'to keep at' something means to continue doing something even if it is hard or unpleasant.
- Learning phrasal verbs is hard so you have to keep at it every day.
- I had to finish so I kept at it for an hour before I took a break.
'to keep back' means to not use or give away all of something.
- Keep back some of the cream to put on top of the dessert.
- We kept back some of the money for emergencies.
'to keep down' means to stop the number or level of something from rising.
- If we want to make a profit this year, we have to keep costs down as much as possible.
- I try to eat well and exercise regularly to keep my weight down.
'to keep off' = means to not go onto an area.
- Please keep off the grass.
- Keep off the motorway in the morning, there are always traffic jams at that time.
'to keep off' can also mean to not talk about a particular subject.
- He started talking about it. I tried to keep off the subject.
- He can't keep off the question of climate change, he never stops talking about.
'to keep on' means to continue with something.
- All night long he kept on asking me questions about it.
- I asked him to stop but he just kept on.
'to keep out of' means to not get involved in something.
- It's not my business. I keep out of their arguments.
- It's got nothing to do with me. I'm keeping out of it.
'to keep to' means stay on a subject when talking.
- We don't have much time so can we keep to the agenda, please?
- Can you keep to the point, please?
'to keep up' means to go at the same speed as someone or something.
- She was walking so fast that I couldn't keep up with her.
- I've got too much to do at the moment. I can't keep up with my work.