📚 KS3 Maths: Drawing Graphs Practice | KS3 数学：绘图练习
Drawing graphs is a core skill in the KS3 Cambridge Lower Secondary mathematics curriculum. It connects algebra, coordinates and real-life data, helping students visualise the relationship between two variables. This article provides guided practice on reading and plotting coordinates, building tables of values, drawing straight lines and curves, and interpreting the most common graph types examined at KS3 level.
绘图是 KS3 剑桥初中数学课程的核心技能。它把代数、坐标与实际数据联系起来，帮助学生直观理解两个变量之间的关系。本文提供系统练习：读坐标与描点、制作数值表、绘制直线与曲线，以及解读 KS3 阶段最常见的几类图像。
1. Coordinates and the Cartesian Plane | 坐标与笛卡尔平面
The Cartesian plane is formed by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). A point is written as an ordered pair (x, y), where x gives the horizontal distance from the origin and y gives the vertical distance. The order matters: (3, 5) and (5, 3) are two completely different points.
笛卡尔平面由两条互相垂直的数轴组成：x 轴（水平）和 y 轴（垂直）。一个点用有序数对 (x, y) 表示，其中 x 表示距原点的水平距离，y 表示垂直距离。顺序非常重要：(3, 5) 与 (5, 3) 是两个完全不同的点。
The origin is (0, 0), the point where the axes cross. The two axes divide the plane into four quadrants. In the first quadrant both coordinates are positive; in the second quadrant x is negative and y is positive; in the third quadrant both are negative; in the fourth quadrant x is positive and y is negative.
原点是 (0, 0)，即两条轴相交的点。两条轴把平面分成四个象限。第一象限两个坐标均为正；第二象限 x 为负、y 为正；第三象限两个坐标均为负；第四象限 x 为正、y 为负。
| Quadrant 象限 | Sign of x x 的符号 | Sign of y y 的符号 | Example 例子 |
|---|---|---|---|
| First 第一 | positive 正 | positive 正 | (4, 3) |
| Second 第二 | negative 负 | positive 正 | (-4, 3) |
| Third 第三 | negative 负 | negative 负 | (-4, -3) |
| Fourth 第四 | positive 正 | negative 负 | (4, -3) |
ave x = 0, for example (0, 6); points on the x-axis always have y = 0, for example (6, 0).
恰好落在坐标轴上的点不属于任何象限。y 轴上的点总有 x = 0，例如 (0, 6)；x 轴上的点总有 y = 0，例如 (6, 0)。
2. Plotting Points from Coordinates | 根据坐标描点
To plot a point such as (3, -2), start at the origin. Move 3 units right because x is positive, then move 2 units down because y is negative. Mark the position with a small cross and label the point if the question requires it.
要描出 (3, -2) 这样的点，从原点开始。因为 x 为正，向右移动 3 个单位；因为 y 为负，向下移动 2 个单位。用小叉标出位置，若题目要求则标上字母。
Always check the scale on each axis before plotting, because one square may represent 1, 2, 5 or 10 units. If the scale is not one square to one unit, do not count squares as single units.
描点前务必检查每个轴的刻度，因为一个方格可能代表 1、2、5 或 10 个单位。如果刻度不是"一格一单位"，就不要把一个方格当作一个单位来数。
- Read the coordinates in the correct order: x first, then y.
按正确顺序读坐标：先 x，后 y。 - Move along the x-axis first, then move vertically, not diagonally.
先沿 x 轴移动，再竖直移动，不要斜着走。 - Use a sharp pencil and a ruler for the axes so points stay visible.
用削尖的铅笔和直尺画坐标轴，保证点迹清晰。 - Write coordinates with brackets and a comma: (7, 2), never 7, 2 or (7.2).
坐标要带括号和逗号：(7, 2)，不要写成 7, 2 或 (7.2)。
Working backwards is equally examinable: given a marked point on a grid, describe its coordinates by counting from the origin. Count left or down for negative values and record the two numbers in the order (x, y).
反向练习同样常见：给出网格上已标出的点，通过从原点起数格来确定其坐标。向左或向下要记为负值，并始终按 (x, y) 的顺序写出两个数字。
3. Tables of Values for Linear Equations | 线性方程的数值表
To draw the graph of a linear equation such as y = 2x + 1, first choose a sensible range of x values, for example from -3 to 3. Substitute each x into the equation to find the matching y value, then record the results in a table of values.
要绘制 y = 2x + 1 这样的线性方程图像，首先选择一个合适的 x 值范围，例如从 -3 到 3。把每个 x 代入方程求出对应的 y 值，再把结果记录到数值表中。
ply before adding or subtracting. For x = -3, work out 2 × (-3) + 1 = -6 + 1 = -5, not -7.
代入时必须遵循运算顺序：先乘除后加减。当 x = -3 时，应算 2 × (-3) + 1 = -6 + 1 = -5，而不是 -7。
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| y = 2x + 1 | -5 | -3 | -1 | 1 | 3 | 5 | 7 |
A quick check: the y values must go up by the same amount each time x increases by 1. Here they rise by 2, which is exactly the coefficient of x, so the table is correct.
快速检验：x 每增加 1，y 值应按相同的量变化。这里的 y 每次增加 2，正好等于 x 的系数，说明表格正确。
Three points are enough to draw a straight line, but five to seven points give a safety net: if one point is wrong, it will clearly fall off the line and you can spot the error before drawing.
画直线其实三个点就够了，但列五到七个点更保险：如果某个点算错，它会明显偏离直线，你在画线之前就能发现错误。
4. Drawing Straight-Line Graphs from Tables | 根据数值表绘制直线图像
Once the table is complete, plot each pair of coordinates carefully, then join the points with a single straight line drawn with a ruler. Extend the line across the whole grid unless the question restricts the domain, and always write the equation of the line next to it.
表格填完后，仔细描出每一组坐标，然后用直尺把这些点连成一条直线。除非题目限定了定义域，否则要把直线延伸到整个网格，并在直线旁写上它的方程。
- Line up the ruler so that all points sit on the same straight edge before drawing.
摆放直尺时，先确认所有点都落在同一条直边上，再落笔画线。 - Extend the line to the edge of the grid and add arrowheads if the graph continues.
把直线延伸到网格边缘，如果图像继续延伸就加上箭头。 - Label the line as y = 2x + 1 so the marker can identify it instantly.
在直线旁标注 y = 2x + 1，方便阅卷老师快速识别。 - Never join plotted points with separate little segments - a linear graph is one line.
绝不要用小线段把点一段一段地连起来——线性图像就是一条直线。
If a plotted point does not lie on the line formed by the others, do not bend the line to reach it. Recalculate the y value instead; the calculation is almost always the source of the mistake.
如果某个描出的点不在其余各点连成的直线上，不要为了迁就它而把线弯过去，而应重新计算该 y 值——错误几乎总是出在计算上。
5. Gradient and Intercept | 斜率与截距
Every straight-line graph can be written in the form y = mx + c, where m is the gradient (steepness) and c is the y-intercept, the value of y where the line crosses the y-axis.
每条直线图像都可以写成 y = mx + c 的形式，其中 m 是斜率（倾斜程度），c 是 y 轴截距，即直线与 y 轴交点处的 y 值。
gradient m = change in y ÷ change in x 斜率 m = y 的变化量 ÷ x 的变化量
A positive gradient means the line rises from left to right; a negative gradient means it falls. A horizontal line has gradient 0, and a vertical line has a gradient that is undefined because the change in x is zero and division by zero is not allowed.
斜率为正表示直线从左向右上升；斜率为负表示下降。水平线的斜率为 0；竖直线的斜率不存在，因为 x 的变化量为 0，而除以 0 是不允许的。
| Equation 方程 | Gradient 斜率 | y-intercept y 轴截距 |
|---|---|---|
| y = 3x + 2 | 3 | 2 |
| y = -x + 5 | -1 | 5 |
| y = ½x | ½ | 0 |
| y = 4 | 0 | 4 |
he gradient from a graph, choose two points that lie exactly on lattice crossings and count the vertical and horizontal steps between them.
平行线的斜率相同但截距不同，例如 y = 2x + 1 与 y = 2x - 6。要从图像上求斜率，就选两个正好落在格点上的点，数出它们之间竖直与水平的步数。
6. Drawing Quadratic Curves | 绘制二次曲线
A quadratic equation contains x² and produces a smooth curve called a parabola. It is never drawn with a ruler. Build a table of values first, plot the points, then join them with a single smooth curve.
二次方程含有 x²，其图像是一条平滑曲线，称为抛物线。它绝不能用直尺画。先列数值表，再描点，最后用一条平滑曲线把它们连起来。
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| y = x² | 9 | 4 | 1 | 0 | 1 | 4 | 9 |
controls how narrow the curve is: y = 3x² is narrower than y = x², while y = ½x² is wider.
y = x² 的图像关于 y 轴对称，最低点（最小值点）在原点。x² 的系数控制曲线的宽窄：y = 3x² 比 y = x² 更窄，而 y = ½x² 更宽。
nger symmetric about the y-axis but about a vertical line through its turning point. The table below is used to sketch it.
对于同时含 x² 项和 x 项的二次式，如 y = x² - 4x + 3，曲线不再关于 y 轴对称，而是关于过顶点的一条竖直线对称。下表用于绘制它的草图。
| x | -1 | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|---|
| y = x² - 4x + 3 | 8 | 3 | 0 | -1 | 0 | 3 | 8 |
equation x² - 4x + 3 = 0, and reading them off a graph is a standard KS3 and IGCSE skill.
这条曲线在 x = 1 和 x = 3 处与 x 轴相交，此时 y = 0。这些交点称为方程 x² - 4x + 3 = 0 的根（解），从图像上读出它们，是 KS3 与 IGCSE 的标准技能。
7. Cubic and Reciprocal Graphs | 三次与反比例图像
no maximum or minimum point. Because the values grow quickly, a scale such as 2 or 5 units per square is usually needed.
三次方程含有 x³，其图像呈 S 形。y = x³ 的图像经过原点，没有最大值点或最小值点。由于函数值增长很快，通常需要采用每格 2 或 5 个单位的刻度。
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| y = x³ | -27 | -8 | -1 | 0 | 1 | 8 | 27 |
is impossible, so the curve splits into two separate branches and never touches either axis.
反比例图像的形式是 y = 1 ÷ x，更一般地写作 y = k ÷ x。它在 x = 0 处没有定义，因为不能除以 0，所以曲线分成两条互不相连的分支，并且永远不会与坐标轴相交。
| x | -3 | -2 | -1 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|
| y = 1 ÷ x | -0.33 | -0.5 | -1 | 1 | 0.5 | 0.33 |
Notice the mirror symmetry: for every positive value there is a matching negative value. When k is positive, one branch lies in the first quadrant and the other in the third quadrant; when k is negative, the branches lie in the second and fourth quadrants.
注意镜像对称性：每一个正值都对应一个负值。当 k 为正时，一条分支在第一象限，另一条在第三象限；当 k 为负时，两条分支分别位于第二和第四象限。
own very steeply. Remember that the curve must approach the axes without ever crossing or touching them.
在 x = 0 附近，两条分支会非常陡地向上和向下延伸。请记住，曲线只能无限接近坐标轴，绝不能穿过或接触坐标轴。
8. Real-Life Graphs and Conversion Graphs | 实际情境图像与换算图像
Real-life graphs turn word problems into pictures. A conversion graph is a straight line through the origin that changes one unit into another, such as kilometres into miles or pounds into dollars. The gradient of this line is the conversion rate.
实际情境图像把文字应用题变成图形。换算图像是一条过原点的直线，用来把一种单位转换成另一种，例如千米换算成英里、英镑换算成美元。这条直线的斜率就是换算比率。
To use a conversion graph, find the known value on its axis, move vertically or horizontally to the line, then move straight to the other axis and read the answer. A graph that converts 5 km to 3.1 miles can convert any distance without a formula, which is why examiners love them.
使用换算图像的方法：在已知量所在的轴上找到该值，竖直或水平移动到直线上，再垂直移动到另一条轴读出答案。一张把 5 km 换算为 3.1 英里的图，无需公式就能换算任意距离，这也是考官偏爱它的原因。
On a distance-time graph the vertical axis shows distance from the starting point and the horizontal axis shows time. A horizontal section means the object is stationary, not that it is travelling slowly, and the gradient of each sloping section gives the speed.
在距离—时间图像中，竖直轴表示距出发点的距离，水平轴表示时间。图形中的水平线段表示物体静止，而不是走得慢；每段斜线的斜率则给出速度。
speed = distance ÷ time 速度 = 距离 ÷ 时间
On a speed-time graph the roles change: the gradient measures acceleration and the area under the line measures the distance travelled. A flat section here means constant speed, which is very different from the flat section of a distance-time graph.
在速度—时间图像中，角色互换：斜率表示加速度，直线下方的面积表示行驶的距离。这里的水平线段表示匀速运动，这与距离—时间图像中的水平线段含义完全不同。
Filling-container graphs are another favourite: the height of water rises quickly in a narrow part of a container and slowly in a wide part, so the shape of the graph mirrors the shape of the container.
注水容器图像同样常见：水位在容器较窄的部分上升很快，在较宽的部分上升较慢，因此图像形状与容器形状相互对应。
9. Choosing Suitable Scales and Labelling Axes | 选择合适的刻度与轴标签
Before plotting anything, decide the scale. Look at the largest value you need to show, count how many squares the grid provides, and choose a step that divides neatly: 1, 2, 5, 10 or 20 units per square. Awkward scales such as 3 or 7 make accurate plotting difficult.
动手描点之前，先决定刻度。看看需要表示的最大数值，数一数网格能提供多少格，然后选择一个能整除的步长：每格 1、2、5、10 或 20 个单位。像 3 或
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