Absolute Value Functions: Properties and Graphs | 绝对值函数的性质与图像

📚 Absolute Value Functions: Properties and Graphs | 绝对值函数的性质与图像

The absolute value function is a fundamental concept in the IB Mathematics curriculum, appearing in topics from functions and equations to calculus. Mastering its algebraic definition and graphical behaviors is essential for solving a wide range of problems.

绝对值函数是 IB 数学课程中的一个基本概念，出现在函数、方程以及微积分等多个主题中。掌握其代数定义和图形行为，对于解决各类问题至关重要。

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1. Definition and Notation | 定义与记号

The absolute value of a real number x, written as |x|, represents its distance from 0 on the number line. For example, |3| = 3 and |-3| = 3. Because distance cannot be negative, the absolute value of a number is always non-negative.

实数 x 的绝对值记作 |x|，表示它在数轴上到 0 的距离。例如，|3| = 3，|-3| = 3。因为距离不能为负，所以一个数的绝对值总是非负的。

Formally, |x| is defined piecewise as:

正式地，|x| 分段定义为：

|x| = x, if x ≥ 0; |x| = -x, if x < 0

This two-branch definition is the foundation for solving any equation or inequality that contains an absolute value.

这个二分定义是解一切含有绝对值的方程或不等式的基础。

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2. Piecewise Representation | 分段表示

Any absolute-value expression can be expanded by finding the critical points where the expression inside is zero. For |x – 3|, the critical point is x = 3, so we write:

任何绝对值表达式都可以通过找出其中表达式为零的临界点来展开。对于 |x – 3|，临界点是 x = 3，因此我们写成：

|x – 3| = x – 3, for x ≥ 3; |x – 3| = -(x – 3) = 3 – x, for x < 3

This piecewise representation allows us to handle absolute values algebraically by splitting into cases. It is also the starting point for graphing more complicated absolute-value functions.

这种分段表示让我们能够通过分类讨论在代数上处理绝对值。它也是绘制更复杂绝对值函数图像的起点。

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3. Basic Graph of y = |x| | y = |x| 的基本图像

The graph of y = |x| is a V-shaped curve with its vertex at the origin (0, 0). The graph lies entirely above the x-axis because absolute values are non-negative.

y = |x| 的图像是一条 V 形曲线，顶点在原点 (0, 0)。由于绝对值非负，图像完全位于 x 轴之上。

For x > 0, the graph follows y = x; for x < 0, it follows y = -x. Thus the slopes are +1 on the right branch and -1 on the left branch. The function is symmetric with respect to the y-axis.

当 x > 0 时，图像沿 y = x；当 x < 0 时，沿 y = -x。因此右支斜率为 +1，左支斜率为 -1。该函数关于 y 轴对称。

Some key points are: (-2, 2), (-1, 1), (0, 0), (1, 1), (2, 2).

一些关键点包括：(-2, 2)、(-1, 1)、(0, 0)、(1, 1)、(2, 2)。

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4. Transformations of y = |x| | y = |x| 的图像变换

You can apply standard transformations to the basic graph y = |x| using function notation:

你可以利用函数记号对基本图像 y = |x| 进行标准变换：

• y = |x| + c shifts the graph vertically. If c > 0 it moves up, if c < 0 it moves down.

  y = |x| + c 使图像垂直平移。当 c > 0 时向上移动，c < 0 时向下移动。

• y = |x – h| shifts the graph horizontally. If h > 0 it moves right, if h < 0 it moves left.

  y = |x – h| 使图像水平平移。当 h > 0 时向右移动，h < 0 时向左移动。

• y = a|x| stretches or compresses the graph vertically, depending on |a|. If a is negative, the graph is reflected across the x-axis.

  y = a|x| 使图像垂直拉伸或压缩，取决于 |a|。若 a 为负，图像关于 x 轴翻转。

For example, the graph of y = 2|x – 3| + 1 has a vertex at (3, 1) and opens upward with a steeper V shape.

例如，y = 2|x – 3| + 1 的图像顶点在 (3, 1)，开口向上且 V 形更陡。

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5. Graphs of y = |f(x)| and y = f(|x|) | y = |f(x)| 与 y = f(|x|) 的图像

When f(x) is any function, the graph of y = |f(x)| is obtained from y = f(x) by reflecting the parts below the x-axis upward, while preserving the portions above the x-axis.

当 f(x) 是任意函数时，y = |f(x)| 的图像可由 y = f(x) 将 x 轴以下部分向上翻折得到，而 x 轴以上的部分保持不变。

On the other hand, the graph of y = f(|x|) is constructed by deleting the left half of the graph of y = f(x) (for x < 0) and reflecting the right half across the y-axis. The resulting graph is even.

另一方面，y = f(|x|) 的图像是删去 y = f(x) 图像的左半部分（x < 0），并将右半部分关于 y 轴翻折而得到。所得图像是偶函数。

For instance, let f(x) = x – 2. Then |f(x)| = |x – 2| has a V shape with vertex at (2, 0). The function f(|x|) = |x| – 2 has a V shape with vertex at (0, -2) and retains symmetry.

例如，令 f(x) = x – 2，则 |f(x)| = |x – 2| 是顶点在 (2, 0) 的 V 形。而 f(|x|) = |x| – 2 是顶点在 (0, -2) 的 V 形，并保持对称。

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6. Solving Absolute Value Equations | 解绝对值方程

To solve an equation of the form |ax + b| = c, where c > 0, we use the two cases:

对于形如 |ax + b| = c（c > 0）的方程，我们利用两种情形求解：

ax + b = c or ax + b = -c

Example: Solve |2x – 3| = 5.

例：解 |2x – 3| = 5。

Case 1: 2x – 3 = 5 gives 2x = 8, so x = 4. Case 2: 2x – 3 = -5 gives 2x = -2, so x = -1. Therefore x = 4 or x = -1.

情形 1：2x – 3 = 5，得 2x = 8，所以 x = 4。情形 2：2x –

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