Question

which of these graphs have rotational symmetry about the origin? (select all that apply.)

Explanation:

Step1: Recall rotational - symmetry definition

A graph has rotational symmetry about the origin if rotating it 180 - degrees about the origin results in the same graph. Mathematically, if \((x,y)\) is on the graph, then \((-x,-y)\) is also on the graph.

Step2: Analyze graph A

For graph A, if we rotate it 180 - degrees about the origin, it will coincide with itself. So graph A has rotational symmetry about the origin.

Step3: Analyze graph B

For graph B, when we rotate it 180 - degrees about the origin, it will not coincide with the original graph. So graph B does not have rotational symmetry about the origin.

Step4: Analyze graph C

For graph C, rotating it 180 - degrees about the origin will not result in the same graph. So graph C does not have rotational symmetry about the origin.

Step5: Analyze graph D

For graph D, if we rotate it 180 - degrees about the origin, it will coincide with itself. So graph D has rotational symmetry about the origin.

Step6: Analyze graph E

For graph E, rotating it 180 - degrees about the origin will not give the original graph. So graph E does not have rotational symmetry about the origin.

Step7: Analyze graph F

For graph F, rotating it 180 - degrees about the origin will not result in the same graph. So graph F does not have rotational symmetry about the origin.

Answer:

A. Graph A, D. Graph D