1. The infinitive

2. Infinitive forms bare infinitive (e.g. sit) I'd rather sit at the back
to-infinitive (e.g. to sit).
I'd prefer to sit at the back.

3. Simple
play to play

4. Perfect
have played
to have played

5. It's a pity I missed that programme.
I'd like to have seen it.
You'd better have finished by tomorrow

6. Continuous
be playing
to be playing

7. I'd rather be lying on the beach than stuck in a traffic jam.
The weather seems to be getting worse.
I'd rather be lying on the beach than stuck in a traffic jam.

8. Perfect + continuous
have been playing
to have been playing

9. The man appeared to have been drinking.

10. In the negative, not comes before the infinitive.
I'd rather not sit at the front.
I'd prefer not to sit at the front.

11. The to-infinitive as subject

12. To be or not to be.
That is the question

13. To defrost this fridge takes ages.
To turn down the invitation seems rude.
Not to take a holiday now and then is a great mistake.

14. To err is human; to forgive divine. To find fault with others is easy.
To make mistakes is easy.
common in older English

15. It takes ages to defrost this fridge.
More often we use it as an 'empty subject'
It takes ages to defrost this fridge.
Would it seem rude to turn down the invitation?
It's a great mistake not to take a holiday now and then.

16. A to-infinitive clause can be
a complement after be.
Melanie's ambition is to go to Australia.
The important thing is not to panic.
The idea was to surprise everybody.

17. The to-infinitive expressing purpose and result

18. A to-infinitive clause can express purpose
Laura has gone to town to do some shopping.
I'm writing to enquire about activity holidays.
To get a good seat, you need to arrive early.

19. In order = He came here in order to study English
He came here to study English

20. To express purpose use In order to, to NOT for
He came here for study English
He came here for to study
He came here to (in order to) study

21. I went to the store for some bread (noun)
I went to the store to buy some bread

22. We can sometimes use a to-infinitive clause to express result, although this use is rather literary
Laura came home to find her house on fire.
He grew up to be a handsome young man.

23. I found my keys only to lose them again.
Charles arrived for the concert (only) to find it had been cancelled.

24. An infinitive clause can also express a comment on the sentence.
To be frank, you didn't make a very good impression.
I'm a bit tired of sightseeing, to tell you the truth.