Question

answer
a reflection over the line (x = - 4)
a rotation of (90^{circ}) clockwise about the origin
a reflection over the line (y = 1)
a rotation of (180^{circ}) counterclockwise about the origin

Explanation:

Step1: Recall transformation rules

For reflection over line $x = a$, the rule for a point $(x,y)$ is $(2a - x,y)$. For rotation of $90^{\circ}$ clock - wise about the origin, the rule is $(y,-x)$. For reflection over line $y = b$, the rule is $(x,2b - y)$. For rotation of $180^{\circ}$ counter - clockwise about the origin, the rule is $(-x,-y)$.

Step2: Analyze each option visually

• Reflection over $x=-4$: Points on the figure would be mirrored across the vertical line $x = - 4$.
• Rotation of $90^{\circ}$ clockwise about the origin: The orientation of the figure would change in a way that the $x$ and $y$ coordinates of each point follow the $(y,-x)$ rule.
• Reflection over $y = 1$: Points on the figure would be mirrored across the horizontal line $y=1$.
• Rotation of $180^{\circ}$ counter - clockwise about the origin: Each point $(x,y)$ on the original figure would be transformed to $(-x,-y)$. By observing the figure and the orientation and position of the pre - image and potential image under each transformation, we can see that a reflection over the line $y = 1$ would map the given figure to a new position consistent with the transformation rules for reflection over a horizontal line.

Answer:

a reflection over the line $y = 1$