Straight Line Graphs and Gradient | 直线图像与斜率

📚 Straight Line Graphs and Gradient | 直线图像与斜率

In IGCSE Mathematics, straight line graphs are one of the most important topics in coordinate geometry. They appear in algebra, geometry, and real-life modelling questions, and examiners often combine gradients, intercepts, and equations with parallel and perpendicular lines. This unit revises the core skills needed to find, interpret, and use the equation of a straight line.

在 IGCSE 数学中，直线图像是坐标几何里最重要的主题之一。它出现在代数、几何和实际生活建模题中，考官经常把斜率、截距和方程与平行线、垂直线结合起来考查。本单元复习求取、解释和使用直线方程所需的核心技能。

1. What is a Linear Equation? | 什么是线性方程？

A linear equation produces a straight line when it is drawn on a coordinate grid. The highest power of x is 1, so the graph has no curves or turning points.

线性方程在坐标格上绘制时会生成一条直线。x 的最高次数是 1，因此图像没有弯曲或转折点。

For example, y = 2x + 3, 3x + y = 6, and x = 4 are all linear equations. Equations such as y = x² + 1 or y = 2x² − 3x + 5 are not linear because they contain an x-squared term.

例如，y = 2x + 3、3x + y = 6 和 x = 4 都是线性方程。像 y = x² + 1 或 y = 2x² − 3x + 5 这样的方程不是线性方程，因为它们含有 x 的平方项。

In an IGCSE exam, you may be asked to decide whether a given equation is linear. Check both variables: if either variable has a power higher than 1, or if the variables are multiplied together, the graph will not be a straight line.

在 IGCSE 考试中，你可能需要判断一个给定方程是否为线性方程。检查两个变量：如果任一变量的次数高于 1，或者两个变量相乘，图像就不会是直线。

2. The General Form y = mx + c | 一般式 y = mx + c

The equation of any straight line can be written in the form y = mx + c, where m represents the gradient and c represents the y-intercept. This form is essential because it tells you the steepness and vertical position of the line.

任何直线的方程都可以写成 y = mx + c 的形式，其中 m 表示斜率，c 表示 y 轴截距。这个形式非常重要，因为它告诉你直线的倾斜程度和竖直位置。

y = mx + c

For example, in the line y = 3x − 2, the gradient m is 3 and the y-intercept c is −2. In y = 4 − x, you should first rearrange it as y = −x + 4, so m = −1 and c = 4.

例如，在直线 y = 3x − 2 中，斜率 m 是 3，y 轴截距 c 是 −2。在 y = 4 − x 中，你应该先将其改写为 y = −x + 4，因此 m = −1，c = 4。

If the equation is not in the form y = mx + c, rearrange it to make y the subject. This will help you identify the gradient and y-intercept quickly.

如果方程不是 y = mx + c 的形式，请将其变形为以 y 为主语的形式。这能帮助你快速识别斜率和 y 轴截距。

3. Finding the Gradient | 求斜率

The gradient measures how steep a straight line is. It is calculated as the vertical change divided by the horizontal change between two points on the line, often remembered as ‘rise over run’.

斜率衡量直线的倾斜程度。它等于直线上两点之间的竖直变化量除以水平变化量，通常记作“上升量除以水平量”。

m = (y₂ − y₁) ÷ (x₂ − x₁)

For two points (1, 2) and (4, 11), the gradient is (11 − 2) ÷ (4 − 1) = 9 ÷ 3 = 3. If the y-values decrease as x increases, the gradient will be negative.

对于两点 (1, 2) 和 (4, 11)，斜率为 (11 − 2) ÷ (4 − 1) = 9 ÷ 3 = 3。如果 y 值随着 x 增大而减小，斜率将为负。

A horizontal line has gradient 0 because the vertical change is zero. A vertical line has an undefined gradient because the horizontal change is zero, and division by zero is not possible.

水平线的斜率为 0，因为竖直变化量为零。垂直线的斜率未定义，因为水平变化量为零，除以零是不可能的。

4. The y-intercept | y 轴截距

The y-intercept is the point where the line crosses the y-axis. At this point, the value of x is always 0, so you can find c by substituting x = 0 into the equation.

y 轴截距是直线与 y 轴相交的点。在这个点上，x 的值始终为 0，因此你可以通过将 x = 0 代入方程来求 c。

For y = 2x + 5, when x = 0, y = 2(0) + 5 = 5, so the y-intercept is (0, 5). On a graph, this is the starting height of the line above or below the origin.

对于 y = 2x + 5，当 x = 0 时，y = 2(0) + 5 = 5，因此 y 轴截距为 (0, 5)。在图像上，这是直线位于原点上方或下方的起始高度。

In real-life linear models, c often represents a fixed or initial value, such as a base fee or starting distance. Do not confuse the y-intercept with the x-intercept, which is found by setting y = 0.

在实际生活的线性模型中，c 通常表示固定值或初始值，例如基础费用或起始距离。不要把 y 轴截距与 x 轴截距混淆，x 轴截距通过令 y = 0 求得。

5. Plotting a Straight Line Graph | 绘制直线图像

To plot a straight line graph, you only need two points because a straight line is determined by two distinct points. A third point can be used as a check.

要绘制一条直线图像，你只需要两个点，因为直线由两个不同的点确定。第三个点可以用作检验。

Choose two x-values, such as x = 0 and x = 2.

选择两个 x 值，例如 x = 0 和 x = 2。
Substitute each x-value into the equation to find the corresponding y-value.

将每个 x 值代入方程，求出对应的 y 值。
Plot the coordinate pairs and draw a straight line through them.

标出坐标对，并通过它们画一条直线。

For y = 2x − 1, when x = 0, y = −1; when x = 3, y = 5. Plot (0, −1) and (3, 5), then use a ruler to draw the line carefully.

对于 y = 2x − 1，当 x = 0 时，y = −1；当 x = 3 时，y = 5。标出 (0, −1) 和 (3, 5)，然后用直尺小心地画出直线。

Always label the axes, use a sensible scale, and extend the line across the full grid. In IGCSE graphs, marks are often awarded for accuracy and correct labelling.

务必给坐标轴标上名称，使用合理的刻度，并将直线贯穿整个网格。在 IGCSE 图像题中，通常会根据准确度和正确标注给分。

6. Finding the Equation from a Graph | 从图像求方程

To find the equation of a straight line from its graph, first read the y-intercept directly from the point where the line crosses the y-axis. This gives the value of c.

要从图像中求直线方程，首先直接从直线与 y 轴的交点读取 y 轴截距。这就是 c 的值。

Next, choose two clear points on the line and calculate the gradient using m = (y₂ − y₁) ÷ (x₂ − x₁). Make sure the points are exactly on grid intersections to avoid reading errors.

接下来，在直线上选取两个清晰的点，用 m = (y₂ − y₁) ÷ (x₂ − x₁) 计算斜率。确保这些点正好在网格交点上，以避免读数误差。

Once you have m and c, write the equation in the form y = mx + c. For example, if a line crosses the y-axis at 4 and has gradient 2, its equation is y = 2x + 4.

求出 m 和 c 后，将方程写成 y = mx + c 的形式。例如，如果一条直线与 y 轴相交于 4，且斜率为 2，则其方程为 y = 2x + 4。

7. Parallel Lines | 平行线

Parallel lines never meet and always remain the same distance apart. In coordinate geometry, two straight lines are parallel if and only if they have the same gradient.

平行线永不相交，且始终相隔相同的距离。在坐标几何中，两条直线平行当且仅当它们的斜率相同。

For example, y = 3x + 1 and y = 3x − 5 are parallel because both have gradient 3. Their y-intercepts are different, so they are distinct lines.

例如，y = 3x + 1 和 y = 3x − 5 是平行的，因为两者的斜率都是 3。它们的 y 轴截距不同，因此是两条不同的直线。

When an IGCSE question asks for a line parallel to y = 2x + 7 passing through (1, 4), keep m = 2 and substitute the point into y = 2x + c to find c. Here, 4 = 2(1) + c, so c = 2, giving y = 2x + 2.

当 IGCSE 题目要求求一条与 y = 2x + 7 平行且经过点 (1, 4) 的直线时，保持 m = 2，将该点代入 y = 2x + c 求 c。此时 4 = 2(1) + c，所以 c = 2，得到 y = 2x + 2。

8. Perpendicular Lines | 垂直线

Perpendicular lines meet at a right angle of 90°. If the gradient of one line is m, the gradient of a line perpendicular to it is the negative reciprocal of m.

垂直线相交成 90° 直角。如果一条直线的斜率为 m，那么与其垂直的直线的斜率是 m 的负倒数。

m₁ × m₂ = −1

For a line with gradient 3, the perpendicular gradient is −1 ÷ 3 = −⅓. For a gradient of −2, the perpendicular gradient is 1 ÷ 2 = ½.

对于斜率为 3 的直线，垂直斜率为 −1 ÷ 3 = −⅓。对于斜率为 −2 的直线，垂直斜率为 1 ÷ 2 = ½。

Remember two special cases: a horizontal line has gradient 0, so its perpendicular line is vertical with undefined gradient; a vertical line has undefined gradient, so its perpendicular line is horizontal with gradient 0.

记住两个特殊情况：水平线的斜率为 0，因此它的垂直线是斜率未定义的竖直线；竖直线的斜率未定义，因此它的垂直线是斜率为 0 的水平线。

9. Real-life Contexts: Speed, Cost, and Conversion | 实际情境：速度、费用与换算

Straight line graphs are used to model many real-life relationships. In a distance-time graph, a straight line represents constant speed, and the gradient gives the speed.

直线图像用于建立许多实际生活关系的模型。在距离-时间图中，直线代表匀速运动，斜率给出速度。

In a cost equation such as C = 5n + 20, C is the total cost, n is the number of items, 5 is the cost per item, and 20 is the fixed charge. The gradient is 5 and the y-intercept is 20.

在诸如 C = 5n + 20 的费用方程中，C 是总费用，n 是物品数量，5 是每件物品的费用，20 是固定收费。斜率为 5，y 轴截距为 20。

Conversion graphs between currencies or units are also straight lines. For example, if 1 pound = 1.25 dollars, the equation is y = 1.25x, where the gradient is the exchange rate and the y-intercept is 0.

货币或单位之间的换算图也是直线。例如，如果 1 英镑 = 1.25 美元，则方程为 y = 1.25x，其中斜率是汇率，y 轴截距为 0。

10. Exam-style Questions and Common Mistakes | 考试型题目与常见错误

IGCSE exam questions often ask you to find a gradient, write an equation, or identify a missing coefficient. Follow a clear method and always substitute values carefully.

IGCSE 考试题经常要求你求斜率、写方程或找出缺失的系数。请遵循清晰的方法，并始终仔细代入数值。

Common mistake	Correct approach
Forgetting to rearrange into y = mx + c before reading m	First make y the subject, then identify the coefficient of x as m.
Using the wrong sign for a negative gradient	Check whether y increases or decreases as x increases.
Taking the reciprocal but forgetting the negative sign for perpendicular lines	Use m₁ × m₂ = −1 to test your answer.
Reading points that are not exactly on grid intersections	Choose two clear lattice points on the line.

Practise past-paper questions under timed conditions. Write your final equation in the form y = mx + c unless the question asks for a different form.

在计时条件下练习历年真题。除非题目要求使用其他形式，否则将最终方程写成 y = mx + c 的形式。

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Straight Line Graphs and Gradient | 直线图像与斜率