Question

a hexagon is shown on the coordinate grid.
which transformations produce a hexagon with the same vertices?
move the green rectangles to select three correct answers.
$(x, y) \to (-x, y)$
$(x, y) \to (x, -y)$
$(x, y) \to (-y, x)$
$(x, y) \to (-x, -y)$
$(x, y) \to (y, -x)$

Explanation:

Step1: Analyze the hexagon's symmetry

The hexagon is symmetric about the \(y\)-axis, \(x\)-axis, and the origin. Let's check each transformation:

Step2: Check \((x, y) \to (-x, y)\)

This is a reflection over the \(y\)-axis. Since the hexagon is symmetric about the \(y\)-axis, this transformation will map the hexagon to itself (same vertices).

Step3: Check \((x, y) \to (x, -y)\)

This is a reflection over the \(x\)-axis. The hexagon is symmetric about the \(x\)-axis, so this transformation will also map the hexagon to itself (same vertices).

Step4: Check \((x, y) \to (-x, -y)\)

This is a reflection over the origin (rotation by \(180^\circ\)). The hexagon is symmetric about the origin, so this transformation will map the hexagon to itself (same vertices).

Step5: Check the other transformations

\((x, y) \to (-y, x)\) is a rotation by \(90^\circ\) counterclockwise, and \((x, y) \to (y, -x)\) is a rotation by \(90^\circ\) clockwise. These will not map the hexagon to itself as the shape's orientation will change, and vertices will not match.

Answer:

The three correct transformations are:
\((x, y) \to (-x, y)\), \((x, y) \to (x, -y)\), \((x, y) \to (-x, -y)\)