Question

question which transformation would take figure a to figure b? answer

Explanation:

Step1: Analyze reflection properties

A reflection over the x - axis changes the sign of the y - coordinate of each point \((x,y)\) to \((x, - y)\), and a reflection over the y - axis changes the sign of the x - coordinate of each point \((x,y)\) to \((-x,y)\). A rotation of \(180^{\circ}\) about the origin changes a point \((x,y)\) to \((-x,-y)\).
Looking at the figures, if we consider the transformation from Figure A to Figure B, we can check the coordinates (by observing the grid). The shape of Figure A and Figure B are related by a \(180^{\circ}\) rotation about the origin (or we can also think of it as a combination of reflection over x - axis and then y - axis, which is equivalent to \(180^{\circ}\) rotation). Another way: the transformation that maps Figure A to Figure B is a \(180^{\circ}\) rotation about the origin (or we can also describe it as a reflection over the x - axis followed by a reflection over the y - axis, or vice - versa). But the most straightforward transformation here is a \(180^{\circ}\) rotation about the origin (or we can also say rotation by \(180^{\circ}\) around the origin). Also, we can check the symmetry: if we rotate Figure A \(180^{\circ}\) around the origin, each point \((x,y)\) in A will map to \((-x,-y)\), and when we look at the grid, this maps to the coordinates of Figure B. Alternatively, reflection over the line \(y=-x\) is not the case here. The key is that the transformation is a \(180^{\circ}\) rotation (or a combination of two reflections: over x and y axes).

Step2: Confirm the transformation

Let's take a vertex of Figure A. For example, if we take a vertex of A with coordinates (let's assume from the grid) say (5, - 2) (approximate, looking at the grid). After a \(180^{\circ}\) rotation about the origin, the coordinates become \((- 5,2)\), which is a vertex of Figure B. Similarly, other vertices will follow the same pattern. So the transformation that takes Figure A to Figure B is a \(180^{\circ}\) rotation about the origin (or we can also describe it as a reflection over the x - axis and then a reflection over the y - axis, since reflecting over x - axis changes \((x,y)\) to \((x, - y)\) and then reflecting over y - axis changes \((x, - y)\) to \((-x,-y)\) which is the same as \(180^{\circ}\) rotation). But the standard transformation here is a \(180^{\circ}\) rotation (or we can say rotation by \(180\) degrees around the origin) or a combination of reflection over x - axis and y - axis. However, the most appropriate transformation is a \(180^{\circ}\) rotation about the origin (or we can also call it a central symmetry with respect to the origin) or a reflection over the x - axis followed by a reflection over the y - axis (or vice - versa). But the common way to describe this is a \(180^{\circ}\) rotation about the origin (or we can say rotation of \(180\) degrees around the origin) or a reflection over the line \(y = - x\) is not correct, the correct is \(180^{\circ}\) rotation or two reflections (x and y). But the answer is a \(180^{\circ}\) rotation about the origin (or we can also say reflection over x - axis and then y - axis, but \(180^{\circ}\) rotation is more direct).

Answer:

\(180^{\circ}\) rotation about the origin (or reflection over the x - axis and then reflection over the y - axis, or vice - versa)