Question

which sequence of transformations maps fgh onto fgh?
a rotation 180° around the origin followed by a translation right 3 units and up 3 units
a rotation 90° clockwise around the origin followed by a reflection across the y - axis
a rotation 90° counterclockwise around the origin followed by a reflection across the y - axis

Explanation:

Step1: Analyze rotation of 90° clockwise

For a point \((x,y)\) rotated \(90^{\circ}\) clockwise around the origin, the transformation rule is \((x,y)\to(y, -x)\).
For \(F(- 4,6)\), after \(90^{\circ}\) clockwise rotation, it becomes \((6,4)\). For \(G(0,3)\), after \(90^{\circ}\) clockwise rotation, it becomes \((3,0)\). For \(H(-4,3)\), after \(90^{\circ}\) clockwise rotation, it becomes \((3,4)\).

Step2: Analyze reflection across the y - axis

The rule for reflection across the \(y\) - axis is \((x,y)\to(-x,y)\).
For the point \((6,4)\) (after rotation), after reflection across the \(y\) - axis, it becomes \((-6,4)\) (incorrect).

Step3: Analyze rotation of 90° counter - clockwise

For a point \((x,y)\) rotated \(90^{\circ}\) counter - clockwise around the origin, the transformation rule is \((x,y)\to(-y,x)\).
For \(F(-4,6)\), after \(90^{\circ}\) counter - clockwise rotation, it becomes \((-6,-4)\). For \(G(0,3)\), after \(90^{\circ}\) counter - clockwise rotation, it becomes \((-3,0)\). For \(H(-4,3)\), after \(90^{\circ}\) counter - clockwise rotation, it becomes \((-3,-4)\).

Step4: Analyze reflection across the y - axis for counter - clockwise rotation result

The rule for reflection across the \(y\) - axis is \((x,y)\to(-x,y)\).
For the point \((-6,-4)\) (after \(90^{\circ}\) counter - clockwise rotation), after reflection across the \(y\) - axis, it becomes \((6,-4)\) (incorrect).

Step5: Analyze rotation of 180°

For a point \((x,y)\) rotated \(180^{\circ}\) around the origin, the transformation rule is \((x,y)\to(-x,-y)\).
For \(F(-4,6)\), after \(180^{\circ}\) rotation, it becomes \((4,-6)\). For \(G(0,3)\), after \(180^{\circ}\) rotation, it becomes \((0,-3)\). For \(H(-4,3)\), after \(180^{\circ}\) rotation, it becomes \((4,-3)\).

Step6: Analyze translation after 180° rotation

The translation rule is \((x,y)\to(x + 3,y+3)\).
For \((4,-6)\) (after \(180^{\circ}\) rotation), after translation \((4 + 3,-6+3)=(7,-3)\) (incorrect).

Answer:

a rotation \(90^{\circ}\) clockwise around the origin followed by a reflection across the \(y\) - axis.