Graph Transformations: y = |f(x)| and y = f(|x|) | 绝对值图像变换：y = |f(x)| 与 y = f(|x|)

📚 Graph Transformations: y = |f(x)| and y = f(|x|) | 绝对值图像变换：y = |f(x)| 与 y = f(|x|)

In Edexcel A-Level Mathematics, the absolute value function creates two common but easily confused transformations: y = |f(x)| and y = f(|x|). Although both use the modulus symbol, they affect a graph in very different ways. This article explains the rules, compares the two transformations, and shows how to sketch graphs and solve related equations.

在 Edexcel A-Level 数学中，绝对值函数产生两种常见但容易混淆的图像变换：y = |f(x)| 和 y = f(|x|)。虽然两者都使用绝对值符号，但它们对图像的影响完全不同。本文将解释规则、对比两种变换，并展示如何画图以及求解相关方程。

1. The Two Transformations at a Glance | 两种变换速览

The expression y = |f(x)| applies the absolute value to the output of the function. For any x in the domain of f, the y-coordinate is forced to be non-negative. The expression y = f(|x|) applies the absolute value to the input before f acts on it, so negative inputs are replaced by positive inputs before the function is evaluated.

y = |f(x)| 对函数的输出取绝对值。对于定义域内的任意 x，y 坐标都被强制为非负。y = f(|x|) 则在函数作用前对输入取绝对值，因此负输入在代入函数前被替换为正输入。

A quick memory aid is: outside the function changes y-values, inside the function changes x-values. Therefore y = |f(x)| is a vertical effect, while y = f(|x|) is a horizontal effect. This distinction is essential for sketching.

一个快速记忆法是：绝对值在函数外部改变 y 值，在函数内部改变 x 值。因此 y = |f(x)| 是垂直方向的影响，而 y = f(|x|) 是水平方向的影响。这一区别对绘图至关重要。

2. Key Graph Rules | 关键绘图规则

For y = |f(x)|, keep all parts of y = f(x) that lie on or above the x-axis unchanged. Any part below the x-axis is reflected in the x-axis. The resulting graph never goes below the x-axis.

对于 y = |f(x)|，保留 y = f(x) 位于 x 轴及其上方的部分不变。x 轴下方的任何部分都关于 x 轴反射。所得图像不会出现在 x 轴下方。

For y = f(|x|), start with y = f(x) for x ≥ 0, then reflect this right-hand side in the y-axis to produce the left-hand side. The graph is even, so it is symmetric about the y-axis.

对于 y = f(|x|)，先取 y = f(x) 在 x ≥ 0 的部分，然后将右侧图像关于 y 轴反射，得到左侧图像。图像是偶函数，因此关于 y 轴对称。

For y = |f(x)|: y = f(x) if f(x) ≥ 0; y = -f(x) if f(x) < 0

对于 y = |f(x)|：若 f(x) ≥ 0，则 y = f(x)；若 f(x) < 0，则 y = -f(x)

The table summarises the core difference: one transformation changes vertical signs, while the other creates horizontal symmetry.

上表总结了核心区别：一种变换改变竖直方向的正负号，另一种则产生水平方向的对称性。

3. Worked Example: f(x) = x – 2 | 例题：f(x) = x – 2

Let f(x) = x

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