Question

determining vertices after a sequence of transformations. analyze the pre - image abcd. what are the vertices of the final image if $t_{-1,-2}circ r_{y = x}$ is applied to figure abcd? a\ b\(3, 2) c\ d\

Explanation:

Step1: Recall transformation rules

The transformation $r_{y = x}$ swaps the $x$ and $y$ - coordinates of a point $(x,y)$ to get $(y,x)$. The transformation $T_{-1,-2}$ translates a point $(x,y)$ to $(x - 1,y - 2)$.

Step2: Assume pre - image coordinates

Let's assume the pre - image coordinates of the parallelogram $ABCD$ are $A(1,4)$, $B(3,4)$, $C(2,2)$, $D(0,2)$ (by observing the grid).

Step3: Apply reflection $r_{y = x}$

For point $A(1,4)$: After reflection $r_{y = x}$, $A'=(4,1)$. For point $B(3,4)$: $B'=(4,3)$. For point $C(2,2)$: $C'=(2,2)$. For point $D(0,2)$: $D'=(2,0)$.

Step4: Apply translation $T_{-1,-2}$

For $A'=(4,1)$: $A''=(4 - 1,1 - 2)=(3,-1)$. For $C'=(2,2)$: $C''=(2 - 1,2 - 2)=(1,0)$. For $D'=(2,0)$: $D''=(2 - 1,0 - 2)=(1,-2)$.

Answer:

$A''(3,-1)$
$C''(1,0)$
$D''(1,-2)$