Der zweite Tagesordnungspunkt betrifft die Hausarbeit, und dazu gibt es auf jeder Folie ein Beispiel.

Because this noun phrase is the subject of the first clause, so it is in the nominative case.

• der = nominative masculine singular
• zweite = adjective/ordinal with the weak ending -e after der
• Tagesordnungspunkt is masculine, so the whole phrase is nominative masculine singular

So:

• Der zweite Tagesordnungspunkt = The second agenda item

If it were a direct object, you would expect den zweiten Tagesordnungspunkt instead.

Tagesordnungspunkt is a very typical German compound noun.

It is made of:

• Tagesordnung = agenda
• Punkt = point / item

Together, Tagesordnungspunkt means agenda item.

A helpful detail:
Tagesordnung itself is also a compound:

• Tag = day
• Ordnung = order

Historically, Tagesordnung is something like order of the day, which became agenda.

German often creates long nouns this way, and the last part determines the gender. Since Punkt is masculine (der Punkt), der Tagesordnungspunkt is masculine too.

betreffen means to concern, to affect, or to relate to.

In this sentence:

• Der zweite Tagesordnungspunkt = subject
• betrifft = verb
• die Hausarbeit = direct object

So betreffen takes the accusative case.

That is why we get:

• die Hausarbeit

This form could be nominative or accusative, because Hausarbeit is feminine:

• nominative: die Hausarbeit
• accusative: die Hausarbeit

So the form does not change, but the function does: here it is the accusative object of betrifft.

It can, but Hausarbeit is a word with a few possible meanings depending on context.

Common meanings include:

• homework
• term paper
• written assignment
• sometimes even housework in other contexts, though that is less likely in modern academic language

In a context involving slides, agenda items, and examples, many learners would suspect it means something like:

• written assignment
• term paper
• course paper

So Hausarbeit is a word you usually interpret from context rather than translate mechanically the same way every time.

Here dazu means something like:

• about that
• on that
• for that
• regarding that

It refers back to what was just mentioned, probably die Hausarbeit or the topic of that agenda item.

So:

• dazu gibt es auf jeder Folie ein Beispiel

means roughly:

• there is an example for that on every slide
• there is an example about that on every slide

A useful way to think about da- words is:

• da- = referring back to something already mentioned
• zu = the preposition involved

So dazu = to it / for it / about that, depending on context.

This is a word order question.

In a German main clause, the finite verb must be in second position. The first position can be taken by many different elements, not just the subject.

Here, dazu is placed first for emphasis or smooth connection to the previous clause:

• und dazu gibt es auf jeder Folie ein Beispiel

Structure:

• position 1: dazu
• position 2: gibt
• then: es
• then the rest

So dazu gibt es is completely normal because German is following the verb-second rule.

You could also say:

• und es gibt dazu auf jeder Folie ein Beispiel

but the version with dazu first sounds very natural because it links strongly to the previous idea.

In es gibt, es is a fixed part of the expression meaning there is / there are.

So:

• es gibt ein Beispiel = there is an example
• es gibt Beispiele = there are examples

This es does not really mean it in the normal sense. It is part of the standard German way to express existence.

Because the sentence begins with dazu, the verb comes second, and es follows it:

• dazu gibt es ...

That is just normal German word order with the expression es gibt.

Because auf can take either accusative or dative, depending on meaning.

• accusative = movement toward a destination
• dative = location, where something is

Here the meaning is location:

• the example is on each slide
• it is not moving onto each slide

So German uses the dative:

• auf jeder Folie

Since Folie is feminine singular:

• dative feminine singular = jeder Folie

Compare:

• Ich lege das Blatt auf die Folie. = movement, so accusative
• Das Blatt liegt auf der Folie. = location, so dative

Because jede changes its ending according to case, gender, and number.

Here:

• Folie is feminine singular
• the phrase is in the dative
• so jede becomes jeder

That gives:

• nominative feminine: jede Folie
• accusative feminine: jede Folie
• dative feminine: jeder Folie

So auf jeder Folie is exactly what we expect for a feminine noun after a two-way preposition used in the locational sense.

Because Beispiel is a neuter noun:

• das Beispiel

With a neuter noun, the indefinite article in nominative or accusative singular is:

• ein

So:

• ein Beispiel = correct

You would use einen only with a masculine accusative noun, for example:

• einen Punkt

A small extra point: after es gibt, the noun phrase is grammatically in the accusative:

• es gibt ein Beispiel

But because Beispiel is neuter, the accusative form is still ein Beispiel, not einen Beispiel.

Yes. The structure es gibt takes an accusative object.

Examples:

• Es gibt einen Plan.
• Es gibt eine Folie.
• Es gibt ein Beispiel.
• Es gibt viele Fragen.

In your sentence, ein Beispiel is therefore accusative. It just happens to look the same as nominative because it is neuter singular.

German punctuation can be a little flexible here.

In general, when und connects two main clauses, a comma is often possible, especially if it makes the sentence easier to read. That is what is happening here:

• Der zweite Tagesordnungspunkt betrifft die Hausarbeit
• und dazu gibt es auf jeder Folie ein Beispiel

These are two full clauses, so the comma helps separate them clearly.

You may also see similar sentences without a comma before und, especially in simpler writing. But with longer clauses, many writers prefer the comma for clarity.

Yes. Both are natural.

• der zweite Tagesordnungspunkt = a compact compound noun
• der zweite Punkt der Tagesordnung = a more expanded version

They mean essentially the same thing:

• the second agenda item

German often prefers compounds in formal or administrative language, so Tagesordnungspunkt sounds especially typical in meetings, presentations, minutes, and official contexts.

Yes, it sounds fairly neutral-to-formal, especially because of words like:

• Tagesordnungspunkt
• betrifft
• dazu gibt es
• Folie

It sounds like something you might hear in:

• a presentation
• a meeting
• a seminar
• a lecture
• official or academic discussion

It is not overly stiff, but it is definitely more formal than casual everyday conversation.