📚 A-Level Maths: Transformations of y=|f(x)| and y=f(|x|) | A-Level 数学：y=|f(x)|与y=f(|x|)的图像变换
Absolute value transformations are among the most frequently tested topics in A-Level Mathematics, appearing in both Pure Mathematics and Calculus papers. Understanding how to sketch y=|f(x)| and y=f(|x|) from a given graph or equation is not just a mechanical exercise—it demands a clear grasp of how the modulus operator interacts with the underlying function.
绝对值图像变换是 A-Level 数学中高频考点之一，在纯数学与微积分试卷中均常出现。掌握如何从已知图形或函数表达式绘制 y=|f(x)| 与 y=f(|x|)，不仅是机械操作，更需要对绝对值算符与函数本身之间互动关系有清晰理解。
1. The Definition of the Modulus Function | 绝对值函数的定义
Before exploring transformations, we must first recall the formal definition of the modulus (absolute value) function. For any real number x, the modulus of x is defined as its distance from zero on the number line, which is always non-negative.
在探讨图像变换之前，我们必须先回顾绝对值函数的正式定义。对于任意实数 x，x 的绝对值定义为它在数轴上到零点的距离，该距离永远非负。
|x| = x, if x ≥ 0; and |x| = −x, if x < 0
This piecewise definition is the foundation for all transformation rules. It tells us that the modulus operator ‘folds’ negative values into positive ones, while leaving positive values unchanged.
这个分段定义是一切变换规则的基石。它告诉我们，绝对值算符将负值“折叠”为正值，而正值保持不变。
Two fundamental identities follow immediately from the definition: |a × b| = |a| × |b|, and |a ÷ b| = |a| ÷ |b| (for b ≠ 0). However, it is crucial to note that |a + b| ≠ |a| + |b| in general—this is a common source of student errors.
两个基本恒等式可直接从定义推导：|a × b| = |a| × |b|，以及 |a ÷ b| = |a| ÷ |b|（其中 b ≠ 0）。但必须注意，一般来说 |a + b| ≠ |a| + |b|——这是学生常见错误的一大来源。
2. Transformation Rules for y=|f(x)| | y=|f(x)| 的变换规则
The transformation y=|f(x)| follows one simple geometric principle: every part of the curve y=f(x) that lies below the x-axis (where f(x)<0) is reflected vertically upwards across the x-axis. The parts of the curve that already lie above or on the x-axis remain completely unchanged.
y=|f(x)| 的变换遵循一条简单的几何原理：曲线 y=f(x) 位于 x 轴下方的所有部分（即 f(x)<0 的区域）将关于 x 轴向上翻折；而位于 x 轴上方或恰好落在 x 轴上的部分则完全保持不变。
To sketch y=|f(x)| from the graph of y=f(x):
要由 y=f(x) 的图像绘制 y=|f(x)| 的图像：
- Step 1: Identify all x-intercepts (roots) of f(x), where f(x)=0. These points remain fixed under the transformation.
- 步骤1：找出 f(x) 的所有 x 轴截距（根），即 f(x)=0 的点。这些点在变换下保持不动。
- Step 2: Trace the portions of the curve where f(x)>0 exactly as they are.
- 步骤2：将 f(x)>0 的曲线部分原样保留。
- Step 3: Reflect the portions where f(x)<0 across the x-axis so that they appear above the x-axis.
- 步骤3：将 f(x)<0 的曲线部分关于 x 轴翻折到 x 轴上方。
The resulting graph is always entirely on or above the x-axis (y ≥ 0 for all x in the domain). This is the single most important check for correctness.
所得图像总是完全位于 x 轴上或上方（对于定义域内所有 x，有 y ≥ 0）。这是检验正确性最重要的一条标准。
3. Transformation Rules for y=f(|x|) | y=f(|x|) 的变换规则
The transformation y=f(|x|) is fundamentally different. Here, the modulus acts on the input variable, not the output. Since |x| is always non-negative, the function y=f(|x|) only ever ‘sees’ non-negative inputs.
y=f(|x|) 的变换有本质不同。这里，绝对值作用于输入变量，而非输出值。由于 |x| 永远非负，函数 y=f(|x|) 只能“看到”非负的输入。
The construction of this graph involves three steps:
该图形的构造涉及三个步骤：
- Step 1: Discard entirely the portion of y=f(x) that lies to the left of the y-axis (x<0).
- 步骤1：完全丢弃 y=f(x) 位于 y 轴左侧（x<0）的部分。
- Step 2: Keep the portion of y=f(x) to the right of the y-axis (x≥0) unchanged.
- 步骤2：保留 y=f(x) 在 y 轴右侧（x≥0）的部分，不做任何改变。
- Step 3: Reflect the right-side portion across the y-axis to create the left-side portion.
- 步骤3：将右侧部分关于 y 轴翻折，以生成左侧部分。
The final graph is always symmetric about the y-axis. This is an even function: f(|−x|)=f(|x|).
最终图像总是关于 y 轴对称。这是一个偶函数：f(|−x|)=f(|x|)。
4. Key Differences Between the Two Transformations | 两种变换的核心区别
Students often confuse these two transformations, so a side-by-side comparison is essential. The table below summarises the critical distinctions.
学生经常混淆这两种变换，因此并排对比至关重要。下表总结了关键区别。
| Feature | 特征 | y=|f(x)| | y=f(|x|) |
| Modulus acts on | 绝对值作用于 | Output (y-values) | 输出值（y 值） | Input (x-values) | 输入值（x 值） |
| Graph shift | 图像变化 | Parts below x-axis fold upwards | x 轴下方的部分向上翻折 | Left side deleted; right side mirrored leftwards | 左侧删除；右侧向左镜像 |
| Symmetry | 对称性 | No guaranteed symmetry | 无保证对称性 | Always symmetric about y-axis | 始终关于 y 轴对称 |
| Range | 值域 | y ≥ 0 always | 始终满足 y ≥ 0 | Same as f(x) restricted to x ≥ 0 | 与 f(x) 在 x ≥ 0 上的值域相同 |
| x-intercept preservation | x 轴截距保持 | All original x-intercepts preserved | 所有原 x 轴截距保持不变 | Only x ≥ 0 intercepts preserved; new intercepts appear on the left | 仅保留 x ≥ 0 的截距；左侧会出现新的截距 |
An easy memory aid: y=|f(x)| flips the graph up (vertical reflection); y=f(|x|) flips the graph horizontally (horizontal reflection). One acts on the output, the other on the input.
一个易用记忆法：y=|f(x)| 将图像向上翻转（垂直反射）；y=f(|x|) 将图像水平翻转（水平反射）。一个作用于输出，一个作用于输入。
5. Worked Example 1: Quadratic Function | 例1：二次函数
Consider the function f(x)=x²−6x+5. First, factorise: f(x)=(x−1)(x−5). The graph is a parabola opening upwards, crossing the x-axis at x=1 and x=5, with vertex at (3,−4).
考虑函数 f(x)=x²−6x+5。首先因式分解：f(x)=(x−1)(x−5)。图像是开口向上的抛物线，x 轴截距为 x=1 和 x=5，顶点为 (3,−4)。
For y=|f(x)|, the portions of the parabola above the x-axis remain unchanged, and the portion below the x-axis (between x=1 and x=5, where f(x)<0) is reflected upward. The vertex (3,−4) moves to (3,4), and the x-axis intercepts remain at x=1 and x=5.
对于 y=|f(x)|，抛物线上位于 x 轴上方的部分保持不变，而位于 x 轴下方的部分（即 x=1 与 x=5 之间的区域，在此区间 f(x)<0）向上翻折。顶点 (3,−4) 移动到 (3,4)，x 轴截距仍然在 x=1 和 x=5。
For y=f(|x|), we discard the left half of the parabola (x<0), keep the right half (x≥0), and reflect the right half across the y-axis. The resulting graph is a W-shaped curve that is symmetric about the y-axis. For instance, f(|−3|)=f(3)=9−18+5=−4, so the point (−3,−4) lies on the graph.
对于 y=f(|x|)，我们丢弃抛物线的左半部分（x<0），保留右半部分（x≥0），再将右半部分关于 y 轴镜像。所得图形是一条 W 形曲线，关于 y 轴对称。例如，f(|−3|)=f(3)=9−18+5=−4，因此点 (−3,−4) 位于图像上。
When solving equations such as |x²−6x+5|=2, it is often easier to sketch y=|f(x)| first and then draw the horizontal line y=2 to find intersection points.
在解方程如 |x²−6x+5|=2 时，通常先画出 y=|f(x)| 的图像，再画水平线 y=2 找交点，这样更为简便。
6. Worked Example 2: Trigonometric Function | 例2：三角函数
Trigonometric functions provide excellent illustrations of these transformations because of their periodic and oscillatory nature.
三角函数的周期性与振荡性使其成为说明这些变换的绝佳例子。
Consider f(x)=sin x over the interval −2π ≤ x ≤ 2π. The graph of y=sin x oscillates between −1 and 1, crossing the x-axis at multiples of π.
考虑区间 −2π ≤ x ≤ 2π 上的 f(x)=sin x。y=sin x 的图像在 −1 与 1 之间振荡，在 π 的整数倍处与 x 轴相交。
For y=|sin x|: all negative ‘valleys’ of the sine wave (where sin x < 0) are reflected upward. The result is a series of positive arches resembling a 'rectified' sine wave. The range becomes [0,1] instead of [−1,1]. This is often described as 'folding the wave upward'.
对于 y=|sin x|：正弦波的所有负“波谷”（即 sin x < 0 的区域）被向上翻折。结果是一系列正拱形，类似“整流”后的正弦波。值域从 [−1,1] 变为 [0,1]。这常被描述为“将波形向上折叠”。
For y=sin(|x|): the left half of the sine wave is deleted, the right half is kept, and then mirrored across the y-axis. The resulting function is even, with period still 2π but with different behaviour on the negative side. Note that sin(|x|) is differentiable everywhere except at x=0.
对于 y=sin(|x|)：删除正弦波的左半部分，保留右半部分，然后关于 y 轴镜像。所得函数为偶函数，周期仍为 2π，但在负半轴的行为有所不同。注意 sin(|x|) 在除 x=0 外的所有点均可导。
These two graphs look completely different: y=|sin x| has sharp ‘V’ cusps at every x=nπ (where the sign changes), while y=sin(|x|) has a smooth even shape with no cusps except possibly at x=0.
这两条图像看起来完全不同：y=|sin x| 在每个 x=nπ 处有尖锐的“V”形尖端（因为符号在此改变），而 y=sin(|x|) 是光滑的偶函数形状，除 x=0 外没有尖端。
7. Worked Example 3: Exponential and Logarithmic Functions | 例3：指数函数与对数函数
Exponential and logarithmic functions highlight the domain considerations in these transformations.
指数函数与对数函数最能体现这些变换中定义域方面的考量。
Take f(x)=eˣ. The domain of f is all real numbers, and the range is (0,∞). Since eˣ>0 for all x, we have |eˣ|=eˣ, so y=|f(x)| is identical to y=f(x).
取 f(x)=eˣ。f 的定义域为全体实数，值域为 (0,∞)。因为对于所有 x 有 eˣ>0，所以 |eˣ|=eˣ，即 y=|f(x)| 与 y=f(x) 的图像完全相同。
For y=e^|x|: we keep the right half of the exponential curve, discard the left half, and reflect the right half across the y-axis. The result is an even function that decays to 1 as x→0⁻ and grows as |x|→∞. In fact, e^|x| = eˣ for x ≥ 0, and e^|x| = e⁻ˣ for x < 0.
对于 y=e^|x|：保留指数曲线的右半部分，丢弃左半部分，并将右半部分关于 y 轴镜像。所得函数为偶函数，当 x→0⁻ 时函数值衰减至 1，当 |x|→∞ 时增长。事实上，e^|x| = eˣ（当 x ≥ 0），且 e^|x| = e⁻ˣ（当 x < 0）。
Now consider f(x)=ln x, which has domain x>0. For y=|ln x|: the portion where ln x < 0 (i.e., 0
现在考虑 f(x)=ln x，其定义域为 x>0。对于 y=|ln x|：ln x < 0 的部分（即 0
8. Solving Equations and Inequalities Involving Modulus | 涉及绝对值的方程与不等式求解
Graphical interpretations of |f(x)| and f(|x|) are powerful tools for solving equations and inequalities.
|f(x)| 与 f(|x|) 的图形解释是解方程与不等式强有力的工具。
To solve |f(x)| = g(x): sketch y=|f(x)| and y=g(x) on the same axes, then find x-coordinates of intersection points. This avoids the need to consider separate cases algebraically.
解 |f(x)| = g(x)：在同一坐标系中画 y=|f(x)| 与 y=g(x) 的图像，然后找到交点的 x 坐标。这避免了用代数方法分情况讨论的繁琐。
For inequalities such as |f(x)| ≥ g(x): identify the intervals on the x-axis where the graph of y=|f(x)| lies above or touches y=g(x).
对于形如 |f(x)| ≥ g(x) 的不等式：找出 x 轴上 y=|f(x)| 的图像位于 y=g(x) 上方或与之相切的区间。
Algebraic methods serve as verification: for |f(x)|=g(x), one must solve both f(x)=g(x) (with condition f(x) ≥ 0) and f(x)=−g(x) (with condition f(x) < 0). The graphical method naturally captures all valid solutions and automatically discards extraneous ones.
代数方法可用于验证：对于 |f(x)|=g(x)，需要分别解 f(x)=g(x)（条件是 f(x) ≥ 0）和 f(x)=−g(x)（条件是 f(x) < 0）。图形方法能自然捕捉全部有效解，并自动排除增解。
9. Common Mistakes and How to Avoid Them | 常见错误及避免方法
Several recurring errors plague students when handling these transformations. Being aware of them is the first step to avoiding them.
学生在处理这些变换时屡次犯下几类典型错误。认识到这些错误是避免它们的第一步。
- Mistake 1: Treating y=|f(x)| and y=f(|x|) as identical transformations. Remember: one affects y-values, the other affects x-values.
- 错误1：将 y=|f(x)| 与 y=f(|x|) 视为相同变换。切记：一个影响 y 值，另一个影响 x 值。
- Mistake 2: Forgetting to check the domain. For y=f(|x|), the domain becomes symmetric—if x must satisfy certain conditions in the original function, those conditions apply to |x| instead.
- 错误2：忘记检查定义域。对于 y=f(|x|)，定义域变为对称——如果原始函数中 x 必须满足某些条件，那么这些条件应施加于 |x|。
- Mistake 3: Drawing y=|f(x)| with points below the x-axis. After the transformation, the graph must lie entirely above or on the x-axis.
- 错误3：绘制 y=|f(x)| 时仍保留了 x 轴下方的点。变换后图像必须完全位于 x 轴上方或恰好落在轴上。
- Mistake 4: Assuming |a+b|=|a|+|b| when solving modulus equations. This identity is false in general; the triangular inequality gives only |a+b| ≤ |a|+|b|.
- 错误4：在解绝对值方程时假定 |a+b|=|a|+|b|。该恒等式一般不成立；三角不等式仅给出 |a+b| ≤ |a|+|b|。
- Mistake 5: Confusing the cusp points. In y=|f(x)|, cusps occur at roots of f(x); in y=f(|x|), a cusp may occur at x=0 but not necessarily elsewhere.
- 错误5：混淆尖点位置。在 y=|f(x)| 中，尖点出现在 f(x) 的根处；在 y=f(|x|) 中，尖点可能出现在 x=0 处，但不一定出现在其他位置。
A reliable self-check: after sketching, test two or three x-values by direct substitution into the transformed equation to verify that your graph matches the computed y-values.
一个可靠的自我检查方法：画完草图后，选取两三个 x 值直接代入变换后的方程中，验证图像与计算所得的 y 值是否一致。
10. Practice Problems | 练习题
The following problems test your understanding of both transformations. Attempt them graphically first, then confirm algebraically.
以下题目考查你对两种变换的理解。先尝试用图像法求解，再用代数方法验证。
- Problem 1: Sketch y=|x²−4| on the domain −3 ≤ x ≤ 3. Identify the coordinates of all turning points.
- 题1：在定义域 −3 ≤ x ≤ 3 上绘制 y=|x²−4| 的图像，并标出所有转折点坐标。
- Problem 2: The graph of y=f(x) passes through the points (−4,−2), (0,3), and (5,−1). Determine through which points the graph of y=f(|x|) passes.
- 题2：函数 y=f(x) 的图像经过点 (−4,−2)、(0,3) 和 (5,−1)。确定 y=f(|x|) 的图像经过哪些点。
- Problem 3: Solve the equation |2x−3|=x+1 both graphically and algebraically, stating the domain restrictions for each method.
- 题3：分别用图像法和代数法解方程 |2x−3|=x+1，并说明每种方法的定义域限制。
- Problem 4: Write down the range of y=|3 sin(2x)+1| and state the minimum and maximum values.
- 题4：写出 y=|3 sin(2x)+1| 的值域，并指出最小值和最大值。
- Problem 5: For f(x)=x³−x, state how many solutions exist to the equation |f(x)|=c for different values of c>0.
- 题5：对于 f(x)=x³−x，说明对于不同的 c>0，方程 |f(x)|=c 有多少个解。
11. Summary and Key Takeaways | 总结与核心要点
Mastering these two transformations requires memorising one simple geometric mantra: y=|f(x)| folds the graph upward across the x-axis; y=f(|x|) folds the graph leftward across the y-axis.
掌握这两种变换的核心在于记住一条简单的几何口诀：y=|f(x)| 是将图像沿 x 轴向上折叠；y=f(|x|) 是将图像沿 y 轴向左折叠。
Always apply the transformations step by step: mark the fixed points first (roots for y=|f(x)|; the y-intercept and right-hand side for y=f(|x|)), then perform the reflection, and finally check symmetry and range conditions.
务必按步骤实施变换：先标出不动点（y=|f(x)| 取根；y=f(|x|) 取 y 轴截距与右侧部分），再执行翻折，最后检验对称性与值域条件。
Graphical reasoning is not just a sketching skill—it is a problem-solving strategy that simplifies equations, inequalities, and calculus questions involving modulus terms. Integrate it into your daily practice, and you will find these questions become some of the most straightforward marks on the paper.
图形推理不仅是一项绘图技能，更是一种问题求解策略，能简化涉及绝对值项的方程、不等式与微积分问题。将其融入日常练习，你会发现这类题目成为试卷中最容易拿分的一部分。
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