Question

which transformation would take figure a to figure b?
answer
a counterclockwise rotation of 270° about the origin
a reflection over the y - axis
a counterclockwise rotation of 90° about the origin
a reflection over the x - axis

Explanation:

Step1: Analyze reflection over y - axis

A reflection over the \(y\) - axis changes the sign of the \(x\) - coordinate of a point \((x,y)\) to \((-x,y)\). Let's consider the general shape of Figure A and Figure B. If we reflect Figure A over the \(y\) - axis, the left - right orientation would flip, but the vertical position (above/below \(x\) - axis) would remain the same. But Figure B is below the \(x\) - axis while Figure A is above, so reflection over \(y\) - axis is not correct.

Step2: Analyze reflection over x - axis

A reflection over the \(x\) - axis changes the sign of the \(y\) - coordinate of a point \((x,y)\) to \((x, - y)\). If we reflect Figure A (above \(x\) - axis) over \(x\) - axis, it should be below \(x\) - axis, but the shape's orientation (left - right) would be the same as Figure A. But Figure B has a different rotational orientation compared to the reflection of Figure A over \(x\) - axis, so reflection over \(x\) - axis is not correct.

Step3: Analyze counterclockwise rotation of 90° about the origin

The rule for counterclockwise rotation of \(90^{\circ}\) about the origin is \((x,y)\to(-y,x)\). Let's take a point from Figure A. If Figure A is in the second quadrant (above \(x\) - axis, left of \(y\) - axis), after a \(90^{\circ}\) counterclockwise rotation, the point would move to the first quadrant, which is not where Figure B is (fourth quadrant - like), so this is not correct.

Step4: Analyze counterclockwise rotation of 270° about the origin

The rule for counterclockwise rotation of \(270^{\circ}\) about the origin is equivalent to a clockwise rotation of \(90^{\circ}\), and the transformation rule is \((x,y)\to(y, - x)\). Let's consider the position of Figure A (above \(x\) - axis, left of \(y\) - axis). After a \(270^{\circ}\) counterclockwise rotation about the origin, the figure will be transformed to a position below \(x\) - axis (since \(y\) - coordinate sign changes in a way that moves it down) and with the correct rotational orientation to match Figure B.

Answer:

A counterclockwise rotation of \(270^{\circ}\) about the origin