Auxiliary verbs are also called helping verbs and they are used together with a main verb to show the verb’s tense or to form a negative or question sentence.

The most common auxiliary verbs are have, be, and do.

Examples;

Does your sister liven in London?
We have completed the project just on time.
Jack is cleaning the house himself. 

Function of auxiliary verbs

Auxiliary verbs add functional or grammatical meaning to the sentence in different ways. 

- By expressing tense (providing a time reference, i.e. past, present, or future)

- Grammatical aspect (expresses how verb relates to the flow of time)

- Modality (quantifies verbs)

- Voice (describes the relationship between the action expressed by the verb and the participants identified by the verb’s subject, object, etc.)

- Adds emphasis to a sentence

► The verb "be"

The verb be can be used as an auxiliary and a full verb. As an auxiliary we use this verb for compound tenses and the passive voice. 

- Simple Present:
I am, he/she/it is, we/you/they are

- Simple Past:
I/he/she/it was, we/you/they were

- Past Participle: been

In the following sentences be is an auxiliary because it is followed by another verb. Look at the examples below;

- Progressive Forms

Present Progressive: He is playing tennis.
Past Progressive: He was playing tennis.
Present Perfect Progressive: He has been playing tennis.
Past Perfect Progressive: He had been playing tennis.

- Passive Form

Simple Present/Past: The window is/was broken.
Present/Past Perfect: The window has/had been broken.
Future I: The window will be broken.

- "Be" as a full verb

The verb be can also be a full verb and it's not followed by another verb. If "be" is used as a full verb, we do not need an auxiliary in negative sentences or questions.

Positive sentence: They are very happy.
Negative sentence: They are not very happy.
Question: Are they very happy?

► The verb "have"

The verb have can also be used both as an auxiliary and as a full verb. As an auxiliary we use this verb to form compound tenses in active and passive voice. We use the past participle of the full verb in this form.

- Compound Tenses - Active Voice

Present Perfect Simple: He has played tennis.
Past Perfect Simple: He had played tennis.
Present Perfect Progressive: He has been playing tennis.
Past Perfect Progressive: He had been playing tennis.

- Compound Tenses - Passive Voice

Present/Past Perfect: The window has/had been broken.

Simple Present: I/we/you/they have, he/she/it has
Simple Past: I/he/she/it/we/you/they had
Past Participle: had

- "Have" in positive sentences

Have, as a full verb indicates possession. In British English, "have got" can also be used to indicate possession. 

Full verb: I have a house. 
Auxiliary verb: I have got a house.

- "Have" in negative sentences and questions

When we use "have" as a full verb, we must use the auxiliary "do" in negative sentences and questions. If we use have got, we do not need another auxiliary.

Have as a full verb: I do not have a car / Do I have a car?
Have as an auxiliary verb: I have not got a car / Have I got a car?

► The verb "will"

- The verb "will" can only be used as an auxiliary. We use it to form the future tenses.

Future I: He will not play tennis.
Future II: He will have played tennis.

- The verb will remains the same for all forms (no "s" for 3rd person singular). The short form for negative sentences is won't.'

I will, he will
I will not = I won't

► The verb "do"

The verb do can be both an auxiliary and a full verb. As an auxiliary we use do in negative sentences and questions for most verbs (except not for be, will, have got and modal verbs) in Simple Present and Simple Past. 

- The auxiliary "do" in negative sentences

Simple Present: He does not play football.
Simple Past: He did not play football.

- The auxiliary "do" in questions

Simple Present: Does he play tennis?
Simple Past: Did he play tennis?

- The verb do is irregular:

Simple Present: I/we/you/they do, he/she/it does
Simple Past: I/he/she/it/we/you/they did

- The full verb "do"

As a full verb we use do in certain expressions. If we want to form negative sentences or questions using do as a full verb, we need another do as an auxiliary.

Positive sentence: She does her homework every day.
Negative sentence: She doesn't do her homework every day.
Question: Does she do her homework every day?