Question

which graphs have rotational symmetry? check all of the boxes that app

Explanation:

Step1: Recall the definition of rotational symmetry

A graph has rotational symmetry if it looks the same after a rotation of \(180^{\circ}\) (or some other non - full - turn rotation) about a point (usually the origin in coordinate - plane problems).

Step2: Analyze the first graph

For the first graph (an odd - function - like curve, assume it is \(y = x^{3}\) - type). If we rotate it \(180^{\circ}\) about the origin, the point \((x,y)\) on the graph will be mapped to \((-x,-y)\). For a function \(y = f(x)\) that satisfies \(f(-x)=-f(x)\) (odd function), the graph has rotational symmetry about the origin.

Step3: Analyze the second graph

For the second graph (a "V" - shaped graph, assume it is \(y = |x|\)). If we rotate it \(180^{\circ}\) about the origin, a point \((x,y)\) on \(y = |x|\) (where \(y\geq0\)) will be mapped to \((-x,-y)\). But \(y = |x|\) has \(y\geq0\) for all \(x\) in its domain, and the rotated graph (with \(y\leq0\)) is not the same as the original graph. So it does not have rotational symmetry about the origin.

Step4: Analyze the third graph

For the third graph (an ellipse - like shape). If we rotate it \(180^{\circ}\) about its center (assume the center is on the \(y\) - axis). The point \((x,y)\) on the graph will be mapped[SSE Completed, Client Connection Error][LLM SSE On Failure]

Answer:

Step1: Recall the definition of rotational symmetry

A graph has rotational symmetry if it looks the same after a rotation of \(180^{\circ}\) (or some other non - full - turn rotation) about a point (usually the origin in coordinate - plane problems).

Step2: Analyze the first graph

For the first graph (an odd - function - like curve, assume it is \(y = x^{3}\) - type). If we rotate it \(180^{\circ}\) about the origin, the point \((x,y)\) on the graph will be mapped to \((-x,-y)\). For a function \(y = f(x)\) that satisfies \(f(-x)=-f(x)\) (odd function), the graph has rotational symmetry about the origin.

Step3: Analyze the second graph

For the second graph (a "V" - shaped graph, assume it is \(y = |x|\)). If we rotate it \(180^{\circ}\) about the origin, a point \((x,y)\) on \(y = |x|\) (where \(y\geq0\)) will be mapped to \((-x,-y)\). But \(y = |x|\) has \(y\geq0\) for all \(x\) in its domain, and the rotated graph (with \(y\leq0\)) is not the same as the original graph. So it does not have rotational symmetry about the origin.

Step4: Analyze the third graph

For the third graph (an ellipse - like shape). If we rotate it \(180^{\circ}\) about its center (assume the center is on the \(y\) - axis). The point \((x,y)\) on the graph will be mapped[SSE Completed, Client Connection Error][LLM SSE On Failure]