It's a corn plant, Dracaena fragrans massangeana. They prefer to dry out between waterings but are pretty adaptable. They also adjust well to most light conditions, though bright indirect light or filtered sun are best (full sun will burn it). It's hard to say what might be the specific problem here. Burnt tips often mean mineral buildup in the soil: it's a little soon for this, if you only got the plant in August, but maybe. I wouldn't change the soil now, or anytime soon: it's probably okay at least until May or June. You might consider flushing the soil with lots and lots of water, letting the water run through the soil pretty thoroughly, the next time it's dry (and only the next time it's dry: don't do it now because you feel like you should be doing something): this will remove some mineral buildup, if there is any. (This works best if you set it in the shower or something and pour water through with a watering can or pitcher. You should use at least three gallons of water, and twice that much wouldn't hurt anything. Then let the plant drain well before moving it back to the decorative pot it's in.) Corn plants do not like to stand in water: whether you're flushing it out or just watering it routinely, make sure that you dump out any excess after about fifteen minutes. If you leave the pot to stand in water, the water will wick up into the soil until it's all gone, leaving the soil that much wetter for that much longer, which will eventually cause it to rot. I don't think the heat was probably the plant's main problem, though I agree that it probably wasn't doing the plant any favors, either. If it was a heat or watering problem, you'll probably be forgiven in a few months; corn plants are nice that way.

I don't think any Gardenias can live permanently in the ground that far north unfortunately. It needs to come inside to escape the cold of winter, then should enjoy taking a vacation outside for summer. There are discussions about potted Gardenias on the fragrant forum. I think it's too soon to call the lack of objections to my suggestion that it's a Gardenia a confirmation. Not much to go on here. Whatever it is, good luck! Don't be too hard on yourself if this doesn't make it. We can't know what's happened to it before you got it, and it looks barely alive.

Yes, I think you are spot on. Thanks a lot.

I know that it's tiny. I must have tried 15 times to upload the picture. I kept getting a message saying that the size exceeded the limit and it wouldn't let me upload it. I didn't realize it was so small but the system wouldn't let up upload anything larger. Don't know what I did wrong. Maybe there is a practice section somewhere on this forum?