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

Absolute value functions are a fundamental topic in IB Mathematics, appearing in both Analysis and Approaches (AA) and Applications and Interpretation (AI) courses. Mastering their graphs and properties is essential for solving equations, inequalities, and optimization problems.

绝对值函数是IB数学中的基础主题，出现在分析与方法（AA）以及应用与解释（AI）课程中。掌握其图像与性质对于求解方程、不等式和优化问题至关重要。

1. Definition and Notation | 定义与符号

The absolute value of a real number x, written as |x|, is the distance from x to 0 on the number line. It is always non-negative.

实数x的绝对值，记作|x|，是x在数轴上到0的距离。它永远是非负的。

Formally:

正式定义为：

|x| = x, if x ≥ 0; |x| = -x, if x < 0

For example, |5| = 5 and |-5| = 5. The notation uses vertical bars on both sides of the expression.

例如，|5| = 5，而|-5| = 5。符号在表达式两侧使用竖线。

2. Basic Graph y = |x| | 基本图像 y = |x|

The graph of y = |x| is a V-shaped curve composed of two rays meeting at the origin (0,0).

y = |x|的图像是一个V形曲线，由两条在原点(0,0)相交的射线组成。

For x ≥ 0, the graph follows the line y = x. For x < 0, it follows the line y = -x. The point where the direction changes is called the vertex.

当x ≥ 0时，图像沿直线y = x；当x < 0时，沿直线y = -x。方向改变的点称为顶点。

Key features: domain is all real numbers; range is y ≥ 0; the graph is symmetric about the y-axis.

关键特征：定义域为全体实数；值域为y ≥ 0；图像关于y轴对称。

3. Key Properties | 主要性质

Non-negativity: |x| ≥ 0 for all x.

非负性：对所有x，|x| ≥ 0。
Evenness: |x| = |-x|, so the function is even.

偶函数性：|x| = |-x|，因此该函数是偶函数。
Multiplicativity: |ab| = |a||b|.

乘法性：|ab| = |a||b|。
Triangle inequality: |a + b| ≤ |a| + |b|.

三角不等式：|a + b| ≤ |a| + |b|。

These properties are frequently tested in IB exams, especially the triangle inequality.

这些性质在IB考试中经常考到，尤其是三角不等式。

4. Transformations of Absolute Value Functions | 绝对值函数的变换

General form: y = a|x – h| + k. The vertex is at (h, k), and a controls the steepness and direction.

一般形式：y = a|x – h| + k。顶点在(h, k)，a控制陡峭程度和方向。

If a > 0, the V opens upward. If a < 0, it opens downward. The graph is stretched vertically by factor |a|.

若a > 0，V形向上开口；若a < 0，向下开口。图像在垂直方向拉伸|a|倍。

Example: y = 2|x – 1| + 3 has vertex (1,3), opens upward, and is steeper than y = |x|.

例如：y = 2|x – 1| + 3的顶点为(1,3)，向上开口，并且比y = |x|更陡。

5. Piecewise Representation | 分段函数表示

Any absolute value function can be written as a piecewise function. For f(x) = |x|:

任何绝对值函数都可以写成分段函数。对于f(x) = |x|：

f(x) = x, x ≥ 0; f(x) = -x, x < 0

In general, to write y = |ax + b|, find the zero of ax + b, then split the domain.

一般地，要写出y = |ax + b|，先求ax + b的零点，然后分割定义域。

For example, |2x – 6| becomes 2x – 6 when x ≥ 3, and -2x + 6 when x < 3.

例如，|2x – 6|在x ≥ 3时为2x – 6，在x < 3时为-2x + 6。

6. Solving Absolute Value Equations | 绝对值方程求解

To solve |f(x)| = a, where a > 0, set f(x) = a and f(x) = -a.

求解|f(x)| = a（其中a > 0），令f(x) = a和f(x) = -a。

Example: Solve |x – 2| = 5.

例：解|x – 2| = 5。

x – 2 = 5 → x = 7; x – 2 = -5 → x = -3

Both solutions are valid because the absolute value of each equals 5.

两个解都有效，因为它们的绝对值都等于5。

If a < 0, the equation has no solution, since absolute value is never negative.

如果a < 0，方程无解，因为绝对值永远不为负。

7. Solving Absolute Value Inequalities | 绝对值不等式求解

Inequalities of the form |f(x)| < a mean -a < f(x) < a (and condition).

形如|f(x)| < a的不等式意味着-a < f(x) < a（且关系）。

Inequalities of the form |f(x)| > a mean f(x) > a or f(x) < -a (or condition).

形如|f(x)| > a的不等式意味着f(x) > a或f(x) < -a（或关系）。

Example: Solve |2x + 1| ≤ 3.

例：解|2x + 1| ≤ 3。

-3 ≤ 2x + 1 ≤ 3 → -4 ≤ 2x ≤ 2 → -2 ≤ x ≤ 1

For strict inequalities, use open circles on the number line.

对于严格不等式，在数轴上使用空心圆圈。

8. Graphs of More Complex Absolute Value Functions | 复杂绝对值函数图像

Functions like y = |x – 2| + |x + 3| have piecewise linear graphs with multiple vertices.

像y = |x – 2| + |x + 3|这样的函数具有多顶点的分段线性图像。

To graph such functions, identify the critical points where each absolute value changes sign.

要绘制此类图像，需找出每个绝对值改变符号的临界点。

For y = |x – 2| + |x + 3|, the critical points are x = 2 and x = -3, which divide the real line into three intervals.

对于y = |x – 2| + |x + 3|，临界点为x = 2和x = -3，它们将实轴分成三个区间。

On each interval, remove the absolute value signs and graph the resulting linear function.

在每个区间上去掉绝对值符号，绘制所得线性函数图像。

9. Applications and Word Problems | 应用与文字题

Absolute value is used to express errors and tolerances, such as |measured – true| ≤ tolerance.

绝对值用于表示误差和容差，例如|测量值 – 真值| ≤ 容差。

In kinematics, speed |v| represents magnitude regardless of direction.

在运动学中，速率|v|表示与方向无关的大小。

IB exam questions often involve finding the range of a function or solving real-world problems with absolute value inequalities.

IB考试题常涉及求函数值域或用绝对值不等式解决实际问题。

10. Common Exam Pitfalls | 考试常见错误

Forgetting that |x| is always non-negative.

忘记|x|永远是非负的。
Missing one solution when solving |f(x)| = a by only considering f(x) = a.

解|f(x)| = a时只考虑f(x) = a，漏掉一个解。
Incorrectly flipping inequality signs when both sides are negative.

当两边为负时错误地翻转不等号。
Confusing < with > in absolute inequalities; remember “less than” is a single interval, “greater than” is two intervals.

混淆绝对值不等式中的；记住”小于”对应单一区间，”大于”对应两个区间。

11. Summary | 总结

Absolute value functions are characterized by V-shaped graphs, piecewise definitions, and key algebraic properties.

绝对值函数的特征是V形图像、分段定义和关键代数性质。

Understanding transformations allows quick sketching, while equations and inequalities require careful case analysis.

理解图像变换有助于快速画图，而方程和不等式需要仔细的分情况讨论。

Practice with past papers to become familiar with common IB question styles.

通过练习历年真题，熟悉常见的IB题型。

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