Learn Hardcore Swahili: Mti mkubwa uliokatwa jana utatumika kutengeneza meza na viti. - The big tree that was cut yesterday will be used to make a table and chairs.

In Swahili, descriptive adjectives normally come after the noun.
So mti mkubwa is literally tree big, but it means a big tree / the big tree.
You cannot normally say mkubwa mti; that would be wrong or at least very odd.
This noun–adjective order is very regular in Swahili: mtu mzuri (good person), nyumba ndogo (small house), kitabu kipya (new book), etc.

What exactly is going on inside the word uliokatwa?

Uliokatwa is a relative verb form meaning that was cut. It is built like this:

u- = subject prefix for noun class 3 (matching mti)
-li- = past tense marker
-o- = relative marker (roughly “that/which”)
katw- = passive verb root from kata (to cut)
-a = final vowel

So mti mkubwa uliokatwa literally is the big tree which was cut.

Why do we use a passive form (uliokatwa) instead of an active one?

The focus here is on the tree undergoing the action, not on who did the cutting.
Uliokatwa (was cut) is the passive of ulikata (cut).
If you wanted an active meaning like the big tree that cut (something), you would use uliokata, from the active root kata without the passive -w-.
In normal contexts for trees, the passive is what makes sense: they are cut by someone.

How does uliokatwa agree with mti mkubwa in terms of noun class?

Mti is in noun class 3 (m-/mi-).
Class 3 uses u- as its subject prefix in the singular, which is why uliokatwa starts with u-.
So: mti → u- in uliokatwa and later utatumika.
If the noun were plural, e.g. miti mikubwa, the agreeing prefixes would change: iliyokatwa / itatumika instead of uliokatwa / utatumika.

What does utatumika mean exactly, and how is it formed?

Utatumika means will be used. It is formed like this:

u- = subject prefix for class 3 (matching mti)
-ta- = future tense marker
tumiki- = verb stem from kutumika (to be used, to serve)
-a = final vowel

So mti ... utatumika = the tree ... will be used (not “will use”).
Compare: atatumia = “he/she will use (something)”, but atatumika = “he/she/it will be used” or “will serve (a function)”.

Why do we have kutengeneza after utatumika, and why is it in the infinitive?

Swahili often uses the infinitive (ku- + verb) after verbs like kutumika, kuanza, kupenda, etc., to express “to do something”.
Here utatumika kutengeneza ... literally means “will be used to make ...”.
Kutengeneza is the infinitive ku- + tengeneza (to make, to manufacture, to repair).
So the pattern is: [be used] + [infinitive] = “be used to do X”.

Why is jana (yesterday) placed after uliokatwa and not somewhere else?

In mti mkubwa uliokatwa jana, jana clearly modifies the cutting, not the future using.
So the structure is: the big tree that was cut yesterday will be used...
You could also say Jana mti mkubwa uliokatwa ... to emphasize “yesterday”, but the original word order is very natural.
If you moved jana to the very end (... utatumika kutengeneza meza na viti jana), it would sound like the using happens “yesterday”, which does not match the future tense utatumika.

How does this sentence express “the big tree” versus “a big tree” when there is no word for “the” or “a”?

Swahili does not use separate words for “the” or “a/an”.
The noun mti mkubwa can mean a big tree or the big tree, depending on context.
In this sentence, the extra information uliokatwa jana (that was cut yesterday) makes it sound like a specific tree, so English naturally uses “the big tree”.
If we were talking about any random big tree, context or extra wording would make that clear instead.

What does na mean in meza na viti, and can it also mean “with”?

Here na means “and”, so meza na viti is “(a) table and (some) chairs”.
Yes, na can also mean “with” (e.g. anakuja na rafiki yake – “he is coming with his friend”).
The meaning and/with is decided by context: between two nouns of the same type (like meza and viti), it’s almost always “and”.

Why doesn’t kutengeneza meza na viti have any object marker on the verb?

Object markers in Swahili are only used when the object is already known, specific, or being emphasized, often like “it/them/him/her”.
Here, meza na viti are just straightforward objects introduced for the first time, so there is no need for an object marker.
If they had been mentioned before, you might say something like kuvitengeneza (to make them) where -vi- refers back to the known objects (class 8).

Could I use ambao instead of the relative form in uliokatwa? For example: Mti mkubwa ambao ulikatwa jana ...?

You can, and people will understand you: Mti mkubwa ambao ulikatwa jana utatumika ... is grammatically possible.
However, everyday Swahili usually prefers the relative prefix inside the verb (uliokatwa) instead of ambao for this kind of simple subject-relative.
Ambayo / ambao / ambayo, etc., are more common in formal writing, emphasis, or complex sentences.
Using uliokatwa is more native-sounding and more concise here.

How would this sentence change in the plural: “The big trees that were cut yesterday will be used ...”?

You would make the noun and all agreeing parts plural:

mti mkubwa → miti mikubwa (class 3 → class 4)
uliokatwa → iliyokatwa (subject prefix i- for class 4, not u-)
utatumika → itatumika (again, i- for class 4)

So the full sentence:
Miti mikubwa iliyokatwa jana itatumika kutengeneza meza na viti.
Notice how mti / miti, mkubwa / mikubwa, uli- / ili-, and uta- / ita- all change together.

Why is the adjective mkubwa and not just kubwa?

Many adjectives in Swahili take a noun-class prefix.
For class 3 singular (mti), the adjective -kubwa takes the prefix m-, giving mkubwa.
Other examples with class 3: mti mrefu (tall tree), mti mnene (thick tree).
In the plural class 4 (miti), the prefix becomes mi-, so you get miti mikubwa (big trees).