Question

what set of transformations are applied to parallelogram abcd to create abcd?
image of coordinate plane with two parallelograms (abcd and abcd)
options:

• reflected over the x-axis and reflected over the y-axis
• reflected over the y-axis and rotated 180°
• reflected over the x-axis and rotated 90° counterclockwise
• reflected over the y-axis and rotated 90° counterclockwise

Explanation:

1. First, analyze the reflection over the y - axis: Reflecting a point \((x,y)\) over the y - axis gives \((-x,y)\). For the original parallelogram ABCD (let's assume coordinates, e.g., A(-4,0), B(-3,2), C(-1,2), D(-2,0) approximately from the graph), reflecting over the y - axis would change the x - coordinates' sign.
2. Then, a \(90^{\circ}\) counter - clockwise rotation: The rule for a \(90^{\circ}\) counter - clockwise rotation about the origin is \((x,y)\to(-y,x)\). Let's check the transformation of the parallelogram. After reflecting over the y - axis and then rotating \(90^{\circ}\) counter - clockwise, we can see that the position of \(A''B''C''D''\) matches the transformation.

• Let's take a point from ABCD, say point A(-4,0). Reflect over y - axis: (4,0). Rotate \(90^{\circ}\) counter - clockwise: (0,4)? Wait, maybe my initial coordinate assumption is wrong. Let's re - examine the graph. The original parallelogram is in the second quadrant (x negative, y positive), and the final parallelogram \(A''B''C''D''\) is in the fourth quadrant (x positive, y negative) with a rotation. Wait, maybe another approach:
• Reflect over y - axis: changes the horizontal position (flips left - right). Then a \(90^{\circ}\) counter - clockwise rotation: the rotation of a figure after reflection over y - axis. Let's consider the orientation. The original parallelogram has a certain slope, and after reflection over y - axis and \(90^{\circ}\) counter - clockwise rotation, the orientation and position match \(A''B''C''D''\).
• Let's check the other options:
• Option 1: Reflect over x - axis (flips top - bottom) and then y - axis (flips left - right). The resulting figure would be in the fourth quadrant but with a different orientation than \(A''B''C''D''\).
• Option 2: Reflect over y - axis and rotate \(180^{\circ}\). A \(180^{\circ}\) rotation rule is \((x,y)\to(-x,-y)\). After reflecting over y - axis \((x,y)\to(-x,y)\), then rotating \(180^{\circ}\) gives \((x,-y)\), which does not match the position of \(A''B''C''D''\).
• Option 3: Reflect over x - axis (flips top - bottom) and rotate \(90^{\circ}\) counter - clockwise. The orientation and position would not match.
• Option 4: Reflect over y - axis (flips left - right) and rotate \(90^{\circ}\) counter - clockwise. This transformation gives the correct position and orientation of \(A''B''C''D''\).

Answer:

D. Reflected over the y - axis and rotated \(90^{\circ}\) counterclockwise