Question

a graphic designer wants to translate rectangle defg using ( t_{-1,2}(x,y) ). the pre - image has coordinates ( d(-1,3) ), ( e(4,3) ), ( f(4,1) ), and ( g(-1,1) ). what is the image of defg?

Explanation:

Step1: Understand the translation rule

The translation \(T_{-1,2}(x,y)\) means subtract \(1\) from the \(x -\) coordinate and add \(2\) to the \(y -\) coordinate. So the formula for translation is \((x,y)\to(x - 1,y + 2)\).

Step2: Translate point \(D(-1,3)\)

For point \(D(-1,3)\), using the formula \(x'=-1-1=-2\) and \(y'=3 + 2=5\). So \(D'(-2,5)\).

Step3: Translate point \(E(4,3)\)

For point \(E(4,3)\), \(x'=4-1 = 3\) and \(y'=3 + 2=5\). So \(E'(3,5)\).

Step4: Translate point \(F(4,1)\)

For point \(F(4,1)\), \(x'=4-1=3\) and \(y'=1 + 2=3\). So \(F'(3,3)\).

Step5: Translate point \(G(-1,1)\)

For point \(G(-1,1)\), \(x'=-1-1=-2\) and \(y'=1 + 2=3\). So \(G'(-2,3)\).

Answer:

The image of rectangle \(DEFG\) has vertices \(D'(-2,5)\), \(E'(3,5)\), \(F'(3,3)\), and \(G'(-2,3)\). Based on the coordinate - based analysis of the translation, the second graph (where \(D'\) is at \((-2,5)\), \(E'\) is at \((3,5)\), \(F'\) is at \((3,3)\) and \(G'\) is at \((-2,3)\)) is the correct image of rectangle \(DEFG\) after the translation \(T_{-1,2}(x,y)\).