Question

santana says the image is formed from the pre - image by a reflection across the y - axis followed by a translation 2 units to the right, in that order. is santanas thinking correct? yes, santana is correct. no, the translation should be 2 units to the left. no, the reflection should be across the x - axis and the translation should be 2 units to the left.

Explanation:

Step1: Analyze the reflection

A reflection across the \(y -\)axis changes the \(x -\)coordinate of a point \((x,y)\) to \((-x,y)\).

Step2: Analyze the translation

A translation of \(2\) units to the right changes the \(x -\)coordinate of a point \((x,y)\) to \((x + 2,y)\). If we first reflect a point \((x,y)\) across the \(y -\)axis to get \((-x,y)\) and then translate \(2\) units to the right, we get \((-x+2,y)\).
Let's assume a general point on the pre - image. Suppose a vertex of the pre - image is \((x,y)\). After reflection across the \(y -\)axis, it is \((-x,y)\), and after translation \(2\) units to the right, it is \((-x + 2,y)\).
If we reverse the order: first translate \(2\) units to the right \((x+2,y)\) and then reflect across the \(y -\)axis \((-(x + 2),y)=(-x-2,y)\), which is a different transformation. But in the given case of the pre - image and image, when we consider the reflection across the \(y -\)axis (which is a symmetry about the \(y -\)axis) followed by a translation of \(2\) units to the right (horizontal shift in the positive \(x -\)direction), we can check the position of corresponding vertices.
Counting the horizontal distance between the pre - image and image (after considering the reflection symmetry), we can see that the translation is \(2\) units to the right.

Answer:

Yes, Santana is correct.