📚 Straight Line Graphs | 直线图像
In IGCSE Edexcel Mathematics, straight line graphs are one of the most important tools for modelling real-life situations and solving coordinate geometry problems. The equation of a straight line reveals two key pieces of information: how steeply the line rises or falls, and where it crosses the vertical axis.
在爱德思 IGCSE 数学中，直线图像是模拟现实情境和解决坐标几何问题最重要的工具之一。一条直线的方程揭示了两项关键信息：直线的陡峭程度，以及它与纵轴相交的位置。
1. The General Equation of a Straight Line | 直线的一般方程
Every straight line, except a vertical line, can be written in the form y = mx + c. Here, m is the gradient of the line and c is the y-intercept.
除竖直线外，任何一条直线都可以写成 y = mx + c 的形式。其中 m 是直线的梯度（即斜率），c 是这条直线在 y 轴上的截距。
For example, the equation y = 2x + 3 means that as x increases by 1 unit, y increases by 2 units, and the line passes through the point (0, 3).
例如，方程 y = 2x + 3 表示当 x 增加 1 个单位时，y 增加 2 个单位，并且直线经过点 (0, 3)。
There is also the alternative form ax + by = d for a straight line. This can be rearranged into the form y = mx + c.
直线还存在另一种一般形式：ax + by = d。这种形式可以改写成 y = mx + c 的形式。
2. Understanding Gradient | 理解梯度（斜率）
The gradient measures the steepness of a line. It is defined as the vertical change divided by the horizontal change between two distinct points on the line.
梯度衡量一条直线的倾斜程度。它的定义为：直线上两点之间的纵向变化量除以横向变化量。
m = (y₂ – y₁) ÷ (x₂ – x₁)
When the line slopes upward from left to right, the gradient is positive. When it slopes downward, the gradient is negative. A horizontal line has a gradient of 0, while a vertical line has an undefined gradient.
当直线从左到右向上倾斜时，梯度为正；当直线从左到右向下倾斜时，梯度为负。水平线的梯度为 0，竖直线的梯度不存在。
To find the gradient from a graph, choose two convenient points and draw a right-angled triangle between them. Then divide the vertical distance by the horizontal distance.
要从图像中求梯度，可以选择两个方便的点，并用它们构造一个直角三角形，然后用纵向距离除以横向距离。
3. The y-Intercept | y 轴截距
In the equation y = mx + c, the constant c is the y-intercept. This is the y-coordinate of the point where the line crosses the y-axis, namely (0, c).
在方程 y = mx + c 中，常数 c 是 y 轴截距。它是直线与 y 轴交点的纵坐标，即点 (0, c)。
For the line y = -4x + 5, the y-intercept is 5. This means the line passes through (0, 5). Even if the line is given in another form, the y-intercept is always found when x = 0.
对于直线 y = -4x + 5，y 轴截距是 5，因此直线经过点 (0, 5)。即使直线以其他形式给出，令 x = 0 总可以求出 y 轴截距。
4. Plotting Straight Line Graphs | 绘制直线图像
To plot the graph of a straight line, create a table of values for x and calculate the corresponding values of y. Then plot the points and join them with a straight line using a ruler.
要绘制直线图像，先列出 x 的值表，并计算对应的 y 值，然后把各点标在坐标平面上，最后用直尺将它们连成直线。
Consider y = 2x – 1. Choose values of x from -1 to 3.
以 y = 2x – 1 为例，选择 x 从 -1 到 3 的数值。
| x | -1 | 0 | 1 | 2 | 3 |
| y | -3 | -1 | 1 | 3 | 5 |
Each pair (x, y) is a point on the line. Because the graph is a straight line, you only actually need two points, but a third point helps to check for mistakes.
每一组 (x, y) 都是直线上的一个点。由于图像是直线，实际上只需两个点即可作图，但第三个点有助于检查错误。
5. Finding the Equation from a Graph | 由图像求直线方程
When a line is drawn on a graph, you can read off its equation in the form y = mx + c.
当直线画在图像上时，你可以读出它的方程，并将其写成 y = mx + c 的形式。
First, find where the line crosses the y-axis. This gives the value of c. Then pick two marked points and calculate the gradient using the formula:
首先，找出直线与 y 轴的交点，从而得到 c 的值。然后选择两个已标出的点，用公式求出梯度：
m = rising step ÷ running step
For instance, if a line crosses the y-axis at 2 and passes through (1, 3) and (2, 4), then the vertical change is 1, the horizontal change is 1, so m = 1. The equation is y = x + 2.
例如，一条直线与 y 轴交于 2，并且经过 (1, 3) 和 (2, 4) 两点，则纵向变化为 1，横向变化为 1，因此 m = 1。该直线的方程是 y = x + 2。
6. Finding the Equation from Two Points | 由两点求直线方程
Given two points (x₁, y₁) and (x₂, y₂), the gradient is calculated first.
已知两点 (x₁, y₁) 和 (x₂, y₂)，首先计算梯度。
m = (y₂ – y₁) ÷ (x₂ – x₁)
Then substitute m and one point into y = mx + c to find c.
然后将 m 和其中一个点代入 y = mx + c，解出 c。
Worked example: Find the equation of the line through (2, 5) and (4, 11).
示例：求经过 (2, 5) 和 (4, 11) 两点的直线方程。
Gradient = (11 – 5) ÷ (4 – 2) = 6 ÷ 2 = 3. Substituting (2, 5) gives 5 = 3 × 2 + c, so c = -1. The equation is y = 3x – 1.
梯度 = (11 – 5) ÷ (4 – 2) = 6 ÷ 2 = 3。将 (2, 5) 代入得 5 = 3 × 2 + c，所以 c = -1。方程为 y = 3x – 1。
7. Parallel Lines | 平行线
Parallel lines always have the same gradient. For example, y = 2x + 1 and y = 2x – 7 are parallel because both have gradient 2.
平行线的梯度始终相等。例如 y = 2x + 1 与 y = 2x – 7 是平行线，因为它们都有梯度 2。
Two lines are parallel if and only if their gradients are equal:
两条直线平行当且仅当它们的梯度相等：
m₁ = m₂
Notice that the y-intercepts may be different; the lines never meet because they have the same steepness but different starting points on the y-axis.
注意两条直线的 y 截距可以不同；由于它们倾斜程度相同，但起点不同，所以永远不会相交。
8. Perpendicular Lines | 垂直线
Perpendicular lines meet at a right angle. If one line has gradient m₁ and a perpendicular line has gradient m₂, then their gradients multiply to give -1.
垂直线以直角相交。如果一条直线梯度为 m₁，与之垂直的直线梯度为 m₂，那么这两条直线的梯度相乘等于 -1。
m₁ × m₂ = -1
For example, a line with gradient 3 is perpendicular to a line with gradient -1/3. Similarly, a line with gradient -1/2 is perpendicular to a line with gradient 2.
例如，梯度为 3 的直线垂直于梯度为 -1/3 的直线。同样，梯度为 -1/2 的直线垂直于梯度为 2 的直线。
One special case is horizontal and vertical lines. A horizontal line has gradient 0, and a vertical line has undefined gradient. These are always perpendicular to each other.
一个特殊情况是水平线和竖直线。水平线的梯度为 0，竖直线的梯度不存在，但水平线与竖直线永远互相垂直。
9. Intercepts with the Axes | 与坐标轴的交点
The y-intercept is the point where the line meets the y-axis, found by setting x = 0. The x-intercept is where the line meets the x-axis, found by setting y = 0.
y 轴截距是直线与 y 轴的交点，令 x = 0 即可求出；x 轴截距是直线与 x 轴的交点，令 y = 0 即可求出。
For the line y = 4x – 8, the y-intercept is -8. To find the x-intercept, set y = 0:
对于直线 y = 4x – 8，y 轴截距是 -8。要求 x 轴截距，令 y = 0：
0 = 4x – 8
4x = 8
x = 2
Therefore the line crosses the x-axis at (2, 0). Knowing both intercepts is often the fastest way to draw a straight line.
因此这条直线在 x 轴上的交点为 (2, 0)。已知两个截距往往是画直线最快的方法。
10. Solving Equations Using Straight Line Graphs | 用直线图像解方程
Straight line graphs can be used to solve linear equations visually. If you draw the line y = 2x – 3 and the line y = 1 on the same axes, the x-coordinate of their intersection is the solution to 2x – 3 = 1.
直线图像可用于直观地求解线性方程。如果把 y = 2x – 3 和 y = 1 画在同一坐标平面内，它们交点的 x 坐标就是方程 2x – 3 = 1 的解。
To solve a linear equation by graphing, rearrange the equation so that one side is y = mx + c and the other side is a horizontal line. Then read off the x-value at the intersection point.
用图解法的步骤是：把原方程重新整理，使一边成为 y = mx + c 的直线，另一边成为一条水平线，然后在交点处读出 x 值。
In an exam, graphical methods may be less accurate than algebra, but they provide a useful check and help to show the meaning of a solution.
在考试中，图解法的精度可能不如代数法，但它能作为有效的检查手段，并帮助展示解的意义。
11. Real-Life Applications of Linear Graphs | 线性图像的实际应用
Linear graphs appear constantly in everyday contexts. For example, the cost of hiring a taxi might be C = 3.5d + 5, where d is the distance travelled and 5 is the initial charge.
直线图像在日常生活中随处可见。例如，出租车费用可表示为 C = 3.5d + 5，其中 d 是行驶距离，5 是起步价。
Another common example is unit conversion. Suppose one pound sterling is equal to 1.15 euros. Then the conversion graph E = 1.15P is a straight line through the origin.
另一个常见例子是单位换算。假设 1 英镑等于 1.15 欧元，那么换算图 E = 1.15P 就是一条经过原点的直线。
Using a straight line model allows us to make predictions. For instance, a mobile phone contract may include a fixed monthly fee plus a cost per gigabyte of data used. This relationship is linear and can be represented and analysed with the techniques above.
使用直线模型可以帮助我们作出预测。例如，手机套餐可能包含固定月租费加上每 GB 流量的费用。这种关系是线性的，可以用上述方法表示和分析。
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