Auxiliaries are what you might have called “helping verbs” when you first learned about grammar: they help a lexical verb by providing grammatical information about a verb’s tense or aspect, or other subtle elements of meaning. There are nine modal auxiliaries, which never change their form because they are never inflected.
Kieran can sing really well.
Laura could climb that rock wall.
We shall decide by drawing straws.
You should take a nap.
The guests will arrive soon.
Malik would like to read that book.
You may leave after you’ve finished the test.
The road might be slippery.
Drivers must obey all traffic laws.
The verbs have, be, and do sometimes behave like auxiliaries and sometimes like ordinary lexical verbs. Unlike the modal auxiliaries, have, be and do get inflected (had, has, having, am, is, are, was, were, been, being, did, done, doing), so even when they are auxiliaries, they are non-modal. Their inflection is not a clue to whether they are auxiliaries or not, so we have to look at their behaviour in the context of a sentence.
If a sentence includes a lexical verb or main verb, then have, be or do in that sentence is likely to be an auxiliary, helping the lexical verb. In the following examples, the auxiliary verbs are underlined and the lexical verbs (also known as main verbs) are bolded:
Arlene is writing a novel.
Beulah has arrived in Saskatoon.
Carmen is planning her vacation.
Doris did not buy any vegetables.
Evlien has been thinking about switching programs.
In addition to their auxiliary functions, have, be and do also have some lexical meaning of their own. If there’s no other verb in the clause, then have, be, or do is probably the main verb of a clause. In these examples the lexical verbs are bolded:
Foster is proud of his sister.
Green vegetables are important for good health.
Harold has an idea for an app.
Ira did his homework before supper.
Javier had a big party.
If have, be or do serves as the lexical verb, then it might also have some auxiliaries helping out:
Foster has been proud of his sister.
Green vegetables might be important for good health.
Harold did have an idea for an app.
Ira could have been doing his homework before supper.
Javier is having a big party.
Notice that not every sentence has an auxiliary, but every sentence does have a lexical verb.