Graphs 1: Linear Graphs and Gradients | 图像基础：线性图像与斜率

📚 Graphs 1: Linear Graphs and Gradients | 图像基础：线性图像与斜率

Linear graphs are one of the most important topics in IGCSE Mathematics. They appear in algebra, coordinate geometry, and even in real-life problems involving rates and trends. Understanding how to plot, interpret, and manipulate straight-line graphs is essential for exam success.

线性图像是IGCSE数学中最重要的主题之一。它出现在代数、坐标几何，甚至涉及速率与趋势的现实问题中。理解如何绘制、解读和操作直线图像，是考试取得好成绩的关键。

1. What Is a Linear Graph? | 什么是线性图像？

A linear graph is a straight-line graph that represents a linear equation. In this graph, the relationship between two variables, usually \(x\) and \(y\), can be written in the form \(y = mx + c\). The plot of such an equation is always a straight line.

线性图像是表示线性方程的直线图形。在这类图像中，两个变量（通常为 \(x\) 和 \(y\)）之间的关系可以写成 \(y = mx + c\) 的形式。这类方程的图像总是一条直线。

For example, the graph of \(y = 2x + 1\) is a straight line. As \(x\) increases, \(y\) increases at a constant rate. This constant rate is what makes the graph linear.

例如，\(y = 2x + 1\) 的图像是一条直线。随着 \(x\) 增大，\(y\) 以恒定速率增大。这个恒定速率正是图像呈线性的原因。

2. The Cartesian Coordinate System | 笛卡尔坐标系

To draw a linear graph, we use the Cartesian coordinate system. It consists of two number lines that intersect at a right angle at the origin \((0,0)\). The horizontal axis is called the \(x\)-axis, and the vertical axis is called the \(y\)-axis.

为了绘制线性图像，我们使用笛卡尔坐标系。它由两条在原点 \((0,0)\) 处垂直相交的数轴组成。水平轴称为 \(x\) 轴，垂直轴称为 \(y\) 轴。

Any point on this plane is written as \((x, y)\). The first number is the horizontal coordinate, and the second number is the vertical coordinate. For example, the point \((3, -2)\) is 3 units right and 2 units down from the origin.

平面上的任意点写作 \((x, y)\)。第一个数是水平坐标，第二个数是垂直坐标。例如，点 \((3, -2)\) 在原点右侧3个单位、下方2个单位处。

\((x, y) \rightarrow \text{horizontal first, vertical second}\)

\((x, y) \rightarrow \text{先水平，后垂直}\)

3. Plotting Linear Graphs | 绘制线性图像

To plot a linear graph, we need at least two points, but it is safer to use three points to avoid mistakes. We can choose values for \(x\), substitute them into the equation, and calculate the corresponding \(y\) values.

要绘制线性图像，至少需要两个点，但为了减少错误，最好使用三个点。我们可以选取 \(x\) 的值，将其代入方程，计算对应的 \(y\) 值。

For the equation \(y = 2x + 1\), let us create a table of values:

对于方程 \(y = 2x + 1\)，让我们建立一张数值表：

\(x\)	0	1	2
\(y\)	1	3	5

Plot the points \((0,1)\), \((1,3)\), and \((2,5)\) on the coordinate grid, then draw a straight line through them. Extend the line across the grid, and label it with its equation.

在坐标格上标出点 \((0,1)\)、\((1,3)\) 和 \((2,5)\)，然后过这些点画一条直线。将直线向两端延伸，并标注其方程。

4. Gradient of a Straight Line | 直线的斜率

The gradient (also called slope) measures how steep a line is. It is the ratio of the vertical change to the horizontal change between any two points on the line.

斜率（gradient）衡量一条线的陡峭程度。它是直线上任意两点之间垂直变化与水平变化的比值。

\(\text{Gradient} = \frac{\text{change in }y}{\text{change in }x} = \frac{\Delta y}{\Delta x}\)

\(\text{斜率} = \frac{y \text{的变化量}}{x \text{的变化量}} = \frac{\Delta y}{\Delta x}\)

For the line passing through points \((1,3)\) and \((3,7)\), the change in \(y\) is \(7 – 3 = 4\), and the change in \(x\) is \(3 – 1 = 2\). So the gradient is \(4 \div 2 = 2\).

对于经过点 \((1,3)\) 和 \((3,7)\) 的直线，\(y\) 的变化量为 \(7 – 3 = 4\)，\(x\) 的变化量为 \(3 – 1 = 2\)。因此斜率为 \(4 \div 2 = 2\)。

A positive gradient means the line rises from left to right. A negative gradient means the line falls from left to right. A horizontal line has gradient 0, and a vertical line has an undefined gradient.

正斜率表示直线从左向右上升。负斜率表示直线从左向右下降。水平线斜率为0，垂直线斜率未定义。

5. Intercepts: Where the Line Meets the Axes | 截距：直线与轴的交点

The y-intercept is the point where the line crosses the y-axis. At this point, \(x = 0\). Similarly, the x-intercept is the point where the line crosses the x-axis, where \(y = 0\).

\(y\) 截距是直线与 \(y\) 轴相交的点。在该点处，\(x = 0\)。类似地，\(x\) 截距是直线与 \(x\) 轴相交的点，此时 \(y = 0\)。

For \(y = 2x + 1\), substitute \(x = 0\) to get \(y = 1\), so the y-intercept is \((0,1)\). To find the x-intercept, set \(y = 0\): \(0 = 2x + 1\), giving \(x = -0.5\). Thus the x-intercept is \((-0.5, 0)\).

对于 \(y = 2x + 1\)，代入 \(x = 0\) 得到 \(y = 1\)，所以 \(y\) 截距是 \((0,1)\)。要求 \(x\) 截距，令 \(y = 0\)：\(0 = 2x + 1\)，解得 \(x = -0.5\)。因此 \(x\) 截距是 \((-0.5, 0)\)。

\(y\text{-intercept: set } x = 0; \quad x\text{-intercept: set } y = 0\)

\(y\) 截距：令 \(x = 0\)；\(x\) 截距：令 \(y = 0\)

6. Equation of a Line: \(y = mx + c\) | 直线方程：\(y = mx + c\)

In the form \(y = mx + c\), \(m\) represents the gradient and \(c\) represents the y-intercept. This is known as the slope-intercept form.

在 \(y = mx + c\) 中，\(m\) 表示斜率，\(c\) 表示 \(y\) 截距。这就是所谓的一次函数一般式。

For example, in the equation \(y = -3x + 5\), the gradient is \(-3\) and the y-intercept is \(5\). The line slopes downwards from left to right because the gradient is negative.

例如，在方程 \(y = -3x + 5\) 中，斜率为 \(-3\)，\(y\) 截距为 \(5\)。由于斜率为负，该直线从左到右向下倾斜。

\(y = mx + c \quad \text{where } m = \text{gradient}, \ c = y\text{-intercept}\)

\(y = mx + c\)，其中 \(m\) 为斜率，\(c\) 为 \(y\) 截距

7. Finding the Equation of a Line from Two Points | 由两点求直线方程

Sometimes you are given two points and asked to find the equation of the line passing through them. The method has two steps: first find the gradient, then find the y-intercept.

有时你会被给定两个点，并要求找出经过这两点的直线方程。方法分两步：先求斜率，再求 \(y\) 截距。

Given points \((2,3)\) and \((6,11)\):

给定点 \((2,3)\) 和 \((6,11)\)：

Find the gradient: \(\frac{11-3}{6-2} = \frac{8}{4} = 2\)

求斜率：\(\frac{11-3}{6-2} = \frac{8}{4} = 2\)
Substitute one point into \(y = mx + c\): \(3 = 2(2) + c\), so \(c = -1\).

将其中一个点代入 \(y = mx + c\)：\(3 = 2(2) + c\)，得 \(c = -1\)。
Write the equation: \(y = 2x – 1\).

写出方程：\(y = 2x – 1\)。

Always check your result by substituting the other point. If both points satisfy the equation, your answer is correct.

务必代入另一个点来检验。如果两个点都满足方程，说明你的答案正确。

8. Parallel and Perpendicular Lines | 平行线与垂直线

Two lines are parallel if they have the same gradient but different y-intercepts. For example, \(y = 3x + 1\) and \(y = 3x – 4\) are parallel because both have gradient 3.

如果两条直线斜率相同但 \(y\) 截距不同，则它们平行。例如，\(y = 3x + 1\) 和 \(y = 3x – 4\) 平行，因为它们的斜率都为3。

Two lines are perpendicular if the product of their gradients is \(-1\). In other words, one gradient is the negative reciprocal of the other. For example, a line with gradient \(2\) is perpendicular to a line with gradient \(-\frac{1}{2}\).

如果两条直线的斜率乘积为 \(-1\)，则它们垂直。换句话说，一条直线的斜率是另一条斜率的负倒数。例如，斜率为 \(2\) 的直线垂直于斜率为 \(-\frac{1}{2}\) 的直线。

Parallel: \(m_1 = m_2\) Perpendicular: \(m_1 \times m_2 = -1\)

平行：\(m_1 = m_2\) 垂直：\(m_1 \times m_2 = -1\)

9. Real-Life Applications of Linear Graphs | 线性图像的实际应用

Linear graphs are used to model many real-world situations where one quantity changes at a constant rate with another. For example, a taxi fare may be a fixed charge plus a constant rate per kilometre. If the fixed charge is 3 units and the rate is 2 units per km, the fare \(F\) is \(F = 2d + 3\), where \(d\) is the distance in km.

线性图像常用于建模一个量相对于另一个量以恒定速率变化的情形。例如，出租车费用可能是固定起步价加上每公里恒定的费用。如果起步价为3单位，每公里费率为2单位，则车费 \(F\) 为 \(F = 2d + 3\)，其中 \(d\) 是以公里为单位的距离。

Another application is temperature conversion. The formula to convert Celsius to Fahrenheit is \(F = \frac{9}{5}C + 32\), which is a linear equation. Its graph is a straight line that helps us compare the two temperature scales.

另一个应用是温度换算。摄氏温度转华氏温度的公式为 \(F = \frac{9}{5}C + 32\)，这是一个线性方程。其图像是一条直线，帮助我们比较两种温标。

Speed-time graphs, distance-time graphs, and currency exchange rates are also everyday examples of linear relationships.

速度-时间图、距离-时间图以及汇率换算也都是线性关系的日常例子。

10. Common Mistakes and How to Avoid Them | 常见错误及避免方法

Students often make a few typical errors when dealing with linear graphs. Here are some important points to remember.

学生在处理线性图像时经常会犯一些典型错误。以下是一些重要提示，请务必记住。

Mixing up the x-intercept and y-intercept. Remember that the y-intercept is found by setting \(x = 0\), and the x-intercept by setting \(y = 0\).

混淆 \(x\) 截距和 \(y\) 截距。记住，求 \(y\) 截距时令 \(x = 0\)，求 \(x\) 截距时令 \(y = 0\)。
Using the wrong formula for gradient. Always use \(\frac{y_2 – y_1}{x_2 – x_1}\), not \(\frac{x_2 – x_1}{y_2 – y_1}\).

使用错误的斜率公式。始终用 \(\frac{y_2 – y_1}{x_2 – x_1}\)，而不是 \(\frac{x_2 – x_1}{y_2 – y_1}\)。
Forgetting that a negative gradient means the line decreases from left to right.

忘记负斜率表示直线从左到右递减。
When finding the perpendicular gradient, make sure you change the sign and take the reciprocal. For gradient \(-\frac{3}{4}\), the perpendicular gradient is \(\frac{4}{3}\).

求垂直斜率时，务必改变符号并取倒数。例如斜率 \(-\frac{3}{4}\) 的垂直斜率为 \(\frac{4}{3}\)。

11. Practice Questions | 练习题

Try these questions to test your understanding. The answers are provided below but attempt them first.

尝试以下问题来测试你的理解。答案在下面，但请先自己动手做。

Find the gradient of the line passing through \((1,2)\) and \((5,14)\).

求经过 \((1,2)\) 和 \((5,14)\) 的直线的斜率。
Write the equation of the line with gradient \(-2\) and y-intercept \(7\).

写出斜率为 \(-2\) 且 \(y\) 截距为 \(7\) 的直线方程。
Find the x-intercept of \(y = 3x – 12\).

求 \(y = 3x – 12\) 的 \(x\) 截距。
Determine which of the lines \(y = 4x – 1\) and \(y = -\frac{1}{4}x + 3\) are perpendicular.

判断直线 \(y = 4x – 1\) 与 \(y = -\frac{1}{4}x + 3\) 是否垂直。

Answers: 1. \(3\) 2. \(y = -2x + 7\) 3. \(x = 4\) 4. Yes, because \(4 \times -\frac{1}{4} = -1\)

答案：1. \(3\) 2. \(y = -2x + 7\) 3. \(x = 4\) 4. 垂直，因为 \(4 \times -\frac{1}{4} = -1\)

12. Summary | 总结

Linear graphs are a fundamental topic in IGCSE Mathematics. You should be able to plot a line from its equation, calculate the gradient, identify intercepts, and find the equation of a line from two points. Parallel and perpendicular relationships, as well as real-life applications, are also essential skills.

线性图像是IGCSE数学的基础主题。你应该能够根据方程绘制直线图像，计算斜率，识别截距，并从两个点求直线方程。平行与垂直关系以及现实应用也是必备技能。

Remember to check your work carefully, especially when substituting values and simplifying fractions. With regular practice, linear graphs will become one of the easiest and most rewarding topics for you.

记得仔细检查计算过程，特别是代入数值和化简分数时。定期练习后，线性图像将成为你最轻松且最容易得分的主题之一。

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Graphs 1: Linear Graphs and Gradients | 图像基础：线性图像与斜率