Question

answer attempt 1 out of 2
a rotation 90° counterclockwise about the origin
a translation 8 units to the right and 4 units down
a rotation 180° counterclockwise about the origin
a rotation 90° clockwise about the origin
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Explanation:

Step1: Recall rotation rules

For a \(90^{\circ}\) clockwise rotation about the origin \((x,y)\to(y, - x)\). For a \(90^{\circ}\) counter - clockwise rotation about the origin \((x,y)\to(-y,x)\). For a \(180^{\circ}\) rotation about the origin \((x,y)\to(-x,-y)\). A translation \((x,y)\to(x + h,y + k)\) where \(h = 8,k=-4\) would be \((x,y)\to(x + 8,y-4)\).

Step2: Analyze the transformation

Let's assume a general point \((x,y)\) on the original figure. If we consider the orientation of the figure. A \(180^{\circ}\) rotation about the origin changes the sign of both \(x\) and \(y\) coordinates. A translation \(8\) units right and \(4\) units down changes \(x\) to \(x + 8\) and \(y\) to \(y-4\). A \(90^{\circ}\) counter - clockwise rotation changes \((x,y)\) to \((-y,x)\) and a \(90^{\circ}\) clockwise rotation changes \((x,y)\) to \((y,-x)\).
By visual inspection of the orientation of the two figures (the pre - image and the image), we can see that if we rotate a point \((x,y)\) \(180^{\circ}\) counter - clockwise (which is the same as \(180^{\circ}\) clockwise) about the origin \((x,y)\to(-x,-y)\). The two figures (the upper - left and the lower - right) are symmetric about the origin.

Answer:

A rotation \(180^{\circ}\) counterclockwise about the origin.