Learn Turkish: Rolü figürana ezberleteyim mi? - Shall I make the extra memorize the role?

This is the golden rule of the causative voice! The base verb ezberlemek (to memorize) is transitive, meaning it takes a direct object. In this sentence, that object is the role (rolü, which takes the accusative case). Because the accusative slot is already taken, the person being made to do the action (the causee, figüran) gets bumped to the dative case (-a).

How do you break down the suffixes in ezberleteyim?

It breaks down like this: ezberle- (to memorize) + -t (causative suffix: to make someone memorize) + -eyim (optative suffix: let me / shall I). Because the verb root ezberle- ends in a vowel, we use the -t causative suffix instead of -dir.

What is the function of -eyim mi together?

The -eyim ending is the first-person singular optative, which on its own means "let me [do it]". But when you pair it with the question particle mi, it becomes a request for input, translating beautifully to "Shall I...?" or "Should I...?". Here, it means "Shall I make [them] memorize?"

Why is it rolü with a front vowel (ü) when the word rol has a back vowel (o)?