Straight-Line Graphs: Mastering y = mx + c for Cambridge KS3 | 直线图像：掌握 y = mx + c

📚 Straight-Line Graphs: Mastering y = mx + c for Cambridge KS3 | 直线图像：掌握 y = mx + c

Straight-line graphs are one of the most important algebra tools in the Cambridge KS3 mathematics course. They help you connect equations, tables of values and visual lines on a coordinate grid. In this article, we focus on the equation y = mx + c, which is often the exact type of question found around page 239 of the Cambridge learner resources.

直线图像是 Cambridge KS3 数学课程中最重要的代数工具之一。它帮助你建立方程、数值表和坐标网格上图像之间的联系。本文重点讲解 y = mx + c 这一方程形式，这也正是 Cambridge 学习资料第 239 页附近常见的题目类型。

1. What Is y = mx + c? | 什么是 y = mx + c？

The equation y = mx + c is called the gradient-intercept form of a straight line. The letter m represents the gradient, or steepness, of the line. The letter c represents the y-intercept, which is the point where the line crosses the y-axis.

方程 y = mx + c 被称为直线的斜截式。字母 m 表示斜率，也就是直线的倾斜程度。字母 c 表示 y 轴截距，即直线与 y 轴交点的纵坐标。

For example, in y = 3x – 2, the gradient m is 3 and the y-intercept c is -2. This means the line rises by 3 units for every 1 unit it moves to the right, and it crosses the y-axis at (0, -2).

例如，在 y = 3x – 2 中，斜率 m 是 3，y 轴截距 c 是 -2。这意味着直线每向右移动 1 个单位就上升 3 个单位，并且它在 (0, -2) 处与 y 轴相交。

2. Identifying the Gradient m | 识别斜率 m

The gradient is the number written in front of x when the equation is in the form y = mx + c. It tells you how fast y changes as x increases. A positive gradient means the line goes uphill from left to right. A negative gradient means the line goes downhill.

斜率是方程写成 y = mx + c 形式时 x 前面的数字。它表示当 x 增大时 y 变化的速度。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。

If the equation is y = -4x + 5, then m = -4. The line falls by 4 units for every 1 unit it moves to the right. If the equation is y = x + 2, then m = 1 because x is the same as 1x.

如果方程是 y = -4x + 5，那么 m = -4。直线每向右移动 1 个单位就下降 4 个单位。如果方程是 y = x + 2，那么 m = 1，因为 x 等同于 1x。

3. Finding the y-Intercept c | 求 y 轴截距 c

The y-intercept is the constant term in the equation y = mx + c. It is the value of y when x equals 0. On a graph, it is the starting point of the line on the vertical axis.

y 轴截距是方程 y = mx + c 中的常数项。它是当 x 等于 0 时 y 的值。在图像上，它是直线在纵轴上的起点。

For y = 2x + 7, the y-intercept is 7, so the coordinate is (0, 7). For y = -x – 3, the y-intercept is -3, so the coordinate is (0, -3). Always write the y-intercept as a coordinate with x = 0.

对于 y = 2x + 7，y 轴截距是 7，因此坐标是 (0, 7)。对于 y = -x – 3，y 轴截距是 -3，因此坐标是 (0, -3)。始终把 y 轴截距写成 x 等于 0 的坐标形式。

4. Rearranging into y = mx + c | 将方程整理为 y = mx + c 形式

Sometimes the equation is not given in the form y = mx + c. You may need to rearrange it first. For example, 2y = 4x + 10 should be divided by 2 on both sides to give y = 2x + 5.

有时方程并不是以 y = mx + c 的形式给出的。你可能需要先进行整理。例如，2y = 4x + 10 应两边同时除以 2，得到 y = 2x + 5。

If you have 3x + y = 6, subtract 3x from both sides to get y = -3x + 6. Now the gradient is -3 and the y-intercept is 6. Practise this rearranging skill because exam questions often hide the gradient this way.

如果有 3x + y = 6，两边同时减去 3x，得到 y = -3x + 6。此时斜率是 -3，y 轴截距是 6。要练习这种整理技巧，因为考试题经常用这种方式隐藏斜率。

5. Making a Table of Values | 制作数值表

To draw a straight-line graph, you usually create a table of values. Choose x values such as -2, -1, 0, 1 and 2. Substitute each x value into the equation to find the matching y value.

要画直线图像，通常先制作数值表。选择 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。

For y = 3x – 2, the table looks like this:

对于 y = 3x – 2，数值表如下：

x	-2	-1	0	1	2
y	-8	-5	-2	1	4

Each pair of x and y values becomes a coordinate point you can plot on a grid.

每一对 x 和 y 值都成为你可以在坐标网格上描出的坐标点。

6. Plotting Points and Drawing the Line | 描点并画直线

Plot each coordinate pair on a labelled coordinate grid. The points should all lie on the same straight line. Use a ruler to join them neatly. Extend the line beyond the plotted points and add arrowheads if required.

在带有标注的坐标网格上描出每个坐标对。所有点都应位于同一条直线上。用直尺将它们整齐连接。把直线延伸到描点之外，并在需要时加上箭头。

For y = 3x – 2, the coordinates are (-2, -8), (-1, -5), (0, -2), (1, 1) and (2, 4). Check that each point satisfies the equation before drawing the line.

对于 y = 3x – 2，坐标是 (-2, -8)、(-1, -5)、(0, -2)、(1, 1) 和 (2, 4)。在画直线之前，检查每个点是否都满足方程。

7. Reading Off the Gradient and Intercept from a Graph | 从图像读取斜率和截距

If you are given a graph instead of an equation, you can find the gradient by choosing two clear points on the line. Divide the vertical change by the horizontal change. Then find the y-intercept where the line crosses the y-axis.

如果给出的是图像而不是方程，你可以选择直线上两个清晰的点来求斜率。用竖直变化量除以水平变化量。然后找出直线与 y 轴交点的 y 轴截距。

If a line passes through (0, 1) and (2, 5), the vertical change is 5 – 1 = 4 and the horizontal change is 2 – 0 = 2. The gradient is 4 ÷ 2 = 2. The y-intercept is 1, so the equation is y = 2x + 1.

如果一条直线经过 (0, 1) 和 (2, 5)，竖直变化量是 5 – 1 = 4，水平变化量是 2 – 0 = 2。斜率是 4 ÷ 2 = 2。y 轴截距是 1，因此方程是 y = 2x + 1。

8. Checking Your Graph | 检查你的图像

Always check your graph by substituting one extra point into the equation. If the point lies on your line, your graph is likely correct. A common mistake is misreading a negative y value or using the wrong scale.

一定要通过将额外一个点代入方程来检查图像。如果该点位于你的直线上，说明图像很可能是正确的。常见错误是读错负的 y 值或使用错误的比例。

For y = 3x – 2, try x = 3. The equation gives y = 3(3) – 2 = 7. Your line should pass through (3, 7). If it does not, check your table and plotted points.

对于 y = 3x – 2，尝试 x = 3。方程给出 y = 3(3) – 2 = 7。你的直线应经过 (3, 7)。如果没有经过，请检查数值表和描出的点。

9. Parallel Lines and Equal Gradients | 平行线与相等斜率

Parallel lines have the same gradient but different y-intercepts. For example, y = 2x + 3 and y = 2x – 5 are parallel because both have m = 2. They never meet.

平行直线具有相同的斜率但不同的 y 轴截距。例如，y = 2x + 3 和 y = 2x – 5 是平行的，因为两者的斜率都是 m = 2。它们永远不会相交。

If a question asks you to write the equation of a line parallel to y = 4x – 1, keep the 4x part the same and change only the constant. Any equation of the form y = 4x + c is parallel to the original line.

如果题目要求写出与 y = 4x – 1 平行的直线方程，保持 4x 部分不变，只改变常数项。任何形如 y = 4x + c 的方程都与原直线平行。

10. Common Mistakes and How to Avoid Them | 常见错误与如何避免

One common error is mixing up the gradient and the y-intercept. Remember that m is the coefficient of x, while c is the number on its own. For y = 5 – 2x, rearranging gives y = -2x + 5, so m = -2 and c = 5.

一个常见错误是混淆斜率和 y 轴截距。请记住，m 是 x 的系数，而 c 是单独的常数项。对于 y = 5 – 2x，整理后得到 y = -2x + 5，因此 m = -2，c = 5。

Another mistake is forgetting that the sign belongs to the coefficient. In y = -3x – 4, the gradient is -3, not 3, and the intercept is -4, not 4. Underline the sign as you identify each part.

另一个错误是忘记符号属于系数。在 y = -3x – 4 中，斜率是 -3，而不是 3，截距是 -4，而不是 4。在识别每一部分时，请把符号标出来。

11. Exam-Style Walkthrough: P239 Q2 Style | 考题风格演练：P239 第 2 题类型

Let us work through a question similar to the type found on Cambridge KS3 page 239, question 2. A line has the equation y = 3x – 2. Write down the gradient and y-intercept, then draw the graph for x values from -2 to 2.

我们来完成一道与 Cambridge KS3 第 239 页第 2 题类型相似的题目。一条直线的方程是 y = 3x – 2。写出它的斜率和 y 轴截距，并画出 x 从 -2 到 2 的图像。

Step 1: Compare with y = mx + c.
步骤 1：与 y = mx + c 作比较。
Step 2: m = 3 and c = -2.
步骤 2：m = 3，c = -2。
Step 3: Make a table of values for x = -2, -1, 0, 1, 2.
步骤 3：为 x = -2、-1、0、1、2 制作数值表。
Step 4: Plot the points (-2, -8), (-1, -5), (0, -2), (1, 1), (2, 4).
步骤 4：描出点 (-2, -8)、(-1, -5)、(0, -2)、(1, 1)、(2, 4)。
Step 5: Draw a straight line through all points.
步骤 5：过所有点画一条直线。

This structured approach earns full marks for method and accuracy.

这种分步方法可以在方法和准确性上拿到满分。

12. Summary and Revision Checklist | 小结与复习清单

Remember that y = mx + c gives you the gradient m and the y-intercept c directly. Use a table of values to plot at least three points, then join them with a ruler. Check parallel lines have equal gradients and always include signs when writing m and c.

请记住，y = mx + c 直接给出斜率 m 和 y 轴截距 c。使用数值表至少描出三个点，然后用直尺连接它们。检查平行线是否具有相等斜率，并且在书写 m 和 c 时始终带上符号。

Before your test, practise identifying m and c from equations, drawing graphs from tables, and writing the equation from a given line. These are the core skills for Cambridge KS3 straight-line graphs.

在考试前，练习从方程中识别 m 和 c、根据数值表画图，以及根据给定直线写方程。这些是 Cambridge KS3 直线图像的核心技能。

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Straight-Line Graphs: Mastering y = mx + c for Cambridge KS3 | 直线图像：掌握 y = mx + c