Reflections - Understanding I have been asked 

Does the set of all reflections in the plane form a group? Explain.

I thought that the answer was that it wouldn't as if I reflected the parabola $y=x^2$ through the y axis, I have a shape that is more of an "n" shape rather than a "u" shape. 
However in my notes I have found that when listing the elements of a symmetry group of the same parabola, a reflection in the y axis exists. 
Would someone be able to help clarify where I am going wrong? It could be with understanding the concept of a group, or what it means to reflect a function, or indeed both. Thanks for any assistance available
 A: It depends on the group action. If the group action is composition (i.e. performing one reflection after another), then you have to answer four questions:


*

*Is the identity operation (i.e. leaving the plane as-is) representable as one of the reflections in the set?

*For any reflection $R$ you make, can you get back to the original function using another reflection (the inverse of $R$).

*If you perform two arbitrary reflections in the set, $R_1$ and $R_2$, one after another, can the net result always be represented as a single reflection $R$ in the set?

*Are compositions of reflections in the group associative?


If the answer to all questions is 'Yes', then the set of reflections you're looking at forms a group under composition.
Based on your initial description, assuming composition, the answer to your question is 'No', since leaving the plane as-is doesn't (typically) constitute a "reflection" and thus (1) is false.
As Dr. Kildetoft indicates, the answer depends on how exactly you define "reflection".
