Question

answer a reflection over the x - axis a translation 8 units to the right and 6 units down a rotation 180° clockwise about the origin a rotation 90° counterclockwise about the origin

Explanation:

Step1: Analyze reflection over x - axis

Reflection over x - axis changes the sign of y - coordinate. But the orientation of the figure (like the slope of the sides) would be different in a reflection compared to the given transformation.

Step2: Analyze rotation 180° clockwise about the origin

Rotation 180° clockwise about the origin changes \((x,y)\) to \((-x,-y)\). The orientation and position do not match the given transformation.

Step3: Analyze rotation 90° counter - clockwise about the origin

Rotation 90° counter - clockwise about the origin changes \((x,y)\) to \((-y,x)\). The orientation and position do not match the given transformation.

Step4: Analyze translation

For a point \((x,y)\) in Figure R, if we consider translation:
The x - coordinate change: Let's take a vertex of Figure R. Suppose a vertex of Figure R is \((-5,0)\). For a vertex of Figure S, say \((5,-3)\). The change in x - coordinate: \(5-(-5)=10\) (incorrect for 8 units right). Wait, let's take another approach.
Count the horizontal and vertical distance between corresponding points.
If we consider the movement from the left - most point of Figure R (say \((-5,0)\)) to the right - most point of Figure S (say \((5,0)\) - no, wrong. Wait, better to use vector approach.
Let’s assume a general point \((x,y)\) in Figure R. After translation \(x\) changes by \(h\) (horizontal) and \(y\) changes by \(k\) (vertical).
By observing the graph (counting the grid units):
The horizontal change (right - ward): If we take a reference point (e.g., the vertex near \((-5,0)\) in Figure R and the corresponding vertex in Figure S). Moving from \(x=-5\) to \(x = 3\) (for a non - x - axis vertex), \(h=3-(-5)=8\) (right - ward).
The vertical change (down - ward): From \(y = 3\) (a vertex of Figure R) to \(y=-3\) (corresponding vertex of Figure S), \(k=-3 - 3=-6\) (6 units down).

Answer:

A translation 8 units to the right and 6 units down