Graph Transformations of |f(x)| and f(|x|) | |f(x)|与f(|x|)的图像变换

📚 Graph Transformations of |f(x)| and f(|x|) | |f(x)|与f(|x|)的图像变换

The absolute value function is one of the most elegant tools in the transformation toolkit. In this article we explore two closely related but frequently confused graph transformations: y = |f(x)| and y = f(|x|). Recognising the difference between applying the absolute value to the output and applying it to the input is essential for solving a wide range of exam questions quickly and correctly.

绝对值函数是图像变换工具箱中最精巧的工具之一。在本文中，我们探讨两种紧密相关却常被混淆的图像变换：y = |f(x)| 与 y = f(|x|)。理解绝对值作用于输出还是作用于输入之间的本质区别，对于快速、正确地解决大量考试题目至关重要。

1. The Absolute Value Function |x| | 绝对值函数 |x|

The absolute value of a number is its distance from zero on the number line, and it is always non-negative. For any real input, the absolute value simply removes the negative sign.

一个数的绝对值是它在数轴上到零点的距离，永远是非负数。对于任意实数输入，绝对值的作用就是把负号去掉。

|x| = x for x ≥ 0; |x| = −x for x < 0

The graph of y = |x| is V-shaped with a sharp corner at the origin. From x = 0, the right branch is the line y = x and the left branch is the line y = −x.

y = |x| 的图像呈 V 字形，在原点处有一个尖角。从 x = 0 出发，右支是直线 y = x，左支是直线 y = −x。

This simple idea extends to whole graphs: wherever a quantity is negative, the absolute value flips it to positive. The challenge in exam questions is deciding which quantity is being made positive: the y-value or the x-value.

这个简单的概念可以推广到整幅图像：只要某个量为负，绝对值就把它翻转成正量。考试题的关键在于判断究竟是哪个量被取正值：是 y 值还是 x 值。

2. y = |f(x)|: Reflect Negative Parts Upwards | y = |f(x)|：将负部分向上翻折

The transformation y = |f(x)| applies the absolute value to the output of the function. It changes the y-coordinates of every point but leaves the x-coordinates untouched.

变换 y = |f(x)| 是将绝对值作用于函数的输出值。它改变每个点的 y 坐标，但 x 坐标保持不变。

• Where f(x) ≥ 0, the graph is unchanged because |f(x)| = f(x).

  当 f(x) ≥ 0 时，图像保持不变，因为 |f(x)| = f(x)。

• Where f(x) < 0, the graph is reflected in the x-axis, so all negative y-values become positive.

  当 f(x) < 0 时，图像沿 x 轴翻折，所有负的 y 值都变为正值。

• The final graph lies entirely on or above the x-axis.

  最终图像完全位于 x 轴上或 x 轴上方。

• Points where f(x) = 0, that is the x-intercepts, remain fixed and are called invariant points.

  f(x) = 0 的点，即 x 截距，保持不动，这些点称为不变点。

In practice, sketch y = f(x) first, then fold the part below the x-axis upwards like paper. The valley becomes a hill, and any negative “dip” becomes a positive “bump”.

实际操作时，先画出 y = f(x)，然后把 x 轴下方的部分像折纸一样向上翻折。原来的谷变成山，所有向下的凹陷都变成向上的凸起。

3. y = f(|x|): Keep the Right Half and Mirror It | y = f(|x|)：保留右半并作镜像

The transformation y = f(|x|) applies the absolute value to the input, not the output. This time the x-coordinates are changed, and the y-coordinates are recalculated.

变换 y = f(|x|) 是将绝对值作用于输入值，而非输出值。这一次改变的是 x 坐标，再重新计算对应的 y 坐标。

• For x ≥ 0, |x| = x, so f(|x|) = f(x). The right half of the graph is unchanged.

  当 x ≥ 0 时，|x| = x，所以 f(|x|) = f(x)。图像的右半部分保持不变。

• For x < 0, we can write x = −t with t > 0, so f(|x|) = f(t). The left half is therefore a mirror image of the right half reflected in the y-axis.

  当 x < 0 时，可令 x = −t（t > 0），则 f(|x|) = f(t)。因此左半部分是右半部分关于 y 轴的镜像。

• The graph of y = f(|x|) is always even, meaning it is symmetric about the y-axis.

  y = f(|x|) 的图像永远是偶函数，即关于 y 轴对称。

• The y-intercept is invariant because f(|0|) = f(0).

  y 截距保持不变，因为 f(|0|) = f(0)。

A clear procedure is: draw y = f(x), erase the portion for x < 0, then reflect the remaining right half across the y-axis. The old left half plays no part in the final graph.

清晰的作图步骤是：画出 y = f(x)，擦去 x < 0 的部分，然后将剩下的右半部分沿 y 轴镜像。原来的左半部分在最终图像中不起任何作用。

4. Comparing |f(x)| and f(|x|) | 对比 |f(x)| 与 f(|x|)

Although the two transformations look similar in notation, they have very different geometric meanings. The table below summarises the key differences.

虽然两种变换在写法上很相似，但它们的几何意义截然不同。下表总结了关键区别。