Lots of things still need answering here.
What temperature is the water. You say you have no heater. If it's too cold, the fish will not like it at all.
Second, you say that you "filled the wtare up". What % of the water did you change ? If you change more than half the water at once, then the fish may well be very distressed as the water chemistry changes too much at once. So MAXIMUM 50% water changes. You say that you put in "the purifier". I'm hoping that is some type of dechlorinator. If not, then it needs to be. Also your water change water (if doing a large water change) also needs to be the same temperature as the water the fish is already in. Any sudden change in water chemistry or temperature will distress or even kill the fish.
Also. you said earlier that you were feeding it, but it never ate. The uneaten food MUST be removed from the tank, as this will rot and fill the tank with ammonia and nitrites, which will kill fish.
The fish may well have been bought with some type of sickness, but it sounds as if you have stressed the fish by giving it sudden changes in water chemistry, temperature, and leaving uneaten food in the bowl.
OK, where to now ? Well I'd take a water sample to your LFS and get it tested to get an idea of the water stats (ammonia, nitite, nitrate). Then, if you are serious about keeping fish, get a heater and a filter. I do understand that you are young, and do not have many resources, but it is actually cruel to keep fish (or any animal) without the correct equipment and conditions to keep it happily and healthily.
Yes, many people do keep these for a while in small bowls with no heating or filtration, but I would bet a lot of money that the water stats, and temperature, are all over the place, and that the fish are frequently stressed and live a relatively short, unhappy life.
Maybe this is a good time to encourage your parents to allow you to become a responsible and mature fish-keeper, by allowing you to get the equipment you need to do so.
As for your current fish, straight away I would advise doing 30-50% daily water changes with de-chlorinated water. The water must be the same temperature as the water in the bowl (get it up to temperature by adding a bit of boiled water from the kettle).
If your fish starts to pick up, offer food daily - remove any that isn't eaten (syphoning it out with a bit of airstone tubing is a good way). Also don't overfeed, as Bettas are extremely greedy when happy.
GOod luck, and i hope you fish picks up.