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 \(x\) - coordinate is decreased by \(1\) (\(x'=x - 1\)) and \(y\) - coordinate is increased by \(2\) (\(y'=y + 2\)).

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

For \(D(-1,3)\):
\(x'=-1-1=-2\), \(y'=3 + 2=5\). So \(D'(-2,5)\)

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

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

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

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

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

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

Looking at the graphs, the second graph has the translated points \(D'(-2,5)\), \(E'(3,5)\), \(F'(3,3)\), \(G'(-2,3)\) which match the translation \(T_{-1,2}(x,y)\) applied to \(D(-1,3)\), \(E(4,3)\), \(F(4,1)\), \(G(-1,1)\)

Answer:

The second graph (with \(D'\), \(E'\), \(G'\), \(F'\) as described in the second - grid)