📚 Graphs of Functions | 函数图像
In IGCSE Mathematics, understanding the graphs of functions is one of the most important skills you will develop. A graph is a visual representation of how the output of a function changes as the input changes. This article will guide you through the key types of function graphs, their properties, and how to sketch them step by step.
在 IGCSE 数学中，理解函数图像是你需要掌握的最重要技能之一。图像是函数输出值随输入值变化的一种可视化表示。本文将引导你了解关键类型的函数图像、它们的性质以及如何逐步绘制草图。
1. What is a Function Graph? | 什么是函数图像？
A function is a rule that assigns exactly one output value to each input value. When we write y = f(x) , the graph of the function is the set of all points (x, y) that satisfy this equation. In coordinate geometry, the horizontal axis is called the x-axis and the vertical axis is called the y-axis. Every point on the graph corresponds to an input–output pair.
函数是一种规则，它将每个输入值恰好对应到一个输出值。当我们写 y = f(x) 时，函数的图像就是满足该方程的所有点 (x, y) 的集合。在坐标几何中，水平轴称为 x 轴，垂直轴称为 y 轴。图像上的每个点都对应一个输入-输出对。
For example, for the function f(x) = 2x + 1 , when x = 3 , f(3) = 7 , so the point (3, 7) lies on its graph. The graph of a function helps us quickly see its key features: where it crosses the axes, whether it increases or decreases, and any turning points or asymptotes.
例如，对于函数 f(x) = 2x + 1 ，当 x = 3 时， f(3) = 7 ，因此点 (3, 7) 位于其图像上。函数图像帮助我们快速看到其关键特征：它与坐标轴的交点、它是递增还是递减，以及是否有极值点或渐近线。
2. Linear Function Graphs | 线性函数图像
A linear function has the general form y = mx + c , where m is the gradient (slope) and c is the y-intercept. The graph of a linear function is always a straight line. If m > 0 , the line slopes upward from left to right; if m < 0 , it slopes downward; if m = 0 , the line is horizontal.
线性函数的一般形式为 y = mx + c ，其中 m 是梯度（斜率）， c 是 y 轴截距。线性函数的图像总是一条直线。若 m > 0 ，直线从左到右上升；若 m < 0 ，直线从左到右下降；若 m = 0 ，直线是水平的。
To sketch a linear graph, you only need two points. A common method is to find the y-intercept (0, c) and the x-intercept by setting y = 0 and solving for x . Alternatively, choose two convenient x-values and plot their corresponding y-values. Let's take y = 2x - 4 : the y-intercept is -4, and setting y = 0 gives x = 2 . Plot (0, -4) and (2, 0), then draw a straight line through them.
要绘制线性图像草图，你只需要两个点。常用方法是找到 y 轴截距 (0, c)，并通过令 y = 0 解出 x 来找到 x 轴截距。另一种方法是选取两个方便的 x 值并绘制它们对应的 y 值。以 y = 2x - 4 为例：y 轴截距为 -4，令 y = 0 得到 x = 2 。描点 (0, -4) 和 (2, 0)，然后穿过它们画一条直线。
3. Quadratic Function Graphs | 二次函数图像
A quadratic function has the general form y = ax2 + bx + c , where a ≠ 0 . Its graph is a curve called a parabola. When a > 0 , the parabola opens upward (a "U" shape); when a < 0 , it opens downward (an "n" shape). The turning point of the parabola is called its vertex.
二次函数的一般形式为 y = ax2 + bx + c ，其中 a ≠ 0 。它的图像是一条称为抛物线的曲线。当 a > 0 时，抛物线开口朝上（呈 "U" 形）；当 a < 0 时，开口朝下（呈 "n" 形）。抛物线的转向点称为顶点。
To sketch a quadratic graph, you should find:
绘制二次函数图像草图时，你应该找到：
- The y-intercept: set x = 0 , so y = c .
- y 轴截距：令 x = 0 ，则 y = c 。
- The x-intercepts (roots): set y = 0 and solve ax2 + bx + c = 0 , by factorising, completing the square, or using the quadratic formula.
- x 轴截距（根）：令 y = 0 并解方程 ax2 + bx + c = 0 ，可用因式分解、配方法或二次公式。
- The vertex: the x-coordinate of the vertex is x = -b2a . Substitute this value into the equation to find y.
- 顶点：顶点的 x 坐标为 x = -b2a 。将此值代入方程求出 y。
For example, for y = x2 - 4x + 3 , the y-intercept is 3. Factorising x2 - 4x + 3 = 0 gives (x-1)(x-3)=0 , so the x-intercepts are 1 and 3. The vertex x-coordinate is x = --42(1) = 2 , and y = 22 - 8 + 3 = -1 . So the vertex is at (2, -1).
例如，对于 y = x2 - 4x + 3 ，y 轴截距为 3。因式分解 x2 - 4x + 3 = 0 得 (x-1)(x-3)=0 ，因此 x 轴截距为 1 和 3。顶点 x 坐标为 x = --42(1) = 2 ，且 y = 22 - 8 + 3 = -1 。所以顶点在 (2, -1)。
Vertex formula: x = −b ⁄ (2a)
顶点公式：x = −b ⁄ (2a)
4. Cubic Function Graphs | 三次函数图像
A cubic function has the general form y = ax3 + bx2 + cx + d , where a ≠ 0 . The graph of a cubic function has one of two basic shapes. If a > 0 , it starts low on the left, rises, and ends high on the right. If a < 0 , it starts high on the left and ends low on the right. A cubic graph can have up to two turning points.
三次函数的一般形式为 y = ax3 + bx2 + cx + d ，其中 a ≠ 0 。三次函数图像有两种基本形状之一。若 a > 0 ，它从左侧低处开始，上升，并在右侧高处结束。若 a < 0 ，它从左侧高处开始，在右侧低处结束。三次图像最多可以有两个转向点。
To sketch a cubic graph, identify the y-intercept (0, d), the x-intercepts by solving ax3 + bx2 + cx + d = 0 (often by factorisation), and the general shape determined by the sign of a . For instance, y = x3 - x = x(x-1)(x+1) has x-intercepts at -1, 0, 1 and a y-intercept of 0. The graph crosses the x-axis three times and has two turning points between these roots.
绘制三次函数图像草图时，确定 y 轴截距 (0, d)，通过解 ax3 + bx2 + cx + d = 0 来找到 x 轴截距（通常用因式分解），并由 a 的符号确定整体形状。例如， y = x3 - x = x(x-1)(x+1) 的 x 轴截距为 -1, 0, 1，y 轴截距为 0。图像三次穿过 x 轴，并在这些根之间有两个转向点。
5. Reciprocal Function Graphs | 反比例函数图像
The reciprocal function has the form y = kx , where k ≠ 0 . Its graph is called a hyperbola and consists of two separate branches. When k > 0 , the branches lie in the first and third quadrants; when k < 0 , they lie in the second and fourth quadrants.
反比例函数的形式为 y = kx ，其中 k ≠ 0 。它的图像称为双曲线，由两条分离的分支组成。当 k > 0 时，分支位于第一和第三象限；当 k < 0 时，分支位于第二和第四象限。
The axes are asymptotes: the graph approaches the x-axis and the y-axis but never touches them. As x → +∞ , y → 0 ; as x → 0+ , y → +∞ . You should also be familiar with the graph of y = k⁄x2 , which lies entirely above the x-axis when k > 0 , approaching the axes without crossing them.
坐标轴是渐近线：图像接近 x 轴和 y 轴但永远不会触及它们。当 x → +∞ 时， y → 0 ；当 x → 0+ 时， y → +∞ 。你还应熟悉 y = k⁄x2 的图像，当 k > 0 时它完全位于 x 轴上方，接近坐标轴但不会穿过它们。
6. Exponential Function Graphs | 指数函数图像
An exponential function has the form y = ax , where a > 0 and a ≠ 1 . Its graph is a smooth curve that shows rapid growth or decay. If a > 1 , it is an exponential growth curve: it increases slowly, then rises very steeply. If 0 < a < 1 , it is an exponential decay curve: it decreases rapidly at first, then levels off.
指数函数的形式为 y = ax ，其中 a > 0 且 a ≠ 1 。它的图像是一条平滑曲线，显示出快速的增长或衰减。若 a > 1 ，它是指数增长曲线：先缓慢增加，然后非常陡峭地上升。若 0 < a < 1 ，它是指数衰减曲线：先快速下降，然后趋于平缓。
All exponential graphs of this form pass through the point (0, 1) because a0 = 1 . The x-axis is a horizontal asymptote: as x → -∞ , ax → 0 for a > 1 . In IGCSE, you may also encounter transformations such as y = 2x + 3 , which shifts the basic graph upward by 3 units, so the asymptote becomes y = 3 .
所有这种形式的指数图像都经过点 (0, 1)，因为 a0 = 1 。x 轴是水平渐近线：当 x → -∞ 时，对于 a > 1 ， ax → 0 。在 IGCSE 中，你还可能遇到诸如 y = 2x + 3 的变换，它将基本图像向上平移 3 个单位，因此渐近线变为 y = 3 。
7. The Absolute Value Function | 绝对值函数
The absolute value function is written as y = |x| . It is defined as y = x when x ≥ 0 and y = -x when x < 0 . Its graph forms a V-shape, with the vertex at the origin (0, 0). The graph is symmetric about the y-axis.
绝对值函数写作 y = |x| 。它的定义为当 x ≥ 0 时 y = x ，当 x < 0 时 y = -x 。它的图像呈 V 形，顶点在原点 (0, 0)。图像关于 y 轴对称。
To sketch y = |f(x)| , draw the graph of f(x) , then reflect the parts below the x-axis upward, keeping the parts above the x-axis unchanged. For example, y = |x - 2| is a V-shape with vertex at (2, 0), sloping up to both sides. It is useful to remember that the absolute value always produces a non-negative output, so the entire graph lies on or above the x-axis.
要绘制 y = |f(x)| 的图像，先画出 f(x) 的图像，然后将 x 轴下方的部分向上翻折，保持 x 轴上方的部分不变。例如， y = |x - 2| 是顶点在 (2, 0) 的 V 形，向两侧向上倾斜。要记住绝对值总是产生非负的输出，因此整个图像位于 x 轴上或上方。
8. Transformations of Graphs: Translations | 图像变换：平移
Graph transformations allow you to sketch new functions from known ones. A translation moves the entire graph without changing its shape. The function y = f(x) + a shifts the graph of f(x) vertically upward by a units (or downward if a < 0 ). The function y = f(x - a) shifts the graph horizontally to the right by a units (or to the left if a < 0 ).
图像变换让你可以根据已知函数绘制新函数的图像。平移是在不改变图像形状的情况下移动整个图像。函数 y = f(x) + a 将 f(x) 的图像垂直向上平移 a 个单位（若 a < 0 则向下）。函数 y = f(x - a) 将图像水平向右平移 a 个单位（若 a < 0 则向左）。
For example, y = x2 shifted up by 3 becomes y = x2 + 3 , with its vertex at (0, 3). Similarly, y = x2 shifted right by 2 becomes y = (x - 2)2 , with its vertex at (2, 0). Notice that inside the bracket, the sign is inverted: (x - 2) means right by 2, not left.
例如， y = x2 向上平移 3 个单位变为 y = x2 + 3 ，其顶点在 (0, 3)。类似地， y = x2 向右平移 2 个单位变为 y = (x - 2)2 ，其顶点在 (2, 0)。注意括号内的符号是反转的： (x - 2) 表示向右 2 个单位，而不是向左。
9. Reflections and Stretches | 反射与伸缩
Reflections flip the graph over a line. The function y = -f(x) reflects the graph of f(x) in the x-axis: points with positive y-values become negative and vice versa. The function y = f(-x) reflects the graph in the y-axis, which is equivalent to reversing the direction of the x-values.
反射将图像关于一条线翻转。函数 y = -f(x) 将 f(x) 的图像关于 x 轴反射：y 值为正的点变为负，反之亦然。函数 y = f(-x) 将图像关于 y 轴反射，这等同于反转 x 值的方向。
Vertical stretches multiply the y-coordinates by a constant: y = k · f(x) stretches the graph vertically by a factor k . If k > 1 , the graph becomes taller; if 0 < k < 1 , it compresses. Horizontal stretches are less common at IGCSE but appear as y = f(kx) . A horizontal stretch by a factor of 1k means that the graph is compressed horizontally when k > 1 .
垂直伸缩将 y 坐标乘以一个常数： y = k · f(x) 将图像垂直伸缩 k 倍。若 k > 1 ，图像变得更高；若 0 < k < 1 ，图像被压缩。水平伸缩在 IGCSE 中较少见，但会以 y = f(kx) 的形式出现。水平伸缩因子为 1k ，意味着当 k > 1 时图像在水平方向被压缩。
10. Reading Key Features from a Graph | 从图像中读取关键特征
Given a graph, you should be able to identify the x-intercepts (roots), the y-intercept, the turning points (maximum and minimum points), and asymptotes. The x-intercepts are the solutions to f(x) = 0 . The y-intercept is the value of f(0) . Turning points are where the graph changes direction, and they correspond to local maxima or minima.
给定一个图像，你应该能够识别 x 轴截距（根）、y 轴截距、转向点（极大点和极小点）以及渐近线。x 轴截距是 f(x) = 0 的解。y 轴截距是 f(0) 的值。转向点是图像改变方向的位置，它们对应局部极大值或极小值。
You may also be asked to estimate the gradient of a curve at a point. To do this, draw a tangent line at that point and calculate its slope using two points on the tangent. For example, if the tangent passes through (1, 2) and (3, 8), the gradient is 8-23-1 = 3 .
你还可能被要求估计曲线上某一点的梯度。为此，在该点画一条切线，并使用切线上两个点来计算其斜率。例如，若切线经过 (1, 2) 和 (3, 8)，则梯度为 8-23-1 = 3 。
Gradient = (y₂ − y₁) ⁄ (x₂ − x₁)
梯度 = (y₂ − y₁) ⁄ (x₂ − x₁)
11. Solving Equations Graphically | 用图像法解方程
A graphical method can be used to solve equations or to estimate roots. For an equation like x2 - 2x - 3 = 0 , you can plot the graph of y = x2 - 2x - 3 and read the x-coordinates where it crosses the x-axis. Those x-values are the solutions.
图像法可用于解方程或估算根。对于像 x2 - 2x - 3 = 0 这样的方程，你可以绘制 y = x2 - 2x - 3 的图像，并读取它与 x 轴相交处的 x 坐标。这些 x 值就是方程的解。
For more complex equations, you can find the intersection point of two graphs. For example, to solve x2 = 2x + 1 , plot y = x2 and y = 2x + 1 on the same axes. The x-coordinates where the two graphs intersect are the solutions. This method is especially useful when the equation cannot be easily factorised.
对于更复杂的方程，你可以找到两个图像的交点。例如，要解 x2 = 2x + 1 ，在同一坐标轴上绘制 y = x2 和 y = 2x + 1 。两个图像相交处的 x 坐标就是解。当方程不容易因式分解时，这种方法特别有用。
- Always label your axes and plot the graphs accurately.
- 始终标出坐标轴并准确绘制图像。
- Check whether the equation has one, two, or no intersections.
- 检查方程有一、两个还是零个交点。
- Give answers to an appropriate degree of accuracy, usually one decimal place.
- 给出适当精度的答案，通常精确到一位小数。
12. Exam Tips and Common Mistakes | 考试技巧与常见错误
One common mistake is confusing the shapes of different functions. Remember: linear graphs are straight lines; quadratic graphs are symmetrical parabolas; cubic graphs have an S-like shape; reciprocal graphs have two branches approaching the axes. Practise sketching each type until you can do it quickly and accurately.
一个常见错误是混淆不同函数的形状。记住：线性图像是直线；二次图像是对称的抛物线；三次图像有类似 S 的形状；反比例图像有两条接近坐标轴的分支。练习绘制每种类型，直到你能快速准确地完成。
When dealing with transformations, be careful with the direction of horizontal shifts. y = f(x+2) moves the graph left by 2, not right. Also, when finding the vertex of a parabola, substitute x = -b2a back into the original equation to get the y-coordinate. Drawing a smooth curve through plotted points is essential; avoid simply connecting the dots with straight lines.
在处理变换时，注意水平平移的方向。 y = f(x+2) 将图像向左移动 2 个单位，而不是向右。此外，在求抛物线顶点时，将 x = -b2a 代入原方程以得到 y 坐标。通过描点绘制平滑曲线至关重要；避免用直线简单地连接各点。
Finally, always show your working. Even if your sketch is not perfect, a correct method with clear axes, labels, and key points can earn you marks in the exam. Review the topic regularly, and use past paper questions to build confidence.
最后，始终展示你的解题过程。即使你的草图并不完美，使用正确的方法、清晰的坐标轴、标签和关键点也能在考试中为你赢得分数。定期复习该主题，并使用过往试卷题目来建立信心。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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