What composition transformation occurs in the following? (x,y)→(y+5,−x)

Question

(1 point)
Responses

a reflection across the y-axis and a translation down by 5
a reflection across the y -axis and a translation down by 5

a 90-degree counterclockwise rotation and a translation down by 5
a 90-degree counterclockwise rotation and a translation down by 5

a reflection across the x-axis and a translation 5 right
a reflection across the x -axis and a translation 5 right

a 90-degree clockwise rotation and a translation to the right by 5

Answer 1

To determine the composition transformation for the mapping \((x,y) \to (y + 5, -x)\), we can break down the transformation into its components.

1. Identifying Components:

   • The first part of the transformation \((x,y) \to (y+5, -x)\) suggests two operations:
     • The \(y\) coordinate is transformed to \(y + 5\), indicating a vertical translation upwards by 5 units.
     • The \(x\) coordinate is transformed to \(-x\), which indicates a reflection across the y-axis.

2. Order of Operations:

   • First, the point \((x,y)\) is reflected across the y-axis, changing it to \((-x, y)\).
   • Second, the \(y\) coordinate of this new point is changed to \(y + 5\), leading to the final point \((-x, y + 5)\).

Thus, the overall transformation consists of a reflection across the y-axis followed by a translation upwards by 5 units.

So, among the given options, the correct transformation is:

a reflection across the y-axis and a translation up by 5.

Note: It appears there's an error in the responses you've provided since "up" is not listed but should be the correct direction for the shift. If there's a choice that closely represents what I've just explained but uses "down" or a different direction, it's important to consider the accurate direction being a shift upwards rather than downwards.