Learn Turkish: Genç öğrencileri resim yapmaya teşvik ettik. - We encouraged the young students to paint.

The verb teşvik etmek (to encourage) requires the dative case (the "to" direction) for the action you are encouraging someone to do. First, we turn yapmak into a noun by swapping -mak for the short infinitive -ma (yapma). Then we add the dative suffix -a using a buffer y, giving us yapmaya (literally "to the doing").

Why is there an -i at the end of öğrencileri?

That is the accusative case suffix. The main verb teşvik ettik needs a direct object to show who we encouraged. Because "the young students" are specific, defined people, we must mark them with the accusative case. So it breaks down as öğrenci (student) + ler (plural) + i (specific direct object).

Why doesn't resim have a suffix like öğrencileri does?

In Turkish, resim yapmak (literally "to do picture") is a general phrase meaning "to paint" or "to draw." Because the students are just making art in general, and not painting one specific, previously mentioned picture, resim acts as an indefinite object and stays in its plain dictionary form.

How does teşvik ettik mean "we encouraged"?