Question

in this graph, which transformation can produce quadrilateral ( abcd ) from quadrilateral ( abcd )?

( \bigcirc ) a translation 4 units down and 3 units to the left

( \bigcirc ) a reflection across the line ( y = x )

( \bigcirc ) a reflection across the line ( y = -x )

( \bigcirc ) a ( 180^{circ} ) counterclockwise rotation about the origin

Explanation:

Step1: Check translation

Assume translation rule \((x,y)\to(x - 3,y - 4)\).
For point \(A(0,1)\), after translation: \((0-3,1 - 4)=(-3,-3)\), but \(A'(0,-1)\) (from graph), so translation is wrong.

Step2: Check reflection across \(y = x\)

Reflection rule \((x,y)\to(y,x)\).
For \(A(0,1)\), \((1,0)
eq A'(0,-1)\), so reflection across \(y = x\) is wrong.

Step3: Check reflection across \(y=-x\)

Reflection rule \((x,y)\to(-y,-x)\).
For \(A(0,1)\), \((-1,0)
eq A'(0,-1)\), so reflection across \(y=-x\) is wrong.

Step4: Check \(180^{\circ}\) counter - clockwise rotation

Rotation rule \((x,y)\to(-x,-y)\).
For \(A(0,1)\): \((0,-1)=A'\), for \(B(2,1)\): \((-2,-1)=B'\), for \(C(2,2)\): \((-2,-2)=C'\), for \(D(1,3)\): \((-1,-3)=D'\).

Answer:

a \(180^{\circ}\) counterclockwise rotation about the origin