Chapter 12.6, Problem 22E

To determine

To graph:the polygon after the ${180}^{\circ }$ rotation and reflect it in the y - image and find their coordinate.

Explanation of Solution

Given : from the figureprovided in the question, it is observed that,

There is an polygon ABCD in quadrant II which coordinates are

  $A\left(-4,4\right),B\left(-2,4\right),C\left(-1,1\right),D\left(-3,1\right)$

Graph:

interpretation : observed from the graph

to rotating the image ${180}^{\circ }$ , there will no switching of coordinates and multiply the both coordinates of image by -1.

  $\begin{array}{l}\left(x,y\right)\to \left(-x,-y\right)\\ A\left(-4,4\right)\to A\text{'}\left(4,-4\right)\\ B\left(-2,4\right)\to B\text{'}\left(2,-4\right)\\ C\left(-1,1\right)\to C\text{'}\left(1,-1\right)\\ D\left(-3,1\right)\to D\text{'}\left(3,-1\right)\end{array}$

After that for reflect it in the y -axis the final image x -axis coordinates will multiply by -1.

  $\begin{array}{l}\left(x,y\right)\to \left(-x,y\right)\\ A\text{'}\left(4,-4\right)\to A\text{'}\text{'}\left(-4,-4\right)\\ B\text{'}\left(2,-4\right)\to B\text{'}\text{'}\left(-2,-4\right)\\ C\text{'}\left(1,-1\right)\to C\text{'}\text{'}\left(-1,-1\right)\\ D\text{'}\left(3,-1\right)\to D\text{'}\text{'}\left(-3,-1\right)\end{array}$