Question

如果先做一个变换再做另一个变换，与先做第二个再做第一个得到同样结果，则称这两个变换可交换。考虑坐标平面的以下四个变换：

• 向右平移 $2$ 个单位；

• 绕原点逆时针旋转 $90^\circ$ ；

• 关于 $x$ 轴反射；

• 以原点为中心、比例因子为 $2$ 的伸缩。

从这四个变换中选取两个不同变换，共有 $6$ 对，其中有多少对可交换？

Two transformations are said to commute if applying the first followed by the second gives the same result as applying the second followed by the first. Consider these four transformations of the coordinate plane:

• a translation $2$ units to the right,

• a $90^\circ$ rotation counterclockwise about the origin,

• a reflection across the $x$ -axis, and

• a dilation centered at the origin with scale factor $2.$

Of the $6$ pairs of distinct transformations from this list, how many commute?

$1$

$2$

$3$

$4$

$5$

答案：C

知识点：变换函数分类讨论

难度评级：1500

解答：

伸缩只是关于原点放大，所以它与绕原点旋转可交换，也与关于横轴反射可交换。这给出 $2$ 对。向右平移与关于 $x$ 轴反射也可交换，因为两种顺序都把 $(x, y) \to (x + 2, -y)$ 。另外三对不交换：平移与旋转、平移与伸缩不交换，旋转与反射不交换。共有 $3$ 对，正确答案是 C。

The dilation just scales about the origin, so it commutes with both the rotation and the reflection. That's $2$ pairs. The translation commutes with the reflection across the $x$ -axis too, since either order sends $(x, y) \to (x + 2, -y).$ The other three pairs fail: the translation clashes with the rotation and with the dilation, and the rotation clashes with the reflection. So $3$ pairs commute. Thus, C is the correct answer.

2024 AMC 10A 第 13 题

答案：C

难度评级：1500

解答：

其他年份的第 13 题