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?

Explanation:

Step1: Find original coordinates

From the graph, assume \(A(1,4)\), \(B(3,4)\), \(C(2,2)\), \(D(0,2)\).

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

The rule for reflecting a point \((x,y)\) over the line \(y=x\) is \((x,y)\to(y,x)\). So \(A(1,4)\to A'(4,1)\), \(B(3,4)\to B'(4,3)\), \(C(2,2)\to C'(2,2)\), \(D(0,2)\to D'(2,0)\).

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

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

Answer:

\(D''(1,-2)\)