Question

which of the following describes the sequence of transformations shown? reflect across the x - axis and then rotate - 90 degrees around the origin reflect across the x - axis and then reflect across the y - axis. rotate - 90 degrees around the origin and then translate down. translate up and then rotate 90 degrees about the origin.

Explanation:

Step1: Analyze the reflection

Reflecting across the \(x -\)axis changes the sign of the \(y -\)coordinate of each point.

Step2: Analyze the reflection across \(y -\)axis

Reflecting across the \(y -\)axis changes the sign of the \(x -\)coordinate of each point. If we first reflect across the \(x -\)axis and then across the \(y -\)axis, it is equivalent to a rotation of \(180^{\circ}\) (but we can also check by visual inspection of the transformation of the figure). When we reflect a figure across the \(x -\)axis and then across the \(y -\)axis, the orientation and position of the figure (in terms of quadrant change) matches the transformation from \(A\) to \(A'\) in the given graph.
For the other options:

• For the rotation - translation option: A rotation of \(- 90^{\circ}\) (clockwise \(90^{\circ}\)) and then translation would change the orientation and position in a way that does not match the given transformation.
• For the translation - rotation option: Translating first and then rotating would not result in the same orientation and position as in the given figure.
• For the reflection - rotation option: Reflecting across \(x -\)axis and then rotating \(-90^{\circ}\) would change the orientation in a non - matching way.

Answer:

Reflect across the \(x -\)axis and then reflect across the \(y -\)axis.