Die Hausarbeit ist noch nicht fertig.

Hausarbeit is a feminine noun, so its basic article is die.

In this sentence, Hausarbeit is the subject, so it is in the nominative case.
That gives us:

• die Hausarbeit = the paper / the homework / the housework (depending on context)

So die here means the for a feminine singular noun in the nominative.

Hausarbeit can mean different things depending on context:

• term paper / essay / written assignment in school or university contexts
• sometimes housework / domestic work in other contexts

A native English speaker often expects Hausaufgabe for homework, and that is often correct for regular school homework.

A useful distinction is:

• Hausaufgabe = homework assignment
• Hausarbeit = often a larger written paper or assignment
• Hausarbeit can also mean housework in the right context

So the exact meaning depends on where the sentence comes from.

German main clauses usually follow the verb-second rule: the finite verb goes in position 2.

Here the sentence is:

• Die Hausarbeit = position 1
• ist = position 2
• noch nicht fertig = the rest

So even though English also says The paper is not finished yet, in German this placement is also following a specific rule: the conjugated verb ist must come second in a normal statement.

Noch nicht means not yet.

It is made of:

• noch = still / yet
• nicht = not

Together, noch nicht expresses that something is not finished now, but may be finished later.

So:

• Die Hausarbeit ist nicht fertig. = The paper is not finished.
• Die Hausarbeit ist noch nicht fertig. = The paper is not finished yet.

The version with noch sounds more natural when the situation is still ongoing.

This is a very common and important question.

• noch nicht = not yet
• nicht mehr = no longer / not anymore

Compare:

• Die Hausarbeit ist noch nicht fertig.
  = The paper is not finished yet.

• Die Hausarbeit ist nicht mehr fertig.
  This would usually sound odd here, because it suggests it used to be finished and now somehow isn't anymore.

So in this sentence, noch nicht is the natural choice.

Fertig is a predicate adjective here. It describes the subject after the verb sein.

German often places this kind of information toward the end of the clause:

• Die Hausarbeit ist fertig. = The paper is finished.
• Die Hausarbeit ist noch nicht fertig. = The paper is not finished yet.

So fertig is not acting like a finished paper; it is the adjective that completes the idea after ist.

Because fertig is being used predicatively, not directly before a noun.

Compare:

• Die Hausarbeit ist fertig.
  Here fertig comes after ist, so it has no ending.

• die fertige Hausarbeit
  Here fertige comes before the noun, so it takes an adjective ending.

This is similar to English:

• The paper is finished.
• the finished paper

German makes that difference very clearly in the adjective form.

Yes, you can. It is grammatically correct.

But the nuance is slightly different:

• Die Hausarbeit ist nicht fertig.
  = The paper is not finished.

• Die Hausarbeit ist noch nicht fertig.
  = The paper is not finished yet.

The version with noch often sounds more natural when the work is still in progress and you expect it to be completed later.

They are related, but not always identical in tone.

• fertig = finished, ready, done
  This is the most everyday and natural word here.
• beendet = ended, finished
  More formal, and often used for events or processes.
• abgeschlossen = completed, concluded
  Often sounds more formal or more final.

For a sentence like this, fertig is the most common and idiomatic choice:

• Die Hausarbeit ist noch nicht fertig.

In German, nicht usually comes before the specific thing it is negating, or near the end of the clause if it negates the main idea.

Here it negates fertig:

• nicht fertig = not finished

That is why the sentence says:

• Die Hausarbeit ist noch nicht fertig.

If you moved nicht elsewhere, the meaning or emphasis could change, and some versions would sound unnatural.

It is singular here.

You can tell because:

• the article is die, which can be feminine singular or plural
• but the verb is ist, which is singular

So:

• Die Hausarbeit ist ... = singular
• Die Hausarbeiten sind ... = plural

German often forms compound nouns by combining smaller words into one bigger word.

Here:

• Haus = house / home
• Arbeit = work

Together:

• Hausarbeit

This is completely normal in German. English often writes similar ideas as two words, but German usually joins them into one.

Also remember: German nouns are capitalized, so Hausarbeit begins with a capital letter.

Yes, you could, but it is less specific.

• die Arbeit = the work / the assignment / the job
• die Hausarbeit = specifically the paper, assignment, or housework, depending on context

So Die Arbeit ist noch nicht fertig means The work isn’t finished yet, but it does not tell us exactly what kind of work you mean.

Hausarbeit is more precise if that is the intended word in the original sentence.