Graphs and Properties of Absolute Value Functions | 绝对值函数的图像与性质

📚 Graphs and Properties of Absolute Value Functions | 绝对值函数的图像与性质

The absolute value function is one of the most fundamental piecewise-defined functions in mathematics. It appears in algebra, geometry, calculus, and real-world modelling. Understanding its graph and properties is essential for solving equations, inequalities, and optimisation problems.

绝对值函数是数学中最基本的分段定义函数之一。它出现在代数、几何、微积分以及现实建模中。理解其图像与性质对于求解方程、不等式和最优化问题至关重要。

1. Definition of Absolute Value | 绝对值的定义

For any real number x, the absolute value of x, denoted |x|, is defined as the distance from x to 0 on the number line. Distance is always non-negative, so |x| ≥ 0 for all x.

对于任意实数 x，x 的绝对值记为 |x|，定义为数轴上 x 到 0 的距离。距离总是非负的，因此对所有 x 都有 |x| ≥ 0。

|x| = x if x ≥ 0; |x| = −x if x < 0

This piecewise definition is the foundation for graphing and analysing the function.

这个分段定义是绘制图像和分析函数的基础。

2. The Graph of y = |x| | y = |x| 的图像

The graph of y = |x| is a V-shaped curve with its vertex at the origin (0, 0). The right branch is the line y = x for x ≥ 0, and the left branch is the line y = −x for x < 0.

y = |x| 的图像是一条 V 形曲线，顶点在原点 (0, 0)。右支是 x ≥ 0 时的直线 y = x，左支是 x < 0 时的直线 y = −x。

Key points on the graph include (−2, 2), (−1, 1), (0, 0), (1, 1), and (2, 2). The graph is symmetric about the y-axis.

图像上的关键点包括 (−2, 2)、(−1, 1)、(0, 0)、(1, 1) 和 (2, 2)。图像关于 y 轴对称。

3. Piecewise Representation | 分段表示

In general, an absolute value function can be written as f(x) = a|x − h| + k, where the vertex is at (h, k). Its piecewise equivalent is:

一般地，绝对值函数可以写成 f(x) = a|x − h| + k，其中顶点为 (h, k)。其分段等价形式为：

f(x) = a(x − h) + k for x ≥ h; f(x) = −a(x − h) + k for x < h

The parameter a controls the steepness and direction of the two branches. If a > 0, the graph opens upward; if a < 0, it opens downward.

参数 a 控制两条支线的陡峭程度和方向。若 a > 0，图像开口向上；若 a < 0，图像开口向下。

4. Transformations: Translation and Scaling | 变换：平移与伸缩

Starting from y = |x|, we can apply transformations to obtain any absolute value function.

从 y = |x| 出发，我们可以通过变换得到任意绝对值函数。

Horizontal shift: y = |x − h| moves the vertex to (h, 0).

水平平移：y = |x − h| 将顶点移到 (h, 0)。
Vertical shift: y = |x| + k moves the vertex to (0, k).

垂直平移：y = |x| + k 将顶点移到 (0, k)。
Vertical stretch/compression: y = a|x| scales both branch slopes by |a|.

垂直伸缩：y = a|x| 将两条支线的斜率按 |a| 缩放。
Reflection: y = −|x| reflects the graph across the x-axis.

反射：y = −|x| 将图像关于 x 轴翻转。

Combining these transformations gives the general form f(x) = a|x − h| + k.

组合这些变换即可得到一般形式 f(x) = a|x − h| + k。

5. Domain and Range | 定义域与值域

The domain of any absolute value function is all real numbers: Domain = ℝ.

任何绝对值函数的定义域都是全体实数：定义域 = ℝ。

The range depends on the vertex and the direction of opening. If a > 0, the minimum value is k, so Range = [k, ∞). If a < 0, the maximum value is k, so Range = (−∞, k].

值域取决于顶点和开口方向。若 a > 0，最小值为 k，因此值域 = [k, ∞)。若 a < 0，最大值为 k，因此值域 = (−∞, k]。

6. Symmetry and Evenness | 对称性与奇偶性

The basic function y = |x| is an even function because |−x| = |x| for all x. Its graph is symmetric about the y-axis.

基本函数 y = |x| 是偶函数，因为对所有 x 都有 |−x| = |x|。其图像关于 y 轴对称。

In general, f(x) = a|x| + k is even. However, f(x) = a|x − h| + k is not even unless h = 0. Its symmetry axis is the vertical line x = h.

一般地，f(x) = a|x| + k 是偶函数。但除非 h = 0，f(x) = a|x − h| + k 不是偶函数。其对称轴是垂直线 x = h。

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

To solve an equation of the form |ax + b| = c, we consider two cases. If c < 0, there is no solution. If c = 0, then ax + b = 0. If c > 0, then:

解形如 |ax + b| = c 的方程时，我们分两种情况讨论。若 c < 0，无解；若 c = 0，则 ax + b = 0；若 c > 0，则：

ax + b = c or ax + b = −c

For example, solve |2x − 3| = 5. Then 2x − 3 = 5 ⇒ x = 4, or 2x − 3 = −5 ⇒ x = −1. The solution set is {−1, 4}.

例如，解 |2x − 3| = 5。则 2x − 3 = 5 ⇒ x = 4，或 2x − 3 = −5 ⇒ x = −1。解集为 {−1, 4}。

8. Solving Absolute Value Inequalities | 解绝对值不等式

Inequalities with absolute values are solved using the distance interpretation.

含绝对值的不等式利用距离意义来求解。

|ax + b| < c means the distance from ax + b to 0 is less than c, so −c < ax + b < c.

|ax + b| < c 表示 ax + b 到 0 的距离小于 c，因此 −c < ax + b < c。
|ax + b| ≤ c gives −c ≤ ax + b ≤ c.

|ax + b| ≤ c 给出 −c ≤ ax + b ≤ c。
|ax + b| > c means ax + b > c or ax + b < −c.

|ax + b| > c 表示 ax + b > c 或 ax + b < −c。
|ax + b| ≥ c gives ax + b ≥ c or ax + b ≤ −c.

|ax + b| ≥ c 给出 ax + b ≥ c 或 ax + b ≤ −c。

Graphically, |x| < 2 corresponds to the region between x = −2 and x = 2 on the number line.

从图像上看，|x| < 2 对应数轴上 x = −2 与 x = 2 之间的区域。

9. The Vertex Formula | 顶点公式

For f(x) = a|x − h| + k, the vertex is exactly the point (h, k). To find h and k from a general linear absolute expression, rewrite the expression in vertex form or locate the x-value where the inside becomes zero.

对于 f(x) = a|x − h| + k，顶点正好是 (h, k)。要从一般的线性绝对值表达式求 h 和 k，可改写为顶点式，或找到内部表达式为零时的 x 值。

Example: For f(x) = 2|x − 3| + 1, the vertex is (3, 1). The graph opens upward with slope 2 on the right branch and slope −2 on the left branch.

例如：对于 f(x) = 2|x − 3| + 1，顶点是 (3, 1)。图像开口向上，右支斜率为 2，左支斜率为 −2。

10. Applications in Real Life | 实际应用

Absolute value functions model situations where only the magnitude matters, such as distance, error tolerance, and temperature deviation.

绝对值函数用于模拟只关心大小的情况，例如距离、误差容限和温度偏差。

Distance: The distance between two points a and b on a number line is |a − b|.

距离：数轴上点 a 与 b 之间的距离是 |a − b|。
Error tolerance: If a machine part must have length L within tolerance ε, we write |x − L| ≤ ε.

误差容限：若机器零件长度 L 的容差为 ε，则写作 |x − L| ≤ ε。
Speed limits: If the speed limit is 60 km/h with a tolerance of 5 km/h, the speed v satisfies |v − 60| ≤ 5.

限速：若限速 60 km/h，容差 5 km/h，则速度 v 满足 |v − 60| ≤ 5。

11. Common Mistakes and Tips | 常见错误与技巧

Students often forget that |x| is never negative or that the graph must be V-shaped with a sharp corner at the vertex.

学生经常忘记 |x| 永远不会为负，或者忘记图像必须是 V 形且在顶点处有尖锐拐角。

Do not write |x| = ±x without conditions. Always state the sign condition.

不要无条件地写 |x| = ±x。务必说明符号条件。
When solving |ax + b| = c, check that the right side c is non-negative.

解 |ax + b| = c 时，检查右边 c 是否为非负数。
When shifting, remember that |x − h| shifts right if h > 0, and left if h < 0.

平移时，记住 h > 0 时 |x − h| 向右平移，h < 0 时向左平移。
Use the vertex and two extra points to sketch the graph quickly.

利用顶点和另外两个点可以快速画出草图。

12. Conclusion | 结语

The absolute value function is a versatile tool in mathematics. By mastering its graph, transformations, equations, and inequalities, you build a strong foundation for more advanced topics such as calculus and complex analysis.

绝对值函数是数学中一个用途广泛的工具。通过掌握其图像、变换、方程和不等式，你可以为微积分和复分析等更高级的主题打下坚实基础。

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Graphs and Properties of Absolute Value Functions | 绝对值函数的图像与性质