Question

which composition transformation occurs in the following? (x,y)→(y + 5, -x) (1 point)
a 180 - degree rotation and a translation to the right by 5
a reflection across the x - axis and a translation to the left by -5
a reflection across the y - axis and a translation to the right by 5
a 90 - degree clockwise rotation and a translation to the right by 5

Explanation:

1. Analyze the x - coordinate transformation: The original x - coordinate changes from \(x\) to \(-x\). A reflection across the y - axis changes the sign of the x - coordinate of a point \((x,y)\) to \((-x,y)\).
2. Analyze the y - coordinate transformation: The original y - coordinate changes from \(y\) to \(y + 5\). A translation to the right (in the x - direction) for a point \((x,y)\) does not affect the y - coordinate in the way described. A translation of a point \((x,y)\) to the right by 5 units in the x - direction has the transformation \((x,y)\to(x + 5,y)\), but here the y - coordinate transformation \(y\to y+5\) is consistent with a vertical - like transformation. Considering the overall transformation \((x,y)\to(y + 5,-x)\), the change in the y - coordinate \(y\to y + 5\) can be thought of as a non - standard way of writing a translation where the new x - coordinate takes the value of the old y - coordinate plus 5. The key is the sign change of \(x\) which is due to reflection across the y - axis and the addition of 5 to the y - value which can be seen as a translation - like operation.

Answer:

C. a reflection across the y - axis and a translation to the right by 5