📚 y=|f(x)| vs y=f(|x|) — Graph Transformations in IB Maths | IB数学：y=|f(x)|与y=f(|x|)的图像
Graphical transformations are a core topic in the IB Mathematics curriculum, tested across both Analysis and Approaches (AA) and Applications and Interpretation (AI). Two of the most commonly confused transformations are y=|f(x)| and y=f(|x|). While they differ by only a single pair of vertical bars, their geometric meanings are fundamentally distinct — one involves reflecting the negative parts of a curve across the x-axis, the other involves reflecting the right half of the curve across the y-axis.
图像变换是IB数学课程的核心考点，在Analysis and Approaches（AA）与Applications and Interpretation（AI）中均会涉及。其中最易混淆的两个变换就是 y=|f(x)| 与 y=f(|x|)。它们虽然只相差一组绝对值符号，但几何含义截然不同——一个涉及将曲线在x轴下方的部分沿x轴翻折，另一个涉及将曲线右半部分沿y轴对称到左侧。
1. Understanding |f(x)|: The Absolute Value of the Output | 理解 |f(x)|：对函数值取绝对值
When we write y=|f(x)|, we take every output value of the function f(x) and make it non-negative. If f(x) is already positive or zero, the graph remains unchanged. If f(x) is negative, we reflect that portion of the graph across the x-axis, making it positive.
当我们写 y=|f(x)| 时，本质上是对函数 f(x) 的每一个输出值取非负值。如果 f(x) 原本就大于或等于零，图像保持不变；如果 f(x) 为负，则将图像在x轴下方的部分沿x轴翻折到上方。
Key property: The portion of the graph above the x-axis stays exactly the same; the portion below the x-axis is reflected upward. The x-axis acts as a “mirror” for all negative outputs.
核心性质：x轴上方的图像完全不变；x轴下方的图像被向上翻折。x轴扮演了一面”镜子”的角色，将一切负输出映射为正。
y = |f(x)| = f(x) when f(x) ≥ 0; y = |f(x)| = −f(x) when f(x) < 0
For example, consider f(x) = x² − 1. The original graph is a parabola opening upward with roots at x = −1 and x = 1. The region between −1 and 1 lies below the x-axis. For y=|x² − 1|, that middle “dip” is reflected above the x-axis, producing a W-shaped curve.
例如，考虑 f(x) = x² − 1。原图像是开口向上的抛物线，零点在 x = −1 和 x = 1。−1 到 1 之间的区域位于x轴下方。对于 y=|x² − 1|，中间的”凹陷”被翻折到x轴上方，形成W形曲线。
2. Understanding f(|x|): The Absolute Value of the Input | 理解 f(|x|)：对自变量取绝对值
When we write y=f(|x|), we replace every x in the function with |x|. Since |x| is always non-negative, the value of f(|x|) depends only on the magnitude of x, not its sign. This means f(|x|) is always an even function, regardless of whether the original f(x) is even or odd.
当我们写 y=f(|x|) 时，将原函数中的每一个 x 替换为 |x|。由于 |x| 永远非负，f(|x|) 的值只取决于 x 的绝对值大小，而与 x 的正负无关。这意味着 f(|x|) 永远是一个偶函数，无论原来的 f(x) 是奇函数还是偶函数。
Key property: The graph of y=f(|x|) for x ≥ 0 is identical to the graph of y=f(x) for x ≥ 0. Then we reflect the right half across the y-axis to obtain the left half.
核心性质：y=f(|x|) 在 x ≥ 0 的部分与 y=f(x) 在 x ≥ 0 的部分完全相同。然后将右半部分沿y轴对称到左侧，得到左半部分。
f(|x|) = f(x) when x ≥ 0; f(|x|) = f(−x) when x < 0
For example, consider f(x) = (x − 1)². The graph of f(|x|) would keep the right branch (x ≥ 0) unchanged and mirror it to the left, resulting in a shape that is symmetric about the y-axis.
例如，考虑 f(x) = (x − 1)²。f(|x|) 的图像保留右侧（x ≥ 0）的部分不变，并将其镜像到左侧，最终得到一个关于y轴对称的图像。
3. Step-by-Step: Sketching y=|f(x)| | 分步作图画 y=|f(x)|
Follow these steps to sketch y=|f(x)| accurately:
按照以下步骤可以准确画出 y=|f(x)| 的图像：
- Step 1: Sketch y=f(x) as usual, marking all x-intercepts clearly. | 第一步：照常画出 y=f(x)，清楚标出所有与x轴的交点。
- Step 2: Identify all regions where f(x) < 0 (portions below the x-axis). | 第二步：找出所有 f(x) < 0 的区域（x轴下方的部分）。
- Step 3: Reflect those negative regions across the x-axis. | 第三步：将x轴下方的区域沿x轴翻折到上方。
- Step 4: The parts above the x-axis remain untouched. | 第四步：x轴上方的部分保持原样。
- Step 5: Double-check that the graph never goes below the x-axis. | 第五步：检查图像永远不落在x轴下方。
Remember: the x-intercepts of y=f(x) become “pinning points” for y=|f(x)| — they remain unchanged because |0| = 0.
记住：y=f(x) 的x轴截距会变成 y=|f(x)| 的”固定点”——它们保持不变，因为 |0| = 0。
4. Step-by-Step: Sketching y=f(|x|) | 分步作图画 y=f(|x|)
To sketch y=f(|x|), use the following approach:
要画出 y=f(|x|) 的图像，可以采用以下方法：
- Step 1: Sketch y=f(x) fully. | 第一步：完整画出 y=f(x)。
- Step 2: Erase (or ignore) entirely the portion of the graph where x < 0. | 第二步：擦掉（或忽略）x < 0 的部分。
- Step 3: Keep the portion where x ≥ 0 exactly as it is. | 第三步：保留 x ≥ 0 的部分不动。
- Step 4: Reflect the right half across the y-axis to produce the left half. | 第四步：将右半部分沿y轴对称到左侧。
- Step 5: Verify the final graph is symmetric about the y-axis. | 第五步：验证最终图像关于y轴对称。
Critical point: the y-intercept of y=f(x) is the point where the reflection “meets” — f(|0|) = f(0), so the y-intercept always remains.
关键点：y=f(x) 的y轴截距是反射的”交汇点”——f(|0|) = f(0)，所以y轴截距始终保持不变。
5. Visual Comparison: |f(x)| vs f(|x|) | 图像对比：|f(x)| 与 f(|x|)
Let us compare the two transformations side by side using a concrete example. Take f(x) = (x − 2)(x + 1) = x² − x − 2, a parabola opening upward with roots at x = −1 and x = 2.
让我们用一个具体例子并排比较这两种变换。取 f(x) = (x − 2)(x + 1) = x² − x − 2，这是一条开口向上的抛物线，零点在 x = −1 和 x = 2。
| Feature | 特征 | y = |f(x)| | y = f(|x|) |
| Reflection axis | 对称轴 | x-axis (horizontal) | x轴（水平） | y-axis (vertical) | y轴（垂直） |
| Symmetry of result | 结果对称性 | Not necessarily symmetric | 不一定对称 | Always even (symmetric about y-axis) | 一定是偶函数（关于y轴对称） |
| Impact on negative outputs | 对负输出的影响 | They become positive | 负值变为正值 | No direct impact on outputs | 不直接影响函数值 |
| Impact on left half (x < 0) | 对左半部分（x < 0）的影响 | Left half may be altered | 左半部分可能被改变 | Left half is overwritten by reflection of right half | 左半部分被右半部分的镜像覆盖 |
| x-intercepts | x轴截距 | Same as f(x) | 与f(x)相同 | May gain extra intercepts | 可能获得新的截距 |
In our example, y=|x² − x − 2| has the negative region between x = −1 and x = 2 reflected upward, while y=(|x|)² − |x| − 2 keeps the right branch and mirrors it leftward, creating a graph that dips below the x-axis on both sides.
在我们的例子中，y=|x² − x − 2| 将 x = −1 到 x = 2 之间的负值区域翻折向上；而 y=(|x|)² − |x| − 2 则保留右侧分支并向左镜像，形成两侧都穿到x轴下方的图像。
6. How the Domain and Range Change | 定义域与值域的变化
Understanding how domain and range are affected is essential for IB exam questions.
理解定义域与值域的变化对IB考试题目至关重要。
- y=|f(x)|: The domain is unchanged — identical to that of f(x). The range, however, is transformed: every negative value becomes positive. If the original range was [−2, 3], the new range becomes [0, 3]. If the range was (−∞, 5], it becomes [0, 5].
- y=|f(x)|：定义域不变——与 f(x) 完全一致。但值域被变换：所有负值变为正值。如果原值域是 [−2, 3]，新值域变为 [0, 3]；如果值域是 (−∞, 5]，则变为 [0, 5]。
- y=f(|x|): The domain is restricted to values where |x| lies within the original domain of f. For example, if f has domain [−1, 3], then we require −1 ≤ |x| ≤ 3, which simplifies to |x| ≤ 3, giving domain −3 ≤ x ≤ 3. The range remains exactly the same as the range of f(x) restricted to x ≥ 0.
- y=f(|x|)：定义域被限制为满足 |x| 落在 f 原定义域内的值。例如，若 f 的定义域为 [−1, 3]，则要求 −1 ≤ |x| ≤ 3，化简得 |x| ≤ 3，即定义域为 −3 ≤ x ≤ 3。值域与 f(x) 在 x ≥ 0 上的值域完全相同。
A common IB question asks: “Find the range of y=|f(x)| given the range of f(x).” The trick is to map every negative value to its absolute value and keep the positive values unchanged.
一个常见的IB考题是：”已知 f(x) 的值域，求 y=|f(x)| 的值域。”技巧是将每一个负值映射为其绝对值，正值保持不变。
7. Composite Transformations | 复合变换
In IB exams, you may be asked to combine these absolute value transformations with other standard transformations such as translations, reflections, and stretches.
在IB考试中，你可能会被要求将这些绝对值变换与平移、反射、伸缩等标准变换结合使用。
Consider the sequence: y = 2|f(x)| + 1. This involves three operations: first stretch vertically by factor 2, then apply the absolute value, then translate upward by 1. Actually, careful — the absolute value applies to f(x) first, then the vertical stretch, then the translation. Let us be precise: y = 2|f(x)| + 1 means: (1) compute |f(x)|, (2) multiply by 2, (3) add 1. The order matters for the shape of the graph.
考虑变换序列：y = 2|f(x)| + 1。这涉及三个操作：先对 f(x) 取绝对值，然后纵向拉伸2倍，最后向上平移1个单位。顺序会影响图像的形状，必须精确理解。
Another composite case: y = f(|x − 2|). Here we first substitute (x − 2) into f, then apply the absolute value. The graph of f(|x − 2|) is the graph of f(|x|) shifted 2 units to the right. Alternatively, you can think of it as: the graph of f(x) for x ≥ 2 is kept, and the portion x < 2 is the reflection of x > 2 about the vertical line x = 2.
另一个复合情形：y = f(|x − 2|)。这里先将 (x − 2) 代入 f，再取绝对值。f(|x − 2|) 的图像就是 f(|x|) 的图像向右平移2个单位。或者可以这样理解：保留 f(x) 在 x ≥ 2 的部分，x < 2 的部分是 x > 2 关于直线 x = 2 的镜像。
General rule: y = f(|x − a|) is symmetric about the line x = a.
一般规律：y = f(|x − a|) 关于直线 x = a 对称。
8. Using Graphs to Solve Inequalities | 利用图像解不等式
Absolute value graphs are often used in IB to solve inequalities graphically. The approach is to sketch both sides of the inequality and read off the regions where one curve lies above the other.
IB考试经常利用绝对值图像来解不等式。方法是分别画出不等式两边的图像，找出一个曲线位于另一个上方的区域。
For example, to solve |f(x)| > g(x), sketch y=|f(x)| and y=g(x) on the same axes. The solution is the set of x-values for which the |f(x)| curve lies strictly above the g(x) curve.
例如，解 |f(x)| > g(x)，在同一个坐标系中画出 y=|f(x)| 和 y=g(x)。解集就是 |f(x)| 曲线严格位于 g(x) 曲线上方时对应的 x 值集合。
Similarly, to solve f(|x|) ≤ 0, sketch y=f(|x|) and identify the intervals where the graph is on or below the x-axis. Since f(|x|) is even, the solution set will be symmetric about the origin.
类似地，解 f(|x|) ≤ 0，画出 y=f(|x|) 并找出图像位于x轴上或下方的区间。由于 f(|x|) 是偶函数，解集将关于原点对称。
9. Worked IB Example | IB真题示例
Let us work through a full exam-style question step by step.
让我们一步步完整解答一道考试风格的题目。
Question: The function f is defined as f(x) = x² − 4x + 3 for x ∈ ℝ. (a) Sketch y=|f(x)|. (b) Sketch y=f(|x|). (c) Find the number of solutions to the equation |f(x)| = f(|x|).
题目：函数 f 定义为 f(x) = x² − 4x + 3，x ∈ ℝ。(a) 画出 y=|f(x)| 的图像。(b) 画出 y=f(|x|) 的图像。(c) 求方程 |f(x)| = f(|x|) 的解的个数。
Solution (a): Factorising: f(x) = (x − 1)(x − 3). The parabola opens upward, crosses the x-axis at x = 1 and x = 3, and its vertex is at x = 2, f(2) = −1. The vertex lies below the x-axis. For y=|f(x)|, the vertex at (2, −1) is reflected to (2, 1). The x-intercepts stay at x = 1 and x = 3.
解答(a)：因式分解：f(x) = (x − 1)(x − 3)。抛物线开口向上，与x轴交于 x = 1 和 x = 3，顶点在 x = 2，f(2) = −1。顶点位于x轴下方。对于 y=|f(x)|，顶点 (2, −1) 被反射到 (2, 1)。x轴截距保持在 x = 1 和 x = 3。
Solution (b): y=f(|x|) = |x|² − 4|x| + 3 = x² − 4|x| + 3. The right half (x ≥ 0) is the original parabola restricted to x ≥ 0, with vertex at (2, −1). Reflecting across the y-axis gives a vertex at (−2, −1). The resulting graph has x-intercepts at x = ±1 and x = ±3, and is symmetric about the y-axis.
解答(b)：y=f(|x|) = |x|² − 4|x| + 3 = x² − 4|x| + 3。右半部分（x ≥ 0）是原抛物线在 x ≥ 0 上的部分，顶点在 (2, −1)。沿y轴对称后得到顶点在 (−2, −1)。最终图像在 x = ±1 和 x = ±3 处与x轴相交，且关于y轴对称。
Solution (c): We set |x² − 4x + 3| = x² − 4|x| + 3. Considering cases: for x ≥ 3, both sides equal f(x), so all x ≥ 3 are solutions. For 1 ≤ x ≤ 3, the left side equals −f(x) = −(x² − 4x + 3), while the right side equals x² − 4x + 3. Setting −(x² − 4x + 3) = x² − 4x + 3 gives −2(x² − 4x + 3) = 0, so x = 1 or x = 3. For 0 ≤ x ≤ 1, both sides equal f(x), so 0 ≤ x ≤ 1 are solutions. By symmetry, the intervals x ≤ −3 and −1 ≤ x ≤ 0 are also solutions. Counting the full solution set: x ∈ [−3, −1] ∪ [1, 3] is not quite right — we must check carefully. Actually, the full solution set is x ∈ (−∞, −3] ∪ [−1, 0] ∪ [1, 3] ∪ [3, ∞). Wait, this is getting complicated — the safer IB method is truly graphical: count the number of distinct x-values where the two graphs intersect. Since both graphs share infinitely many points on the interval [3, ∞) and (−∞, −3] and also on [0, 1] and [−1, 0], the equation has infinitely many solutions. A better version of this question would ask for the intervals. In exam settings, questions like this typically ask for the set of x-values, not the count.
解答(c)：我们令 |x² − 4x + 3| = x² − 4|x| + 3。分情况讨论：当 x ≥ 3 时，两边都等于 f(x)，所以所有 x ≥ 3 都是解。当 1 ≤ x ≤ 3 时，左边等于 −f(x) = −(x² − 4x + 3)，右边等于 x² − 4x + 3。令二者相等得 −(x² − 4x + 3) = x² − 4x + 3，化简得 −2(x² − 4x + 3) = 0，所以 x = 1 或 x = 3。当 0 ≤ x ≤ 1 时，两边都等于 f(x)，所以 0 ≤ x ≤ 1 都是解。由对称性，x ≤ −3 和 −1 ≤ x ≤ 0 也是解。实际上，完整的解集是 x ∈ (−∞, −3] ∪ [−1, 0] ∪ [1, 3] ∪ [3, ∞)。但这里要小心——更稳妥的IB做法是借助图像：由于两个图像在 [3, ∞) 和 (−∞, −3] 上有无限多个重合点，也在 [0, 1] 和 [−1, 0] 上有无限多个重合点，因此方程有无限多个解。考试中这类题通常要求写出x的区间，而非数解的个数。
10. Common Mistakes and How to Avoid Them | 常见错误与避免方法
Students frequently make the following errors. Recognising them early can save valuable marks.
学生经常犯以下错误。及早识别可以避免失分。
- Mistake 1: Thinking |f(x)| and f(|x|) are the same. They are equal only in special cases, such as when f is an even function with f(x) ≥ 0 for all x. | 错误一：认为 |f(x)| 和 f(|x|) 相同。只有当 f 为偶函数且 f(x) 恒非负等特殊情形下它们才相等。
- Mistake 2: Reflecting the entire graph of f(x) for y=|f(x)|, rather than only the negative portions. | 错误二：画 y=|f(x)| 时将 f(x) 的整个图像都反射，而不是只反射负值部分。
- Mistake 3: Erasing the left half of f(x) when sketching f(|x|) and not replacing it with the mirror image of the right half. | 错误三：画 f(|x|) 时只擦掉 f(x) 的左半部分，却忘记用右半部分的镜像填补。
- Mistake 4: Forgetting that y-intercept remains unchanged for both transformations. | 错误四：忘记两种变换下y轴截距都保持不变。
- Mistake 5: When solving equations involving |f(x)|, not considering the piecewise definition properly. | 错误五：解含 |f(x)| 的方程时，没有正确使用分段定义。
The best defence is rigorous graphing practice: sketch the underlying function first, then apply the transformation systematically, checking key points (intercepts, vertices, asymptotes).
最好的防御是严格的作图练习：先画出原函数，然后系统性地应用变换，检查关键点（截距、顶点、渐近线）。
11. Exam Tips and Calculator Skills | 考试技巧与计算器技能
In IB exams, graphing can be done by hand or with a GDC (Graphical Display Calculator). Here are key tips:
在IB考试中，作图可以用手绘或使用GDC（图形计算器）。以下是一些关键技巧：
- Manual sketching: Always mark the scale, intercepts, and vertices. Partial marks are awarded for correctly identifying key features even if the curve is imperfect. | 手动作图：始终标注刻度、截距和顶点。即使曲线画得不完美，正确标出关键特征也能获得步骤分。
- GDC use: Enter y=|f(x)| as abs(f(x)) on your GDC. Enter y=f(|x|) as f(abs(x)). Be aware of the syntax differences on TI-Nspire vs Casio. | 计算器使用：在GDC上输入 y=|f(x)| 使用 abs(f(x)) 语法；输入 y=f(|x|) 使用 f(abs(x)) 语法。注意TI-Nspire和Casio在语法上的差异。
- Checking symmetry: If your sketch of f(|x|) is not symmetric about the y-axis, you made an error — go back and fix it. | 检查对称性：如果你画的 f(|x|) 图像不关于y轴对称，那一定画错了——回头修正。
- Reading solutions: Use the “intersection” function on your GDC to find exact coordinates where |f(x)| meets another curve. | 读取交点：使用GDC上的”intersection”（交点）功能，精确求 |f(x)| 与其他曲线的交点坐标。
Also, in the exam, when asked to “sketch the graph,” draw it in pencil first, then go over in pen. Label at least two points with their exact coordinates.
此外，考试中遇到”画出图像”的题目，先用铅笔画草图，再用签字笔描实。至少标出两点的精确坐标。
12. Summary and Final Thoughts | 总结与要点回顾
Let us consolidate everything into a quick reference.
让我们将所有内容整合为一份快速参考。
| Transformation | 变换 | Geometric meaning | 几何含义 | Effect on negatives | 对负值的影响 | Final symmetry | 最终对称性 |
| y = |f(x)| | Reflect below-x-axis portion upward | 将x轴下方翻折向上 | Negative outputs become positive | 负输出变正 | No guaranteed symmetry | 不保证对称 |
| y = f(|x|) | Keep right half, mirror to left | 保留右半，镜像到左 | No direct effect on outputs | 不直接影响输出 | Always even (y-axis symmetry) | 恒为偶函数（y轴对称） |
Mastering these two transformations will earn you easy marks in the IB examination. The key is to always ask yourself: “Am I changing the x-values or the y-values?” If the absolute value is on the outside (|f(x)|), you are reflecting y-values. If it is on the inside (f(|x|)), you are reflecting x-values.
掌握这两种变换，能帮你在IB考试中轻松拿分。关键是要时刻问自己：”我在改变x值还是y值？”如果绝对值在外面（|f(x)|），你在翻折y值；如果绝对值在里面（f(|x|)），你在翻折x值。
Practice with a variety of functions — polynomials, exponentials, trigonometric functions, and rational functions — and you will build the visual intuition needed for exam success.
用不同类型的函数多加练习——多项式、指数函数、三角函数、有理函数——你就能建立考试成功所需的视觉直觉。
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