Question

identifying point of symmetry
this graph has rotational symmetry about the point
this graph has rotational symmetry about the point

Explanation:

Step1: Recall rotational - symmetry concept

For a function \(y = f(x)\), if rotating the graph of the function \(180^{\circ}\) about a point \((a,b)\) results in the same graph, then the graph has rotational symmetry about the point \((a,b)\). For an odd - function \(y = f(x)\) (i.e., \(f(-x)=-f(x)\)), the graph is symmetric about the origin \((0,0)\).

Step2: Analyze the first graph

The first graph is an odd - function. When we rotate the graph of an odd - function \(180^{\circ}\) about the origin \((0,0)\), it coincides with itself. So the point of rotational symmetry is \((0,0)\).

Step3: Analyze the second graph

For an ellipse - shaped graph centered at \((h,k)\) which has rotational symmetry of order 2 (a \(180^{\circ}\) rotation maps it onto itself), we can find the center of the ellipse. By observing the graph, the center of the ellipse (the point of rotational symmetry) is \((1,2)\) as the graph is symmetric about this point under a \(180^{\circ}\) rotation.

Answer:

First graph: \((0,0)\)
Second graph: \((1,2)\)