Given a point $(a,b)$, what are the coordinates of its image after a reflection across the line $y=-x$?

• A. $(-a,-b)$
• B. $(-b,-a)$
• C. $(b,a)$
• D. $(a,-b)$

Answer: (B)

Verified step by step guidance

1. Understand that reflecting a point across the line \(y = -x\) involves swapping and negating the coordinates of the point in a specific way.
2. Recall the general rule for reflection across the line \(y = -x\): a point \((x, y)\) is transformed to \((-y, -x)\) after reflection.
3. Given the original point \((a, b)\), apply the reflection rule by replacing \(x\) with \(a\) and \(y\) with \(b\), resulting in the image point \((-b, -a)\).
4. Verify the transformation by considering the geometric meaning: the line \(y = -x\) acts like a mirror that swaps the coordinates and changes their signs accordingly.
5. Conclude that the coordinates of the image after reflection across \(y = -x\) are \((-b, -a)\).