Let's analyze the given sequence of transformations and compare it with the options provided.
The given sequence is:
1. Translate triangle RST 2 units left.
2. Reflect the translated triangle over the [tex]$y$[/tex]-axis.
Now we'll determine the net effect of applying these operations.
Starting with the translation:
- Translating 2 units left changes the [tex]$x$[/tex]-coordinates of all points in triangle RST by subtracting 2. If a vertex of the original triangle has coordinates [tex]$(x, y)$[/tex], after the translation it becomes [tex]$(x-2, y)$[/tex].
Next, we reflect the translated triangle over the [tex]$y$[/tex]-axis:
- Reflecting over the [tex]$y$[/tex]-axis changes the [tex]$x$[/tex]-coordinates by changing their sign. So if a point has coordinates [tex]$(x-2, y)$[/tex] after the translation, it will become [tex]$(-(x-2), y)$[/tex] which simplifies to [tex]$(-x + 2, y)$[/tex].
To achieve the same result using a different sequence of transformations, consider the following:
Option: A reflection over the [tex]$y$[/tex]-axis and then a translation 2 units right.
Let's break it down:
1. Reflect triangle RST over the [tex]$y$[/tex]-axis.
- This changes the [tex]$x$[/tex]-coordinates of all points by reversing their signs. So, if a vertex of the original triangle has coordinates [tex]$(x, y)$[/tex], after the reflection it becomes [tex]$(-x, y)$[/tex].
2. Translate the reflected triangle 2 units right.
- Translating 2 units right changes the [tex]$x$[/tex]-coordinates of all points by adding 2. So if a point has coordinates [tex]$(-x, y)$[/tex] after the reflection, it will become [tex]$(-x + 2, y)$[/tex].
Comparing the results:
- Both sequences of transformations result in points with coordinates [tex]$(-x + 2, y)$[/tex].
Thus, the transformation of reflecting over the [tex]$y$[/tex]-axis and then translating 2 units right will indeed result in the same image as the original sequence of translating 2 units left and then reflecting over the [tex]$y$[/tex]-axis.
Therefore, the correct transformation that will result in the same image is:
- a reflection over the [tex]$y$[/tex]-axis and then a translation 2 units right.
This confirms that the sequence provided in the option is correct.
