
7. Plotting a Line from a Table of Values | 由数值表画直线
To plot a line from its equation, build a table of values. Choose at least three x-values, substitute them into the equation, and find the matching y-values.
要根据方程画直线，先建立数值表。选择至少三个 x 值，代入方程，求出对应的 y 值。
For y = 2x + 3, choosing x-values −2, −1, 0, 1 and 2 gives:
对于 y = 2x + 3，选择 x 值 −2、−1、0、1 和 2，得到：
| x | −2 | −1 | 0 | 1 | 2 |
|---|---|---|---|---|---|
| y = 2x + 3 | −1 | 1 | 3 | 5 | 更多咨询请联系16621398022（同微信）
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