📚 y = |f(x)| and y = f(|x|) | y = |f(x)| 和 y = f(|x|)
ons of graphs in A-Level mathematics, two of the most commonly confused operations are y = |f(x)| and y = f(|x| ketching graphs, solving equations, and understanding function behaviour across the Edexcel syllabus.
在学习 A-Level 数学中的图形变换时，y = |f(x)| 和 y = f(|x|) 是最容易被混淆的两种操作。它们都涉及绝对值，但对图形的影响方式却截然不同。掌握这些变换对于 Edexcel 考纲中的图形草图绘制、方程求解以及理解函数行为至关重要。
1. Introduction to Absolute Value Transformations | 绝对值变换简介
The absolute value function, denoted by vertical bars, always returns a non-negative value. For a given function f(x), the notation |f(x)| means we take the absolute value of the output, while f(|x|) means we take the absolute value of the input first, then apply the function. These two distinct operations produce very different visual effects on the Cartesian plane.
绝对值函数用竖线表示，总是返回非负值。对于给定的函数 f(x)，符号 |f(x)| 表示我们对输出值取绝对值，而 f(|x|) 表示我们先对输入值取绝对值，然后再应用函数。这两种不同的操作在笛卡尔平面上会产生截然不同的视觉效果。
2. The Graph of y = |f(x)|: Reflecting Below the x-axis | y = |f(x)| 的图像：x 轴下方的反射
To sketch y = |f(x)| ts of the graph that are already on or above the x-axis remain unchanged. The transformation can be thought of as making all y-coordinates non-negative.
要绘制 y = |f(x)| 的图像，可以从 y = f(x) 的图像入手。图像中位于 x 轴下方的任何部分（即 f(x) < 0 的部分）都会被关于 x 轴反射，变为正值。已经位于 x 轴上或上方的部分保持不变。这种变换可以看作是将所有 y 坐标变为非负值。
sses the x-axis at x = 2. For x < 2, the original function gives negative y-values. In y = |x - 2|, those negative values become positive, creating a V-shaped graph with a vertex at (2,0).
例如，若 f(x) = x - 2，该直线在 x = 2 处穿过 x 轴。当 x < 2 时，原函数给出负的 y 值。在 y = |x - 2| 中，这些负值变为正值，形成一个顶点在 (2,0) 的 V 形图像。
3. The Graph of y = f(|x|): Reflecting the Right Side to the Left | y = f(|x|) 的图像：右侧反射到左侧
For y = f(|x|), the input x is replaced by its absolute value. This means the function only 'sees' d side. The resulting graph is always symmetric about the y-axis (even function), regardless of whether the original f(x) was symmetric or not.
对于 y = f(|x|)，输入 x 被替换为其绝对值。这意味着函数只能“看到”非负的 x 值。要绘制图像，首先画出 x ≥ 0 时 y = f(x) 的图像，然后将右侧部分关于 y 轴反射，得到左侧部分。无论原函数 f(x) 是否对称，最终图像总是关于 y 轴对称（偶函数）。
left side. The graph now consists of two rays: one starting at (2,0) for x ≥ 2, and another reflected ray for x ≤ -2, creating a shape similar to a V but with a flat section near the y-axis.
以 f(x) = x - 2 为例。对于 x ≥ 0，我们只画出从 x = 0 向右的直线 y = x - 2。将右侧部分关于 y 轴反射，得到左侧部分。此时图像由两条射线组成：一条从 (2,0) 出发向右延伸，另一条是反射后的射线，从 (-2,0) 向左延伸，整体形状类似 V 形，但在 y 轴附近有一段水平部分。
4. Key Differences and Similarities | 关键异同点
It is vital to distinguish between these two transformations. The table below summarises their core properties:
区分这两种变换至关重要。下表总结了它们的核心性质：
| Property | y = |f(x)| | y = f(|x|) |
|---|---|---|
| Operation on | Output (y-value) | Input (x-value) |
| Graphical change | Reflect negative y parts in x-axis | Reflect x ≥ 0 part in y-axis; discard x < 0 part of original |
| Symmetry induced | Not necessarily symmetric | Always even: graph symmetric about y-axis |
| Effect on domain | Domain unchanged from f(x) | Domain symmetric about 0; often all real numbers |
| Effect on range | Range becomes subset of [0, ∞) | Range same as right-hand side of f(x) for x ≥ 0 |
Both transformations rely on the property that absolute value makes things non-negative, but they apply this at different stages of the function evaluation. A common mistake is to treat them as interchangeable; they are not.
两种变换都依赖于绝对值使事物非负的性质，但它们在函数求值的不同阶段应用这一性质。一个常见的错误是认为它们可以互换；实际上并非如此。
5. Worked Example 1: Linear Function f(x) = x - 2 | 例题1：线性函数 f(x) = x - 2
solute value transformations in detail. The original straight line has gradient 1 and y-intercept -2.
我们详细分析 f(x) = x - 2 及其两种绝对值变换。原始直线斜率为 1，y 轴截距为 -2。
For y = |f(x)| = |x - 2| arm has equation y = -(x - 2) = 2 - x for x < 2, and the right arm is y = x - 2 for x ≥ 2.
对于 y = |f(x)| = |x - 2|：直线在 x 轴下方的部分（x < 2）被向上反射。图像呈 V 形，顶点在 (2,0)。左臂方程为 y = -(x - 2) = 2 - x（当 x < 2），右臂方程为 y = x - 2（当 x ≥ 2）。
For y = f(|x|) = |x| tute -x into the original formula, giving y = -x - 2. The graph touches the y-axis at (0, -2), which is the lowest point, and then rises on both sides. The vertex is at (0, -2), not (2,0).
对于 y = f(|x|) = |x| - 2：取 f(x) 在 x ≥ 0 时的图像，即从 x ≥ 0 开始的射线 y = x - 2，并将其关于 y 轴反射。得到两条射线。当 x ≥ 0 时，y = x - 2；当 x < 0 时，将 -x 代入原公式，得到 y = -x - 2。图像在 y 轴上的点为 (0, -2)，这也是最低点，然后向两侧升高。顶点在 (0, -2)，而不是 (2,0)。
Observe how the location of the 'corner' changes drastically depending on which absolute value transformation is used.
观察一下，根据所用绝对值变换的不同，“拐角”的位置发生了巨大的变化。
6. Worked Example 2: Quadratic Function f(x) = x² - 4 | 例题2：二次函数 f(x) = x² - 4
Take the quadratic f(x) = x² - 4, a parabola with roots at x = ±2 and a vertex at (0, -4).
取二次函数 f(x) = x² - 4，该抛物线的根在 x = ±2，顶点在 (0, -4)。
y = |f(x)| = |x² - 4| ocal maximum at (0,4) where the original minimum was. The sections outside the interval [-2,2] remain unchanged, so the graph looks like a W-shape.
y = |f(x)| = |x² - 4|：抛物线在 x 轴下方的部分（x = -2 到 x = 2 之间）被向上反射。图像现在在 (-2,0) 和 (2,0) 处接触 x 轴，并在 (0,4) 处有一个局部最大值，该点正是原函数最小值的位置。区间 [-2,2] 之外的部分保持不变，因此图像呈 W 形。
y = f(|x|) = |x|² - 4 = x² - 4, since squaring eliminates the sign. In this particular case, f(|x| bola, with no change. This highlights an important point: if f(x) is already even, then f(|x|) = f(x).
y = f(|x|) = |x|² - 4 = x² - 4，因为平方消除了符号。在这种特殊情况下，f(|x|) 与 f(x) 完全相同，因为 x² 已经是偶函数。因此图像仍然是熟悉的 U 形抛物线，没有任何变化。这突出了一个重要结论：如果 f(x) 已经是偶函数，那么 f(|x|) = f(x)。
7. Worked Example 3: Trigonometric Function f(x) = sin x | 例题3：三角函数 f(x) = sin x
nsformations. Consider f(x) = sin x. Its graph oscillates between 1 and -1 with period 2π.
三角函数的图像经常与这些变换一起考查。考虑 f(x) = sin x，其图像在 1 和 -1 之间振荡，周期为 2π。
y = |sin x| ine wave (where sin x < 0) are flipped above the x-axis. The graph consists of a series of 'bumps' to become π, but technically the function repeats its pattern every π. The range is [0,1].
y = |sin x|：正弦波的所有负值部分（sin x < 0）都被翻转到 x 轴上方。图像由一系列高度为 1 的“拱形”组成，在 π 的整数倍处有尖锐的尖点。视觉上周期似乎变为 π，但严格来说函数的模式每隔 π 重复一次。值域为 [0,1]。
y = sin|x| ulting graph is not periodic in the usual sense; it is symmetric about the y-axis but consists of positive sine waves for x > 0 and inverted sine waves for x < 0. The graph actually represents an even function: it matches sin x on the right and -sin x on the left. Thus it looks like the letter 'S' reflected in the y-axis, creating a point of inflection at the origin.
y = sin|x|：对于 x ≥ 0，它就是 sin x。由于正弦是奇函数，将右侧关于 y 轴反射后，对于 x < 0 得到 sin(-x) = -sin x。所得图像并不是通常意义上的周期函数；它关于 y 轴对称，但在 x > 0 时为正的 sine 波形，在 x < 0 时为倒置的 sine 波形。该图像实际上是一个偶函数：右侧与 sin x 相同，左侧与 -sin x 相同。因此它看起来像是字母“S”在 y 轴中的镜像，在原点处有一个拐点。
8. Worked Example 4: Exponential Function f(x) = eˣ | 例题4：指数函数 f(x) = eˣ
The exponential function eˣ is always positive, so |eˣ| = eˣ for all real x. Thus y = |eˣ| is identical to y = eˣ that if f(x) ≥ 0 for all x in its domain, the absolute value on the output has no effect.
指数函数 eˣ 始终为正，因此对于所有实数 x，有 |eˣ| = eˣ。所以 y = |eˣ| 与 y = eˣ 完全相同。这表明，如果对于其定义域内的所有 x 都有 f(x) ≥ 0，那么对输出取绝对值没有影响。
For y = f(|x|) = e^|x|, we take the curve y = eˣ he graph resembles the shape of a hanging chain or a wide U-shape known as a catenary. The range is [1, ∞). This is a classic example where f(|x|) dramatically changes the shape of the original strictly increasing function.
对于 y = f(|x|) = e^|x|，取 x ≥ 0 时的曲线 y = eˣ 并将其关于 y 轴反射。所得的图像是对称的，当 x < 0 时递减，当 x > 0 时递增。最低点在 (0,1)，图像形状类似于悬链线或一个宽阔的 U 形。值域为 [1, ∞)。这是一个经典例子，说明 f(|x|) 可以极大地改变原本严格单调递增函数的形状。
9. Solving Equations Involving |f(x)| and f(|x|) | 解涉及 |f(x)| 和 f(|x|) 的方程
When solving equations such as |f(x)| = c or f(|x|) = c, it is often useful to consider cases or use graphical reasoning.
在解诸如 |f(x)| = c 或 f(|x|) = c 的方程时，通常可以考虑分情况讨论或利用图像推理。
For |f(x)| = k (with k ≥ 0): this is equivalent to solving f(x) = k or f(x) = -k. For example, |x² - 4| nd yields x² = -1, which has no real solutions. Always check the viability of each branch.
对于 |f(x)| = k（其中 k ≥ 0）：这等价于解 f(x) = k 或 f(x) = -k。例如，|x² - 4| = 5 得到 x² - 4 = 5 或 x² - 4 = -5。前者得到 x² = 9 → x = ±3。后者得到 x² = -1，没有实数解。务必检查每个分支的可行性。
For f(|x| d -x are solutions if one is found, due to the even symmetry. You can solve by setting u = |x| ≥ 0, then solving f(u) = c, and finally taking x = ±u. For |x| - 2 = 0, set u = |x|, then u - 2 = 0 → u = 2, so x = ±2.
对于 f(|x|) = c：由于偶对称性，这意味着如果找到一个解，那么 x 和 -x 都是解。求解时，可设 u = |x| ≥ 0，然后解 f(u) = c，最后取 x = ±u。例如，|x| - 2 = 0，设 u = |x|，则 u - 2 = 0 → u = 2，所以 x = ±2。
It is also possible to solve inequalities like |f(x)| < a by considering where the graph is below the line y = a, or f(|x|) > a by looking at the symmetric graph.
也可以解不等式，比如 |f(x)| < a，只需考虑图像位于直线 y = a 下方的部分；对于 f(|x|) > a，可以观察对称图像的位置。
10. Domain and Range Considerations | 定义域和值域的考量
The domain of y = |f(x)| only modify the output. However, the range becomes the set of all non-negative values that |f(x)| can take. Often the range is [0, ∞) intersected with the possible values.
y = |f(x)| 的定义域与 f(x) 的定义域完全相同，因为我们只修改了输出。然而，值域变为 |f(x)| 可以取到的所有非负值的集合。通常值域是 [0, ∞) 与可能取值集合的交集。
For y = f(|x|), the domain must be symmetric about zero and must ensure that |x| lies within the original domain of f. If the domain of f is [0, ∞), then f(|x|) is defined for all real x because |x| ∈ [0, ∞) always. If the domain of f is [-2, 5], then f(|x|) requires |x| ∈ [-2, 5], which simply means |x| ≤ 5, so x ∈ [-5, 5]. In practice, the domain of f(|x|) is the set of all x such that |x| ∈ f f. The range is the set of y-values attained by f(u) for u ≥ 0 and u in the domain of f.
对于 y = f(|x|)，定义域必须关于零对称，并且必须保证 |x| 落在 f 的原始定义域内。如果 f 的定义域是 [0, ∞)，那么 f(|x|) 对所有实数 x 都有定义，因为 |x| ∈ [0, ∞) 始终成立。如果 f 的定义域是 [-2, 5]，那么 f(|x|) 要求 |x| ∈ [-2, 5]，这仅仅意味着 |x| ≤ 5，所以 x ∈ [-5, 5]。实际上，f(|x|) 的定义域是所有满足 |x| ∈ f 的定义域的 x 的集合。其值域是当 u ≥ 0 且 u 在 f 的定义域内时 f(u) 所能取得的 y 值集合。
11. Exam Tips and Common Mistakes | 考试技巧与常见错误
ching these graphs in an exam, always start with a light sketch of y = f(x) first. For y = |f(x)|, erase or reflect the parts below the x-axis; for y = f(|x| simply put absolute value symbols around x in the equation and leave the graph unchanged.
在考试中绘制这些图像时，务必先轻轻画出 y = f(x) 的草图。对于 y = |f(x)|，擦除或反射 x 轴下方的部分；对于 y = f(|x|)，擦除原始图像左侧 (x < 0) 的部分，然后反射右侧部分。绝不要简单地在方程中的 x 周围加上绝对值符号而保持图像不变。
A common error is to confuse y = |f(x)| with y = f(|x| (horizontal reflection and duplication of the right side). Another mistake is forgetting that f(|x|) always results in an even function; symmetry about the y-axis must be present.
一个常见的错误是将 y = |f(x)| 与 y = f(|x|) 混淆。记住：整个函数外部的竖线影响 y 值（垂直反射），而仅加在 x 上的竖线影响 x 值（水平反射并复制右侧）。另一个错误是忘记 f(|x|) 总是得到偶函数；图像必须关于 y 轴对称。
when solving equations, do not forget to consider both positive and negative branches for |f(x)|, and use the symmetry for f(|x|) to write solutions as ±k once the positive root is found.
此外，在解方程时，不要忘记对 |f(x)| 同时考虑正负分支，而对于 f(|x|)，一旦找到正根，可以利用对称性将解写为 ±k。
12. Practice Questions | 练习题
Test your understanding with these Edexcel-style questions. Solutions should include clear sketches and algebraic verification.
用这些 Edexcel 风格的题目来测试你的理解。解答应包含清晰的草图和代数验证。
Given f(x) = 2x + 3, sketch y = |f(x)| and y = f(|x|). State the coordinates of any vertices or intercepts.
已知 f(x) = 2x + 3，绘制 y = |f(x)| 和 y = f(|x|) 的草图。写出所有顶点或截距的坐标。
Sketch y = |x² - 9| and y = |x|² - 9. Explain why one of these is identical to y = x² - 9.
绘制 y = |x² - 9| 和 y = |x|² - 9 的草图。解释为什么其中一者与 y = x² - 9 完全相同。
Solve the equation |x³ - 4x| = 0. How many distinct real roots does it have? Interpret graphically.
解方程 |x³ - 4x| = 0。它有多少个不同的实数根？从图形上进行解释。
For f(x) = 1 - eˣ, find the range of y = |f(x)| and state whether the function f(|x|) is even, odd, or neither.
对于 f(x) = 1 - eˣ，求 y = |f(x)| 的值域，并说明函数 f(|x|) 是偶函数、奇函数还是两者皆非。
The domain of g(x) is [-4, 2). Write down the domain of g(|x|).
已知 g(x) 的定义域为 [-4, 2)，写出 g(|x|) 的定义域。
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