Question

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

Explanation:

1. First, assume the coordinates of the pre - image vertices:

• Let's assume the coordinates of the vertices of parallelogram \(ABCD\) from the graph are \(A(2,4)\), \(B(4,4)\), \(C(3,2)\), \(D(1,2)\) (by observing the position of the points on the coordinate - grid).

1. Recall the transformation rules:

• Reflection rule: The rule for reflecting a point \((x,y)\) over the line \(y = x\) is \(r_{y = x}(x,y)=(y,x)\).
• Translation rule: The rule for translating a point \((x,y)\) by \(T_{a,b}\) is \(T_{a,b}(x,y)=(x + a,y + b)\), where \(a=-1\) and \(b = - 2\) in our case.

1. Apply the reflection \(r_{y = x}\) first:

• For point \(A(2,4)\):
• After reflection over \(y = x\), \(r_{y = x}(2,4)=(4,2)\).
• Then apply the translation \(T_{-1,-2}\): \(T_{-1,-2}(4,2)=(4-1,2 - 2)=(3,0)\).
• For point \(C(3,2)\):
• After reflection over \(y = x\), \(r_{y = x}(3,2)=(2,3)\).
• Then apply the translation \(T_{-1,-2}\): \(T_{-1,-2}(2,3)=(2-1,3 - 2)=(1,1)\).
• For point \(D(1,2)\):
• After reflection over \(y = x\), \(r_{y = x}(1,2)=(2,1)\).
• Then apply the translation \(T_{-1,-2}\): \(T_{-1,-2}(2,1)=(2-1,1 - 2)=(1,-1)\).

Answer:

A"(3,0)
C"(1,1)
D"(1, - 1)