📚 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。
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| ecause 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。
2. Piecewise Representation | 分段表示
on 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。
3. Basic Graph of y = |x| | y = |x| 的基本图像
The graph of y = |x| (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。
e +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)。
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。
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)| lecting 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| (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。
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。
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 -
Published by TutorHao | IB 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