This is an example of the kind of identification, explanation, and demonstration, or justification, I expect you to provide in individual writing assignments and in group assignments where such explanations and demonstrations are called for. It demonstrates the difference between a transitive phrasal verb and an intransitive verb plus preposition. This is by no means the only topic we will study this semester, but the particular approach implied by the question, contrasting two similar subjects and discovering and explaining the significant difference between the two is characteristic of descriptive grammatical analysis.
Consider the following two sentences that resemble each other superficially but differ in structure. Explain the differences in structure using grammatical terms accurately, justify your explanation, using tests where appropriate, and diagram the sentences.
A. Hank lived down his unfortunate reputation.
B. Hank lived down the road to Blanco.
The sentences have different structures because lived down functions differently in each. In A. lived down is a transitive phrasal verb. In sentence B., however, lived is an intransitive verb and down is the preposition of an adverbial prepositional phrase.
This part is the explanation. It provides a description in grammatical terminology of the difference in structure and uses diagrams as visual illustrations. You don't have to structure your own explanations in precisely this way; for example, another kind of successful explanation might first explain the differences in structure in prose and then present the diagrams. The justification invokes various grammatical tests.
One way to distinguish the two structures is to consider the meaning. In sentence A. lived down functions as a single unit meaning something like "overcame." This can be shown by replacing lived down with a non-phrasal verb, for example: Hank overcame his unfortunate reputation.
In sentence B, by contrast, live and down don't form a single unit of meaning. Instead, the prepositional phrase (P + NP) forms a unit telling where Hank lived. It could be replaced by a single-word adverb (substitution test); for example: Hank lived there.
This is a good start, but although it uses grammatical terms accurately, it leans heavily on meanings we all share in common as English speakers. By supplying the examples that demonstrate that the synonym substitution test works, however, the writer has taken a step beyond simply invoking differences in meaning.
We can also demonstrate the differences in meaning by performing a number of structural tests. We know that sentence A contains a transitive phrasal verb, because the particle down can be moved to the right of the direct object (particle movement test) : Hank lived his unfortunate reputation down.
If you try this with sentence B, the result is an ungrammatical sentence, because down is part of the prepositional phrase and can't be moved out of it: *Hank lived the road to Blanco down.
Since A has a transitive phrasal verb, we can also use the pronominalization test. If you replace the direct object with a pronoun, the particle down automatically shifts to the right of the direct object: Hank lived it down.
When you replace the object of the preposition in B with a pronoun, the preposition stays in place: Hank lived down it.
If you try to move the preposition in B, however, the result is a sentence that seems like it has to have a phrasal verb, and the meaning doesn't correspond to the meaning of sentence B. Instead, it corresponds to the meaning of A: Hank lived it down.
These justifications, which use more sophisticated grammatical tests, support the analysis even more firmly. Note in particular that applying a test like the particle movement test to a structure where you know or suspect it won't work is often a good way to test your analysis. A sentence like *Hank lived the road to Blanco down. is pretty solid proof that lived and down are not part of the same grammatical unit.
We can demonstrate that down the road to Blanco in sentence B is an adverbial prepositional phrase by moving it to the front of the sentence, since adverbial prepositional phrases can often be moved for emphasis or stylistic variation: Down the road to Blanco Hank lived.
Moving lived and his unfortunate reputation, however, produces a sentence that is ungrammatical, because down is part of the phrasal verb, while his unfortunate reputation, the direct object, is a separate grammatical structure: *Down his unfortunate reputation Hank lived.
This final section really clinches the case by demonstrating that a test that identifies adverbials works on sentence B but not on sentence A. The explanation is especially sophisticated because it brings up the idea that down and his unfortunate reputation are separate grammatical structures, or different constituents.