Question

use the diagram above to match the image/pre-image with the transformation. 14 pre-image: shape i image: shape ii a. rotated 180° around the point (0, 0) 15. pre-image: shape ii image: shape iii b. reflected over the line y = -1x + 0 16. pre-image: shape iv image: shape ii c. rotated 270° counter-clockwise around the point (0, 0) 17. pre-image: shape i image: shape iv d. reflected over the x-axis 28. pre-image: shape i image: shape iii e. rotated 90° counter-clockwise around the point (0, 0)

Explanation:

To solve these transformation - matching problems, we analyze each transformation and compare the pre - image and image shapes:

Problem 14

Step 1: Analyze the rotation of 180° around (0,0)

When a point \((x,y)\) is rotated \(180^{\circ}\) around the origin \((0,0)\), the transformation rule is \((x,y)\to(-x,-y)\). We observe the positions of Shape I (pre - image) and Shape II (image). The relative positions of the vertices of Shape I and Shape II are consistent with a \(180^{\circ}\) rotation around the origin. So, the transformation from Shape I to Shape II is a \(180^{\circ}\) rotation around \((0,0)\).

Answer:

a. Rotated \(180^{\circ}\) around the point \((0,0)\) (Pre - image: Shape I, Image: Shape II)

Problem 15

Step 1: Analyze reflection over \(y=-x + 0\) (i.e., \(y=-x\))

The reflection rule over the line \(y = -x\) is \((x,y)\to(-y,-x)\). By comparing the pre - image (Shape II) and the image (Shape III), we find that the positions of the vertices match the reflection over the line \(y=-x\).