Question

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1. b. what transformations map the figure onto itself?

Explanation:

Step1: Analyze rotation

A full - rotation (\(360^{\circ}\)) will always map any figure onto itself. For a circle (the outer part of the smiley - face which is a circle - like shape), a rotation of \(180^{\circ}\) around its center will also map it onto itself. Since the eyes and mouth are symmetrically placed with respect to the center, a \(180^{\circ}\) rotation about the center of the smiley - face maps the figure onto itself. Also, a \(360^{\circ}\) rotation about the center of the figure maps the figure onto itself.

Step2: Analyze reflection

The smiley - face has a horizontal line of symmetry (a line passing through the mid - point of the eyes and the center of the mouth). A reflection across this horizontal line of symmetry will map the figure onto itself.

Answer:

A \(180^{\circ}\) rotation about its center, a \(360^{\circ}\) rotation about its center, and a reflection across its horizontal line of symmetry.