Chapter 12.6, Problem 24E

To determine

To graph:the polygon after the ${90}^{\circ }$ counterclockwise rotation and translate the image according $\left(x,y\right)\to \left(x+3,y+4\right)$ find their coordinate.

Explanation of Solution

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

There is an polygon ABCD which is 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 : to rotating the image ${90}^{\circ }$ counterclockwise, switch the coordinates and multiply the x -coordinate by -1.

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

After that translate it according $\left(x,y\right)\to \left(x+3,y+4\right)$ ,

  $\begin{array}{l}\left(x,y\right)\to \left(x+3,y+4\right)\\ A\text{'}\left(-4,-4\right)\to A\text{'}\text{'}\left(-1,0\right)\\ B\text{'}\left(-4,-2\right)\to B\text{'}\text{'}\left(-1,2\right)\\ C\text{'}\left(-1,-1\right)\to C\text{'}\text{'}\left(2,3\right)\\ D\text{'}\left(-1,-3\right)\to D\text{'}\text{'}\left(2,0\right)\end{array}$