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To the question «Worauf haben Haustiere eine positive Wirkung?», the answer in the book is «Die Haustiere haben eine positive Wirkung auf das Wohlbefinden.».

I thought in the case of "Wo", we use the case "auf" + Dativ. However, in the sentence above, they are using "auf" + Akkusativ. Why is that?

I know that "auf" has Dativ and Akkusativ, depending on the context. I just on this one we were supposed to use the Dativ...

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In the sentence above, auf das Wohlbefinden does not describe the place. It describes on what pets have effect. There is actually direction involved (there is some effect that goes from the pets to the well-being, so to speak). Taking this into account, accusative is the logical case here.

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I think the question relies on a confusion. Namely, altough worauf aludes to wo, it actually has nothing to do with it. Worauf is rather one of the wo+preposition questions like woran, worüber, wonach,... They are not necessarily related to "place", rather wo- is a placeholder that allows you to primarily ask for the piece of information the preposition will be tied to in the answer.

What you have to care about here is the verb+preposition+case (or noun+preposition+case) triple. This means usually first learning by heart, then trying to make sense of why it's like that.

In this case, wirken + auf is always accusative. So remains when you nominalize, Wirkung+auf+Akkusativ or Wirken+auf+Akkusativ.

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Your right that auf is a Wechselpräposition, and normally the rule is that if location is a destination then you use accusative, and if it's the scene of the action then you use dative. With certain phrases and verbs however, this rule goes out the window and you just have to memorize the case since there is no such logic. For example in the prepositional verb antworten auf you use the accusative: Ich antworte auf Ihren Brief. I think that's what is going on here except that in this case it's a noun Wirkung instead of a verb. I hesitate to call it a prepositional phrase because that means something else, prepositional noun maybe? But the point is that this phenomenon occurs with parts of speech other than verbs. I already had stolz auf in my notes, which involves an adjective and which takes the accusative case as well: Ich bin stolz auf dich. I admit that my notes did not include any nouns but I guess it's time to add a new category.

There are dozens if not hundreds of these "prepositional combinations". I wouldn't call them idioms because their meaning is clear from the literal translation, but they have an idiom-like quality in that their grammar can be contrary to the usual rules. I didn't see this particular example documented in the dictionaries I normally look at, but DWDS has a substantial usage database from which you can glean similar examples. This search lists all sentences in their subtitle corpus with the combination Wirkung auf and I'm guessing about half are applicable here.

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