Question

which sequence of transformations constructs the coordinate plane using the first quadrant of the coordinate plane? select all that apply. rotate the first quadrant 180° clockwise three times. translate the first quadrant 1 unit up, 1 unit left, and 1 unit down. translate the first quadrant 1 unit up, 1 unit right and 1 unit down. reflect the first quadrant across the x - axis, across the y - axis, and across the x - axis. reflect the first quadrant across the y - axis and then rotate 90° counter - clockwise. rotate the first quadrant 90° clockwise two times and then reflect across the x - axis.

Explanation:

Step1: Recall transformation rules

Rotations and reflections change the orientation of a shape in the coordinate - plane, while translations just move it. Rotating the first quadrant multiple times and reflecting can cover different parts of the plane. Translating just moves the region within the plane without changing its fundamental position relative to the axes in a non - linear way for the purpose of constructing the whole plane.

Step2: Analyze each option

• Rotating the first quadrant 90° clockwise two times: A 90° clockwise rotation of a point (x,y) in the first quadrant gives (y, - x). Two 90° clockwise rotations will move points to different quadrants and can be used to construct the plane.
• Reflecting the first quadrant across the y - axis and then rotating 90° counter - clockwise: Reflection across the y - axis changes (x,y) to (-x,y), and then a 90° counter - clockwise rotation changes (-x,y) to (-y, - x), covering other quadrants.
• Reflecting the first quadrant across the x - axis, across the y - axis, and across the x - axis again: Reflection across the x - axis changes (x,y) to (x, - y), then across the y - axis to (-x, - y) and then across the x - axis again to (-x,y), covering different quadrants.
• Rotating the first quadrant 180° clockwise three times: A 180° clockwise rotation of (x,y) gives (-x, - y). Three 180° clockwise rotations will cycle through different orientations and cover different parts of the plane.
• Translating the first quadrant 1 unit up, 1 unit right and 1 unit down: This just moves the region within the first - quadrant or adjacent to it in a linear non - covering way for the whole plane.
• Translating the first quadrant 1 unit up, 1 unit left, and 1 unit down: This also just moves the region within the first - quadrant or adjacent to it in a linear non - covering way for the whole plane.

Answer:

Rotate the first quadrant 90° clockwise two times, Reflect the first quadrant across the y - axis and then rotate 90° counter - clockwise, Reflect the first quadrant across the x - axis, across the y - axis, and across the x - axis again, Rotate the first quadrant 180° clockwise three times.