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}cdot r_{y = x} ) is applied to figure abcd?
a\
b\(3, 2)
c\
d\

Explanation:

Step1: Find the coordinates of pre - image

From the graph, \(A(2,4)\), \(B(4,4)\), \(C(3,2)\), \(D(1,2)\)

Step2: Apply the reflection \(r_{y = x}\)

The rule for reflection over the line \(y=x\) is \((x,y)\to(y,x)\)
For \(A(2,4)\): \(A'(4,2)\)
For \(B(4,4)\): \(B'(4,4)\)
For \(C(3,2)\): \(C'(2,3)\)
For \(D(1,2)\): \(D'(2,1)\)

Step3: Apply the translation \(T_{-1,-2}\)

The rule for translation \(T_{-1,-2}\) is \((x,y)\to(x - 1,y - 2)\)
For \(A'(4,2)\): \(A''(4-1,2 - 2)=(3,0)\)
For \(C'(2,3)\): \(C''(2-1,3 - 2)=(1,1)\)
For \(D'(2,1)\): \(D''(2-1,1 - 2)=(1,-1)\)

Answer:

\(A''(3,0)\), \(C''(1,1)\), \(D''(1,-1)\)