My sympathy for your predicament is tempered by the relief that I don't have to deal with these situations anymore. Is it possible to ask your "professor" what reference he's using to inflict the topic on the class? It might help to know so that you're at least using common terminology.

Complements are syntactic structures closely aligned with verbs -- complement here means a completer -- and they're said to be licensed by their associated verbs. That is, a particular verb will require a complement in a particular form. This isn't to say that complements are always mandatory, but when they appear, they appear as licensed.

With the caveat that this isn't an exhaustive discussion, let's identify a starting list of complements:

• Direct object, licensed by transitive verbs as the receiver of the action

  I ate dinner.

• Indirect object, licensed by verbs of giving as the recipient of the direct object

  I gave her a diamond.

• Objective complement, licensed by verbs of labeling or categorization as the classification of the direct oject

  They elected him president

• Nominative complement, licensed by copular verbs to give an equivalent or a description of the subject

  He is her son.
  She seems happy.

• Prepositional complement, licensed for particular prepositions for each verb

  He blamed me for the accident.

The last example illustrates the term licensing. Blame requires for as the preposition in the prepositional phrase that's the complement. You don't say He blamed me with the accident. Note that verbs in the passive voice license a prepositional complement with by to indicate the actor. (The subject is the receiver of the action in sentences with passive-voice verbs.)

I was blamed for the accident by him.

Note also that this complement is optional. It's fine to say

I was blamed for the accident.

Complements have certain syntactic restrictions. For example, direct objects almost always follow the verb (you don't say I dinner ate) and take the object case for pronouns (it's I followed him, not I followed he).

A word about what complements are not. They're not subjects. With few exceptions, all sentences require subjects (although sometimes they're missing but understood). They're not modifiers. Modifiers are freely associated with verbs, add more incidental information, and can move around much more easily. You can't say I dinner ate, with dinner as the direct object, but you can say

Greedily, I ate dinner. I greedily ate dinner. I ate dinner greedily

And you may substitute for greedily any adverb that makes sense -- sloppily, neatly, quickly, etc. Which is not the case with blame and for.

So to get down to your specifics. Eat is transitive, so it licenses a direct object.

I ate dinner.

It's not a verb of giving or categorization, so it doesn't license indirect objects or objective complements:

WRONG: I ate him dinner.
WRONG: I ate dinner restaurant.

It's not a copular verb, so it doesn't license nominative complements. I know you're supposed to be what you eat, but not grammatically. Saying I eat steak does not make me a steak the way saying I am a US citizen makes me an American.

In modern parlance, eat does not license a prepositional complement. But it used to, in the archaic phrasing with of:

You may not eat of the tree of life.

You may certainly use prepositional phrases with eat:

I ate steak at a restaurant.
I ate the dessert beside the car.

But theses phrases are spatial modifiers that just add incidental information. (Try moving them around.)

(The observations are independent of the form of the verb. I.e., the number or tense the verb doesn't matter.)

Let's take a quick look at the three verbs. Everything after the verb is a complement. Complements is an encompassing term. It can include direct objects and indirect objects and adverbial modifiers.

All of the verbs can take adverbial modifiers.

Sleeps does not take a direct object.

Eats takes direct objects.

Puts takes a direct object and an indirect object via a prepositional phrase.

The baby sleeps a lot.

The baby sleeps in a crib.

The baby sleeps soundly.

The baby sleeps mostly during the day.

The parents sleep very little.

She eats mush.

She eats her mush with gusto.

She eats her mush messily.

She eats messily and with gusto.

She eats frequently.

She eats no solid food yet.

We put onesie pajamas on her.

We put pillows over our heads.

We put our hopes in her.