Graphs of Functions | 函数图像

📚 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 \( y = ax^2 + bx + c \), where \( a \neq 0 \). 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 \( y = ax^2 + bx + c \)，其中 \( a \neq 0 \)。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 \( ax^2 + bx + c = 0 \), by factorising, completing the square, or using the quadratic formula.
x 轴截距（根）：令 \( y = 0 \) 并解方程 \( ax^2 + bx + c = 0 \)，可用因式分解、配方法或二次公式。
The vertex: the x-coordinate of the vertex is \( x = -\frac{b}{2a} \). Substitute this value into the equation to find y.
顶点：顶点的 x 坐标为 \( x = -\frac{b}{2a} \)。将此值代入方程求出 y。

For example, for \( y = x^2 – 4x + 3 \), the y-intercept is 3. Factorising \( x^2 – 4x + 3 = 0 \) gives \( (x-1)(x-3)=0 \), so the x-intercepts are 1 and 3. The vertex x-coordinate is \( x = -\frac{-4}{2(1)} = 2 \), and \( y = 2^2 – 8 + 3 = -1 \). So the vertex is at \((2, -1)\).

例如，对于 \( y = x^2 – 4x + 3 \)，y 轴截距为 3。因式分解 \( x^2 – 4x + 3 = 0 \) 得 \( (x-1)(x-3)=0 \)，因此 x 轴截距为 1 和 3。顶点 x 坐标为 \( x = -\frac{-4}{2(1)} = 2 \)，且 \( y = 2^2 – 8 + 3 = -1 \)。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 \( y = ax^3 + bx^2 + cx + d \), where \( a \neq 0 \). 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 \( y = ax^3 + bx^2 + cx + d \)，其中 \( a \neq 0 \)。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 \( ax^3 + bx^2 + cx + d = 0 \) (often by factorisation), and the general shape determined by the sign of \( a \). For instance, \( y = x^3 – x = x(x-1)(x+1) \) 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 \( ax^3 + bx^2 + cx + d = 0 \) 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，\( y = x^3 – x = x(x-1)(x+1) \) 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 \( y = \frac{k}{x} \), where \( k \neq 0 \). 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 \( y = \frac{k}{x} \)，其中 \( k \neq 0 \)。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 \( x \to +\infty \), \( y \to 0 \); as \( x \to 0^+ \), \( y \to +\infty \). You should also be familiar with the graph of \( y = \frac{k}{x^2} \), 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 \( x \to +\infty \) 时，\( y \to 0 \)；当 \( x \to 0^+ \) 时，\( y \to +\infty \)。你还应熟悉 \( y = \frac{k}{x^2} \) 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 \( y = a^x \), where \( a > 0 \) and \( a \neq 1 \). 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 \( y = a^x \)，其中 \( a > 0 \) 且 \( a \neq 1 \)。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 \( a^0 = 1 \). The x-axis is a horizontal asymptote: as \( x \to -\infty \), \( a^x \to 0 \) for \( a > 1 \). In IGCSE, you may also encounter transformations such as \( y = 2^x + 3 \), 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 \( a^0 = 1 \)。x 轴是水平渐近线：当 \( x \to -\infty \) 时，对于 \( a > 1 \)，\( a^x \to 0 \)。在 IGCSE 中，你还可能遇到诸如 \( y = 2^x + 3 \) 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 \( x \geq 0 \) 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 \( x \geq 0 \) 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, \( y = x^2 \) shifted up by 3 becomes \( y = x^2 + 3 \), with its vertex at \((0, 3)\). Similarly, \( y = x^2 \) 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，\( y = x^2 \) 向上平移 3 个单位变为 \( y = x^2 + 3 \)，其顶点在 \((0, 3)\)。类似地，\( y = x^2 \) 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: \( y = k \cdot f(x) \) 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 \( \frac{1}{k} \) means that the graph is compressed horizontally when \( k > 1 \).

垂直伸缩将 y 坐标乘以一个常数：\( y = k \cdot f(x) \) 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 \( \frac{1}{k} \)，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 \( \frac{8-2}{3-1} = 3 \).

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 \( \frac{8-2}{3-1} = 3 \)。

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 \( x^2 – 2x – 3 = 0 \), you can plot the graph of \( y = x^2 – 2x – 3 \) 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 \( x^2 – 2x – 3 = 0 \) 这样的方程，你可以绘制 \( y = x^2 – 2x – 3 \) 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 \( x^2 = 2x + 1 \), plot \( y = x^2 \) 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 \( x^2 = 2x + 1 \)，在同一坐标轴上绘制 \( y = x^2 \) 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 \( x = -\frac{b}{2a} \) 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 \( x = -\frac{b}{2a} \) 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。

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Graphs of Functions | 函数图像