How is the above graph considered symmetrical about the origin. Doesn't symmetry mean that if I were to fold along the origin the two parts of the graph would come together? Instead of symmetry shouldn't it be 180 degree rotation around the origin instead?
There are different concepts of symmetry. The underlying theme is that you do something to an object and end up with the same object again.
The two cases you described are mirror symmetry, also called reflection symmetry (where you can fold the graph) and point symmetry, also called as two-fold rotational symmetry (i.e. rotation by 360°/2) which is a special case of rotational symmetry.
A repeating pattern (e.g. sin function) would also exhibit translational symmetry, and even glide reflections (a combination of translation and mirror reflection).
These four types (reflection, rotation, translation and glide reflection) are the isometries (length-preserving transformations) in the plane, and cover the most commonly considered cases of symmetry. 3d has more variations (you can reflect in a point, a line or in a plane) and special branches of mathematics or physics might have symmetry concepts that appear less intuitive to people new to the subject.