Chapter 12.4, Problem 9E

To determine

To draw: The polygon with the given vertices. Also, predict where in the coordinate plane its image will lie after the specified translation. Then find the coordinates of the vertices and draw the image.

Answer to Problem 9E

The polygon $ABCD$ and image $A\text{'}B\text{'}C\text{'}D\text{'}$ can be represented on a graph as:

Explanation of Solution

Given information:

The coordinates of polygon $ABCD$ :

$A(-2,-1),\text{ B}(-3,4),\text{ C}(-1,3),D(-1,-1)$

The following coordinate notation:

$(x,y)\to (x-4,y-2)$

The polygon can be drawn as follows:

To predict where in the coordinate plane the image will lie after translation, look at the coordinate notation for the translation. $4$ is subtracted from each $x$ coordinate, so the figure shifts to the left. Also, $2$ is subtracted from each $y$ coordinate, so the figure shifts down. Therefore, the image of polygon $ABCD$ will be to the left of and down from polygon $ABCD$ .

To translate $ABCD$ , subtract $4$ from the $x$ coordinate and subtract $2$ from the $y$ coordinate of each vertex

$\begin{array}{l}\text{Original Image}\\ (x,y)\to (x-4,y-2)\\ A(-2,-1)\to A\text{'}(-6,-3)\\ B(-3,4)\to B\text{'}(-7,2)\\ C\text{(}-\text{1,3) }\to C\text{'}\text{(}-\text{5,1)}\\ D\text{(}-\text{1,}-\text{1)}\to D\text{'(}-\text{5,}-\text{3)}\end{array}$

The image $A\text{'}B\text{'}C\text{'}D\text{'}$ using the above coordinates will be as follows:

Observation: The image has moved $4$ units to the left and 2 units down.