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 reflection across y - axis

When a point \((x,y)\) is reflected across the \(y\) - axis, the transformation rule is \((x,y)\to(-x,y)\).

Step2: Analyze translation

After reflection, if we translate a figure \(2\) units to the right. The rule for translation \(2\) units to the right is \((x,y)\to(x + 2,y)\).

Let's assume a point on the pre - image, say \((a,b)\). After reflection across the \(y\) - axis, it becomes \((-a,b)\). Then after translation \(2\) units to the right, it becomes \((-a + 2,b)\).

If we first translate \(2\) units to the right \((a,b)\to(a + 2,b)\) and then reflect across the \(y\) - axis \((a + 2,b)\to(-(a + 2),b)=(-a-2,b)\), which is not the same as the previous result.

But in the given case of the figure (assuming we can track the vertices), reflection across the \(y\) - axis followed by translation \(2\) units to the right is the correct transformation.

Answer:

Yes, Santana is correct.