Different uses of auxiliary verbs
Auxiliary verbs often emphasise things, show interest, or avoid repetition.

If they are used alone, remember they must agree with the tense and person of the main sentence.

1. To emphasise
A: Are you hungry?
B: Yes, I am!

A: I don't think you want to go.
B: I do want to go, but...

2. In questions
You can show interest in what someone said by responding with a short question using an auxiliary verb.
A: Yes, I've got six children.
B: Have you? How lovely!

A: We didn't like the film at all.
B: Really, didn't you, why not?
Notice here that positive questions are used with positive sentences, and negative questions are used with negative sentences.

Question tags at the end of a statement encourage the listener to respond. Here negative questions are used with positive sentences, and positive questions with negative sentences.
You've remembered (+) your keys, haven't (-) you?
It isn't (-) a very nice day, is (+) it?

If there is no doubt about the statement, the voice goes down.
It isn't a very nice day, is it? ↓↓↓

If there is an element of doubt, the voice goes up.
You've remembered your keys, haven't you? ↑↑↑

3. To avoid repetition

We also use auxiliary verbs to avoid repeating whole phrases. 

I enjoyed the film very much, but most of my friends didn't.

(=enjoy the film very much)

The other students in the class don't understand Italian, but Mary does. (=understand Italian)

This is common in answers to yes/no questions.

A. Have you done your homework yet?

B. Yes, actually, I have. (=done my homework)