Question

problem 5: spiral review describe a sequence of transformations that will move the pre-image (shaded) onto the image (unshaded).

Explanation:

Step1: Identify Rotation

First, we can rotate the pre - image (shaded triangle) 90 degrees counter - clockwise about the origin. A 90 - degree counter - clockwise rotation about the origin \((x,y)\to(-y,x)\) will align the triangle's orientation more closely to the image.

Step2: Identify Translation

After the rotation, we need to translate the triangle. Let's assume the coordinates of the vertices of the pre - image after rotation. Then we can find the horizontal and vertical shifts needed. Looking at the grid, after rotating 90 degrees counter - clockwise, we can translate the triangle 4 units to the right (in the positive x - direction) and 1 unit up (in the positive y - direction) to map the pre - image onto the image. (Another possible sequence: First translate and then rotate, but rotation then translation is a common approach. Also, the rotation could be 90 degrees clockwise about a different point, but 90 degrees counter - clockwise about the origin is a straightforward first step here.)

Alternatively, a more precise sequence:

1. Rotate the shaded triangle 90° counterclockwise about the origin. The rule for a 90° counterclockwise rotation about the origin is \((x,y)\to(-y,x)\).
2. Then translate the rotated triangle 4 units to the right (add 4 to the x - coordinate of each vertex) and 1 unit up (add 1 to the y - coordinate of each vertex).

Answer:

One possible sequence of transformations is: Rotate the pre - image 90 degrees counter - clockwise about the origin, then translate it 4 units to the right and 1 unit up. (Other valid sequences may exist, for example, translation followed by rotation or rotation about a different point with appropriate translation.)