📚 GCSE CIE Maths: Graphs – Key Points Explained | CIE GCSE 数学：图表考点精讲
Graphs form a central pillar of the CIE IGCSE Mathematics syllabus, bridging algebra, geometry, statistics, and real‑world problem‑solving. Whether you are sketching a quadratic curve, interpreting a speed‑time graph, or drawing a cumulative frequency diagram, a firm grasp of graphical techniques is essential for achieving a top grade. This article unpacks every key graph type you will encounter, focusing on plotting, reading, and applying graphs efficiently in an exam setting.
图表是 CIE IGCSE 数学课程的核心内容之一，它连接了代数、几何、统计以及实际问题的解决。无论你是要画二次函数曲线、分析速度‑时间图，还是绘制累积频率图，扎实的图形技巧对于取得高分至关重要。本文将逐一讲解你会遇到的所有主要图表类型，重点放在考试的绘图、读图与应用技巧上。
1. The Coordinate Plane and Plotting Basics | 坐标平面与绘图基础
All graphs rely on the Cartesian plane, formed by a horizontal x‑axis and a vertical y‑axis intersecting at the origin (0,0). Coordinates are always written as (x, y). When plotting a point, move along the x‑axis first, then parallel to the y‑axis. Accurate plotting is vital: a single mis‑plotted point can distort a straight line or curve, costing marks in both drawing and interpretation questions.
所有图表都建立在笛卡尔平面上，它由水平的 x 轴和垂直的 y 轴相交于原点 (0,0) 构成。坐标始终写成 (x, y) 的形式。描点时先沿 x 轴移动，再平行于 y 轴移动。精确绘图至关重要：一个点标错就可能使直线或曲线变形，从而在画图与读图题中失分。
- Use a sharp pencil and plot crosses (×) or small dots. | 使用削尖的铅笔，描点用叉号 (×) 或小圆点。
- Label axes clearly with the quantity and unit, e.g. 'Time (s)'. | 坐标轴上要清楚标出物理量及单位，例如 ‘Time (s)’。
- sensible scale: 1 cm = 1, 2, 5, 10, 20, 50, etc. Avoid scales like 1 cm = 3 unless asked. | 选择合理的比例尺：1 cm = 1, 2, 5, 10, 20, 50 等。除非题目要求，不要用 1 cm = 3 这样的刻度。
2. Straight‑Line Graphs and the Equation y = mx + c | 直线图与方程 y = mx + c
of any straight line can be written as y = mx + c, where m is the gradient and c is the y‑intercept (the point where the line crosses the y‑axis). To plot a straight line, you only need three points – a third acts as a check. You can generate points by substituting x‑values into the equation, or by using the gradient‑intercept method: start at (0, c), then use the rise/run from m to find other points.
任何直线方程都可以写成 y = mx + c，其中 m 是斜率，c 是 y 轴截距（即直线与 y 轴的交点）。画一条直线只需要三个点——第三个点用作检验。你可以通过代入 x 值到方程中生成点，或者使用斜截法：从 (0, c) 出发，然后按照 m 给出的纵步/横步寻找其余的点。
- Gradient m = (change in y) / (change in x) = rise / run. | 斜率 m = (y 的变化量) / (x 的变化量) = 纵向变化 / 横向变化。
- Parallel lines have the same gradient. | 平行线具有相同的斜率。
- Perpendicular lines have gradients whose product is −1: m₁ × m₂ = −1. | 互相垂直的直线，斜率之积为 −1：m₁ × m₂ = −1。
If a line passes through (x₁, y₁) and (x₂, y₂), then m = (y₂ − y₁) / (x₂ − x₁).
如果一条直线经过 (x₁, y₁) 和 (x₂, y₂)，则斜率 m = (y₂ − y₁) / (x₂ − x₁)。
3. Quadratic Graphs – Parabolas | 二次函数图像——抛物线
orm y = ax² + bx + c, and its graph is a parabola. If a > 0, the parabola opens upwards (U‑shaped, with a minimum point). If a < 0, it opens downwards (∩‑ d to complete a table of values, plot at least five to seven points smoothly, and label the curve.
二次函数的形式为 y = ax² + bx + c，其图像是一条抛物线。若 a > 0，抛物线开口向上（U 形，有最低点）；若 a < 0，则开口向下（∩ 形，有最高点）。对称轴是直线 x = −b/(2a)，而转折点（顶点）就在这条轴上。你需要完成数值表，至少描出五到七个点并光滑连线，最后标出曲线名称。
- Complete the table of values by substituting x into the expression; show your working for safety. | 把 x 值代入表达式完成数值表；写出计算过程以确保无误。
- Join points with a smooth, freehand curve – never use a ruler. | 用光滑的手绘曲线连接各点——切勿用尺子。
- If asked to solve ax² + bx + c = 0, the solutions are the x‑values where the curve crosses the x‑axis (the roots). | 若要解 ax² + bx + c = 0，解即为曲线与 x 轴交点的 x 值（即方程的根）。
- may also be asked to draw the line y = d and find intersections, solving ax² + bx + c = d. | 你可能还需要画出直线 y = d，然后找出交点，从而解方程 ax² + bx + c = d。
4. Reciprocal and Exponential Graphs | 反比例函数与指数函数图像
simple reciprocal graphs such as y = k/x (a hyperbola) and exponential graphs like y = kaˣ st and third quadrants (if k > 0) or second and fourth (if k < 0). The axes are asymptotes – the curve approaches them but never touches. Exponentials, such as y = 2ˣ, grow rapidly in one direction and approach the x‑axis in the other.
CIE 考试中常出现简单的反比例函数图像，如 y = k/x（双曲线），以及指数函数图像，如 y = kaˣ。对于 y = k/x，若 k > 0，图像有两支分别在第一和第三象限；若 k < 0，则在第二和第四象限。坐标轴是渐近线——曲线无限接近但永不相交。指数函数如 y = 2ˣ 在一端急速增长，在另一端趋近于 x 轴。
- For reciprocal graphs, avoid x = 0 in your table of values – it is undefined. | 反比例函数的数值表中避免取 x = 0——该点无定义。
- Plot enough points to capture the steep change near the asymptotes. | 在渐近线附近要取足够多的点以反映陡峭的变化。
- pes quickly: reciprocal graphs never cross the axes; exponential graphs always cross the y‑axis at (0, a) because y = ka⁰ = k. | 快速辨认形状：反比例曲线永不过轴；指数曲线总在 (0, a) 穿过 y 轴，因为 y = ka⁰ = k。
5. Distance‑Time Graphs | 距离‑时间图
A distance‑ raight, sloping line represents constant speed. The gradient of the line gives the speed: gradient = (change in distance) / (change in time). A horizontal line indicates the object is stationary. A curved line implies acceleration or deceleration – instantaneous speed at a point is found by drawing a tangent and calculating its gradient.
距离‑时间图显示物体离开起点的距离随时间的变化。时间总是在横轴，距离在纵轴。一条倾斜的直线表示匀速运动，斜率给出速度：斜率 = (距离的变化量) / (时间的变化量)。水平线表示物体静止不动。曲线代表加速或减速——某点的瞬时速度可通过作该点的切线并计算切线的斜率得出。
- Always check the scale on both axes before calculating speed. | 计算速度前务必检查两条轴的刻度。
- The steeper the line, the greater the speed. | 直线越陡，速度越大。
- If the graph returns to the time axis, the total distance is read from the highest point; the displacement may be zero if it returns to the start. | 若图像回到时间轴，总距离从最高点读取；若回到起点，位移可能为零。
6. Speed‑Time Graphs | 速度‑时间图
In a speed‑time graph, speed is on the vertical axis and time on the horizontal. The gradient represents acceleration: a steeper gradient means greater acceleration. A horizontal line indicates constant speed. The area under the graph between two times gives the distance travelled during that interval. This is one of the most powerful problem‑solving tools in kinematics: you can find distance by dividing the area into rectangles, triangles, or trapeziums.
在速度‑时间图中，速度为纵轴，时间为横轴。斜率表示加速度：斜率越陡，加速度越大。水平线代表匀速运动。两个时间点之间图像下方的面积即为该段时间内通过的距离。这是运动学中最强大的解题工具之一：可将面积分割为矩形、三角形或梯形来计算距离。
- To find acceleration, pick two points on a straight section and use a = (v − u) / t. | 要求加速度，在直线段上选两点，用 a = (v − u) / t。
- When the graph is curved, acceleration changes – use a tangent to estimate instantaneous acceleration. | 当图像为曲线时，加速度在变化——用切线估算瞬时加速度。
- Always give the units: speed in m/s or km/h, time in s or h, distance in m or km. | 务必写上单位：速度用 m/s 或 km/h，时间用 s 或 h，距离用 m 或 km。
7. Conversion Graphs | 转换图
A conversion graph is a straight line (often passing through the origin) used to change one unit into another, such as miles to kilometres, or Celsius to Fahrenheit. The gradient of the line is the conversion factor. To convert a value, read off directly from the graph, or use the equation derived from the line. These graphs test your ability to interpret scales and extract information accurately.
转换图是一条直线（通常经过原点），用于将一种单位转换为另一种，例如英里转公里、或摄氏度转华氏度。直线的斜率就是换算系数。要转换某个值，直接从图上读出，或使用从直线得到的方程。这类图考察你读懂刻度并准确提取信息的能力。
- Draw a vertical or horizontal line from the given value to the graph line, then across to the other axis. | 从已知值出发作垂直线或水平线到图线，然后平移到另一条轴。
- rigin, the conversion formula has a fixed addition or subtraction, e.g. °F = (9/5)°C + 32. | 若图像不经过原点，转换公式中会有一个恒定加减项，例如 °F = (9/5)°C + 32。
- Be precise: use the edge of a ruler to align with grid lines. | 精确操作：用尺子的边对齐网格线。
8. Statistical Graphs – Bar Charts and Histograms | 统计图——条形图与直方图
Bar charts represent categorical or discrete data with equal‑width bars. The height of each bar shows the frequency. Gaps between bars are essential – , the area of each bar is proportional to the frequency, so for unequal class widths you must calculate frequency density = frequency / class width. The vertical axis is always frequency density (not plain frequency), and there are no gaps between bars.
条形图用等宽的矩形条表示类别型或离散型数据。每个条的高度代表频数。条与条之间的间隔必不可少——它表明各分类是独立的。而直方图则用来展示分的连续数据。在直方图中，每个条的面积与频数成正比，因此当组距不相等时，必须计算频数密度 = 频数 / 组距。纵轴始终是频数密度（不是频数），且条之间没有间隔。
| Feature 特征 | Bar Chart 条形图 | Histogram 直方图 |
|---|---|---|
| Data type 数据类型 | Categorical / discrete 分类/离散 | Continuous 连续 |
| Gaps 间隔 | Yes 有 | No 无 |
| Vertical axis 纵轴 | Frequency 频数 | Frequency density 频数密度 |
| Area represents 面积代表 | Not proportional 非比例关系 | Frequency 频数 |
Common exam mistakes include using frequency instead of frequency density on a histogram with unequal widths, and putting gaps between histogram bars. Always label both axes and give a title.
常见考试错误包括：在组距不等的直方图上仍用频数而非频数密度；在直方图条之间画间隔。一定要标出坐标轴并给出标题。
9. Cumulative Frequency Graphs | 累积频率图
rtical axis against the upper class boundary on the horizontal axis. Join the points with a smooth curve – not straight line segments – y of the first class (where cumulative frequency = 0). You can then find the median (at 50% of the total frequency), the lower and upper quartiles (at 25% and 75% he graph also allows you to estimate how many values lie below or above a given threshold.
累积频率图（又称奥基夫曲线）展示了截至每个区间上限处的频数累计和。以各区间上限为横坐标，累积频率为纵坐标描点。用光滑曲线连接各点——不可用直线段——从第一个区间下限处（累积频率为 0）开始。由此可求得中位数（总频数的 50%）、下四分位数和上四分位数（25% 与 75% 处），以及四分位距（IQR = UQ − LQ）。该图还可用于估算低于或高于某个门槛值的数据个数。
- Always add a point at (first lower boundary, 0) so the curve starts correctly. | 一定要在（第一个区间下限, 0）处补一个点，使曲线正确开始。
- Draw lines across from the percentage of total frequency, then down to the axis to read off values. | 从总频数对应的百分比位置画水平线，再向下投影到横轴读数。
- The interquartile range is a measure of spread, unaffected by extreme values. | 四分位距是一种不受极端值影响的离散程度度量。
10. Scatter Graphs and Correlation | 散点图与相关性
Scatter graphs display the relationship between two sets of numerical data. Each point on the graph represents a paired (x, y). If the points tend to follow an upward trend, there is positive correlation; a downward trend indicates negative correlation. No obvious pattern suggests zero correlation. You may be asked to draw a line of best fit – a straight line that passes through the 'middle' of the points, having roughly as many points above as below. This line can then be used to estimate one value given another.
散点图用于展示两组数值数据之间的关系。图中每一点代表一对 (x, y) 值。若点总体呈上升趋势，即存在正相关；下降趋势为负相关；无明显模式则为零相关。你可能需要画一条最佳拟合线——一条从点群“中间”穿过的直线，使得线两侧的点数量大致相等。然后可利用该线由一个变量估算另一个变量。
- Do not force the line of best fit to pass through the origin unless the context suggests it. | 除非题设暗示，否则不应迫使最佳拟合线经过原点。
- Avoid simply connecting the dots – the line should be straight and represent the trend. | 避免直接将点连起来——该线应为直线，代表整体趋势。
- Estimation within the data range is interpolation (more reliable); outside the data range is extrapolation (less reliable). | 在数据范围内进行估算是内插（更可靠）；超出数据范围是外推（可靠性较低）。
11. Graphs in Practical Contexts – Real‑Life Modelling | 实际情境中的图像——现实建模
CIE frequently embeds graphs in practical scenarios: filling a container with water and plotting depth against time, the motion of a particle, or economic cost‑revenue models. The key is to link the shape of the graph to the rate of change. A curve that becomes less steep indicates a decreasing rate; a straight line indicates a constant rate. Understanding the physical meaning of gradient and area is essential. For example, on a depth‑time graph for a container filling with water, the gradient gives the rate of change of depth; if the container widens, the gradient decreases.
CIE 频繁将图表嵌入实际情境：例如向容器注水并画出深度‑时间图、粒子运动图、或经济成本‑收入模型。关键在于将图形形状与变化率联系起来。曲线变缓代表变化率在减小；直线代表恒定变化率。理解斜率和面积的物理意义至关重要。例如，在注水容器深度‑时间图上，斜率给出了深度变化率；如果容器变宽，斜率就会减小。
- Read the question carefully to identify what each variable represents. | 仔细读题，弄清每个变量代表什么。
- Interpret horizontal sections as 'no change' in the quantity on the vertical axis. | 将水平段解读为纵轴物理量“无变化”。
- When asked to estimate a rate, draw a tangent and construct a right‑angled triangle to measure change in y / change in x. | 被要求估算变化率时，画一条切线并构造直角三角形，计算 y 变化量 / x 变化量。
12. Graphical Solutions of Equations | 利用图像解方程
le, if you have plotted y = x² − 3x + 2, you might be asked to solve x² − 3x − 1 = 0. You would rearrange the required equation to match the left side of the graphed function: x² − 3x + 2 = 3, so you draw the line y = 3 and read the x‑ ts your algebraic manipulation and understanding of the link between equations and graphs.
考试中常见的一种题型是，利用你已经画好的图像去解另一个不完全与图像形式相同的方程。例如，你已画出 y = x² − 3x + 2 的曲线，题目让你解 x² − 3x − 1 = 0。你需要将所需方程变形，使其一端与图像函数匹配：x² − 3x + 2 = 3，于是画出直线 y = 3，然后读取交点处的 x 值。这种方法考查你的代数变形能力以及对方程与图像联系的理解。
- Rearrange the given equation so that one side is exactly the expression that was plotted. | 将给定方程变形，使一侧恰好为已绘制的表达式。
- The other side becomes a simple horizontal or straight line that you can draw easily. | 另一侧则变成易于绘制的水平线或直线。
- Check your solutions by substituting back into the original equation. | 将求得的解代回原方程进行验算。
Published by TutorHao | GCSE CIE Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图