Question

which sequence of transformations describes the movement of the figure on the coordinate grid? horizontal translation, reflection across the x - axis 90° counterclockwise rotation, reflection across the x - axis 90° clockwise rotation, reflection across the y - axis reflection across the y - axis, vertical translation

Explanation:

Step1: Analyze each option

• Option 1: Horizontal translation, reflection across the x - axis
• A horizontal translation would just shift the figure left or right. But the orientation of the figure (the direction of the arrow) is different in the two figures. A horizontal translation alone (without rotation or reflection) cannot change the orientation. Also, a reflection across the \(x\) - axis would flip the figure over the \(x\) - axis (change the \(y\) - coordinates' signs). This option is incorrect.
• Option 2: \(90^{\circ}\) counter - clockwise rotation, reflection across the \(x\) - axis
• A \(90^{\circ}\) counter - clockwise rotation changes the orientation of the figure. But after a \(90^{\circ}\) counter - clockwise rotation and then a reflection across the \(x\) - axis, the final orientation and position do not match the given figures.
• Option 3: \(90^{\circ}\) clockwise rotation, reflection across the \(y\) - axis
• A \(90^{\circ}\) clockwise rotation changes the orientation. Then a reflection across the \(y\) - axis (changes the \(x\) - coordinates' signs) does not result in the given figure transformation.
• Option 4: Reflection across the \(y\) - axis, vertical translation
• Reflection across the \(y\) - axis (if \((x,y)\) is a point on the original figure, \((-x,y)\) is a point on the reflected figure) changes the left - right orientation of the figure (the arrow direction). Then a vertical translation (shifting the figure up or down) can move the reflected figure to the position of the other figure.

Answer:

Reflection across the \(y\) - axis, vertical translation.