Would You Consider This As Empathy?

I have always been a bit intuitive, but lately I have realized I may be empathic. My mother and I are very good at "reading" people and recently I made the connection to having empathy. I have also realized that when other people are feeling extreme emotions, like sadness or grief, I tend to feel like that myself for no reason at all. I even scare myself sometimes because I am a very peaceful and calm person and I tend to get a bit touchy and over-emotional when this happens.

An example of me feeling what someone else is feeling is, the other day a teacher walked in to school and as soon as I saw her I felt a rush of deep sadness. It lasted for a very short time but I was pretty stunned, it was kind of like a slap in the face. I later found out from the teacher there was a death in their family.

That's more of a major case though, normally I can just tell what kind of mood a person is in. I also get really tired when I am around a lot of people or when the people around me are pretty down. I get tired all the time and it doesn't make sense because I make sure I get a good amount of sleep every day as I still am in school. I can't concentrate otherwise because I start to daydream and I get very quiet and thoughtful.

I am seeking other people's opinions, does this seem like empathy? I would also like to know some methods for dealing with it because I am beginning to worry I don't really know myself anymore. This being said, I would like to improve it as well. So if anyone has any advice or any information at all, please comment! Thank you for your time!