This is what I think they are

1."Leaving their native land to colonize in America," they expected to land in the colony of Virginia.
A.prepositional phrase B. appositive phrase c.participial phrase d.gerund phrase e.infinitive phrase f.absolute phrase

2."Steering a northerly course," the captain sailed the ship to the rocky coast of New England instead.
A.prepositional phrase B. appositive phrase c.participial phrase d.gerund phrase e.infinitive phrase f.absolute phrase

3."Landing in New England" caused much disappointment among the Pilgrims.
A.prepositional phrase B. appositive phrase c.participial phrase d.gerund phrase e.infinitive phrase f.absolute phrase

4.Today, homes "on Cape Cod" still display this famous weathered look.
A.prepositional phrase B. appositive phrase c.participial phrase d.gerund phrase e.infinitive phrase f.absolute phrase

I think 1.a 2.a 3.b 4.b

8 answers

All are incorrect, sorry. =(

Notice that 1, 2, and 3 all start with words ending in -ing. No preposition has -ing at the end.

Go back and rethink all of these, please.
http://www.jiskha.com/display.cgi?id=1385695339
So now I think 1,2,3 are all gerund phrases because a gerund is a verb form that ends in -ing
Almost right! Yes, #3 is a gerund phrase. Here's why:
Landing in New England is serving as the subject of the main verb, caused.

In 2 and 3, however, both -ing phrases are describing (modifying) the subjects of the sentences.
Who was "leaving their native land"?
Who was "steering a northerly course"?

Let me know what you think for 2, 3, and 4.
Sorry ... #3 is a gerund phrase. #s 1 and 2, as well as 4, are the ones you need to rethink.
I would say 1.b 2.e 4.e
Infinitive phrases begin with "to" verb forms: to walk, to read, etc.

I don't see any absolute phrases here.
1.e 2.b 4.a
You're pretty much guessing on 1 and 2.

4 is now correct.