The Graphs of y=|f(x)| and y=f(|x|) | 函数 y=|f(x)| 与 y=f(|x|) 的图像

📚 The Graphs of y=|f(x)| and y=f(|x|) | 函数 y=|f(x)| 与 y=f(|x|) 的图像

Understanding how absolute value affects a function’s graph is a core skill in IB Mathematics. The operations y=|f(x)| and y=f(|x|) produce entirely different visual transformations, yet both rely on the idea of making quantities non‑negative. This article unpacks each transformation, works through multiple examples, and provides a clear strategy for sketching these graphs confidently.

理解绝对值如何影响函数图像是IB数学的一项核心技能。运算 y=|f(x)| 与 y=f(|x|) 会产生完全不同的视觉变换，但两者都基于让量变为非负的思想。本文将逐一拆解每种变换，通过多个实例讲解，并提供清晰的方法帮助大家自信地绘制这些图像。

1. Absolute Value Refresher | 绝对值知识回顾

The absolute value of a real number a, written |a|, is defined as a if a ≥ 0, and −a if a < 0. On a graph, this forces every output to be non‑negative, which visually means any part of a curve lying below the x‑axis gets reflected into the upper half‑plane.

实数 a 的绝对值记作 |a|，定义为：当 a ≥ 0 时等于 a，当 a < 0 时等于 −a。在图像上，这会强制每个输出值非负，从视觉上看就是任何位于 x 轴下方的曲线部分都会被翻折到上半平面。

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2. The Graph of y=|f(x)|: Reflecting Negatives Upwards | y=|f(x)| 的图像：将负值部分向上翻折

To obtain y=|f(x)|, we apply the absolute value to the output of f. For any x where f(x) ≥ 0, the point stays exactly where it was. For x where f(x) < 0, the point (x, f(x)) is replaced by (x, −f(x)), which is its mirror image across the x‑axis. The graph never dips below the x‑axis.

要得到 y=|f(x)|，我们将绝对值作用于 f 的输出。对于任何 f(x) ≥ 0 的 x 值，该点保持不变；对于 f(x) < 0 的 x 值，点 (x, f(x)) 被替换为 (x, −f(x))，即它关于 x 轴的镜像点。图像永远不会落到 x 轴下方。

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3. Example 1: y=|mx+c| | 示例1：y=|mx+c|

Start with a line such as f(x)=2x−4. The graph crosses the x‑axis at x=2. For x < 2, f(x) is negative, so that portion is reflected upwards. The result is a V‑shaped graph with a vertex at (2,0). The gradient changes from −2 on the left branch to +2 on the right branch.

以直线 f(x)=2x−4 为例。图像在 x=2 处穿过 x 轴。当 x < 2 时 f(x) 为负，因此该部分向上翻折，最终得到一个顶点在 (2,0) 的 V 形图像。左侧分支的斜率为 −2，右侧分支的斜率为 +2。

x	1	2	3
f(x)=2x−4	−2	0	2
|f(x)|	2	0	2

The equation of the reflected portion is y=−(2x−4)=−2x+4 for x < 2.

翻折部分的方程为 y=−(2x−4)=−2x+4，定义域 x < 2。

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4. Example 2: y=|x²−4| | 示例2：y=|x²−4|

Take the quadratic f(x)=x²−4. Its graph is a standard upward parabola crossing the x‑axis at x=−2 and x=2. Between these roots, f(x) is negative and reaches a minimum of −4 at x=0. The absolute value flips this negative dip upwards, turning the single parabola into a W‑shaped curve. The vertex (0,−4) becomes (0,4).

取二次函数 f(x)=x²−4。其图像是标准的上开口抛物线，与 x 轴交于 x=−2 和 x=2。在这两个根之间，f(x) 为负，并在 x=0 处达到最小值 −4。绝对值将这段向下的凹陷向上翻折，使单支抛物线变为 W 形曲线。顶点 (0,−4) 变成 (0,4)。

Mathematically, for −2 < x < 2 we have y=|x²−4|=4−x², which is an inverted parabola segment, smoothly joining the original parts outside the interval.

数学上，当 −2 < x < 2 时，有 y=|x²−4|=4−x²，这是一段倒置的抛物线，与区间外侧的原有部分光滑连接。

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5. Example 3: y=|sin x| | 示例3：y=|sin x|

The sine function oscillates between −1 and 1. When we apply y=|sin x|, every negative half‑wave is reflected above the x‑axis. The result is a series of arches that touch the x‑axis every π radians instead of every 2π. The period effectively halves because the negative lobes become positive copies of the adjacent positive lobes.

正弦函数在 −1 和 1 之间振荡。当我们应用 y=|sin x| 时，每个负半波都被翻折到 x 轴上方。结果是一系列拱形，每隔 π 弧度触碰 x 轴一次，而非每隔 2π。周期实际上减半，因为负波瓣变成了相邻正波瓣的正向副本。

This transformation is particularly useful in modelling rectified alternating current in physics.

这一变换在物理中建模整流交流电时特别有用。

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6. The Graph of y=f(|x|): Creating Even Symmetry | y=f(|x|) 的图像：产生偶对称

Here the absolute value is applied to the independent variable x before it enters the function. For x ≥ 0, |x| = x, so y = f(x) just as before. For x < 0, |x| = −x, so y = f(−x). This means the shape that existed on the right side of the y‑axis (x ≥ 0) is mirrored exactly onto the left side. The resulting graph is always symmetric about the y‑axis; in other words, y=f(|x|) is an even function.

这里绝对值作用于自变量 x，之后再输入函数。当 x ≥ 0 时，|x| = x，所以 y = f(x)，与原先相同；当 x < 0 时，|x| = −x，所以 y = f(−x)。这意味着 y 轴右侧 (x ≥ 0) 的形状被完全镜像复制到了左侧。最终图像总是关于 y 轴对称；换言之，y=f(|x|) 是一个偶函数。

• Key observation: any part of the original f(x) for x < 0 is discarded and replaced by a reflection of the right‑hand side.
• 关键观察：原函数 f(x) 在 x < 0 的任何部分都被丢弃，并被右侧图像的镜像取代。

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7. Example 4: y=(|x|−2)² | 示例4：y=(|x|−2)²

Consider f(x)=(x−2)². For x ≥ 0, f(|x|) gives (|x|−2)², which traces the original parabola on the right. On the left, we get (−x−2)² = (x+2)². The graph therefore consists of two parabolic arms: one centred at x=2 (for x ≥ 0) and another centred at x=−2 (for x < 0), symmetric about the y‑axis. The minimum points are at (±2, 0) and there is a local maximum at (0, 4), giving a distinct double‑well shape.

考虑 f(x)=(x−2)²。当 x ≥ 0 时，f(|x|) 给出 (|x|−2)²，描绘出右侧原有的抛物线。左侧则得到 (−x−2)² = (x+2)²。因此图像由两支抛物线组成：一支以 x=2 为中心（x ≥ 0），另一支以 x=−2 为中心（x < 0），关于 y 轴对称。最小值点在 (±2, 0)，在 (0, 4) 处有一个局部最大值，形成独特的双阱形状。

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8. Example 5: y=e^{|x|} | 示例5：y=e^{|x|}

Let f(x)=eˣ. Then f(|x|)=e^{|x|}. For x ≥ 0, this is the standard exponential growth curve. For x < 0, we have e^{-x}, which is a decreasing exponential that also grows as x becomes more negative. The graph is symmetric, with a global minimum of 1 at x=0, and it rises sharply on both sides.

令 f(x)=eˣ，则 f(|x|)=e^{|x|}。当 x ≥ 0 时，就是标准的指数增长曲线。当 x < 0 时，我们得到 e^{-x}，这是一个随着 x 变得更负而增长的递减指数曲线。图像对称，在 x=0 处取得全局最小值 1，并向两侧急剧上升。

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9. Example 6: y=ln|x| | 示例6：y=ln|x|

For f(x)=ln x (x > 0), y=ln|x| is defined for all x ≠ 0. The right‑hand branch (x > 0) is the familiar natural log curve. The left‑hand branch (x < 0) is the reflection of the right branch across the y‑axis, because ln(−x) makes sense there. Both branches have a vertical asymptote at x=0 and extend outward symmetrically.

对于 f(x)=ln x (x > 0)，y=ln|x| 对所有 x ≠ 0 有定义。右侧分支 (x > 0) 是熟悉的自然对数曲线。左侧分支 (x < 0) 是右侧分支关于 y 轴的镜像，因为 ln(−x) 在该侧有意义。两侧分支都以 x=0 为垂直渐近线，并对称延伸。

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10. Combining Transformations with Absolute Values | 结合变换与绝对值

More complex cases involve combining a horizontal translation with an absolute value, such as y=|f(x)+k| or y=f(|x−h|). Order matters. For y=|f(x)+2|, first shift f(x) up by 2, then reflect any negative parts. For y=f(|x−1|), first replace x by |x| to obtain f(|x|), then shift the entire graph 1 unit to the right. A useful strategy is to treat the absolute value as the innermost transformation.

更复杂的情况包括将水平平移与绝对值结合，例如 y=|f(x)+k| 或 y=f(|x−h|)。运算顺序很重要。对于 y=|f(x)+2|，先将 f(x) 上移 2 个单位，再将任何负值部分翻折。对于 y=f(|x−1|)，先用 |x| 替换 x 得到 f(|x|)，再将整个图像向右平移 1 个单位。一个有用的策略是将绝对值视为最内层变换。

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11. Step‑by‑Step Sketching Technique | 逐步绘制技巧

For y=|f(x)|: (1) Lightly sketch y=f(x). (2) Identify the x‑intervals where f(x) is negative. (3) Reflect those sections across the x‑axis. (4) Erase the original negative portions. The positive parts remain untouched.

对于 y=|f(x)|：(1) 轻轻勾勒出 y=f(x) 的图像。(2) 标识出 f(x) 为负的 x 区间。(3) 将这些部分关于 x 轴翻折。(4) 擦除原先的负值部分。正值部分保持原样。

For y=f(|x|): (1) Draw y=f(x) for x ≥ 0 only. (2) Reflect this right‑hand part perfectly across the y‑axis to cover x < 0. (3) Discard any original f(x) curve that existed for x < 0. The final graph must be an even function.

对于 y=f(|x|)：(1) 只画出 x ≥ 0 时的 y=f(x)。(2) 将右侧部分完美地关于 y 轴反射，覆盖 x < 0 的区域。(3) 丢弃原有 x < 0 时的任何 f(x) 曲线。最终图像必须是一个偶函数。

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12. Common Mistakes and Key Differences | 常见错误与关键区别

One frequent error is treating |f(x)| and f(|x|) as interchangeable—they are not. y=|x²−4| gives a W‑shape while y=(|x|)²−4 collapses back to x²−4 because the square already makes it even. Another pitfall is forgetting that f(|x|) automatically imposes symmetry, so an originally odd function like f(x)=x³ becomes y=|x|³ which is symmetric and non‑negative on both sides but still cubic in shape.

一个常见错误是认为 |f(x)| 和 f(|x|) 可以互换——它们截然不同。y=|x²−4| 呈现 W 形，而 y=(|x|)²−4 实际上退化回 x²−4，因为平方运算已使其为偶函数。另一个陷阱是忘记 f(|x|) 自动施加对称性，因此原本的奇函数如 f(x)=x³ 变为 y=|x|³，该图像两侧对称且非负，但仍保持三次曲线的形状。

Also, students sometimes incorrectly reflect parts of f(|x|) across the x‑axis—remember, f(|x|) never forces positivity on y; it only modifies the domain symmetry.

此外，学生有时会错误地将 f(|x|) 的部分关于 x 轴翻折——请记住，f(|x|) 从不强制 y 为正；它只改变自变量的定义域对称性。

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