This passage occurs as the Transfiguration Sunday Gospel in the Revised Common Lectionary (Year A) and Narrative Lectionary (Year 1), most recently February 2017.
Obviously a familiar and beautiful passage. Matthew lets us know that this event occurs "six" (hex) days after the first messianic prediction. This is the only time in the Gospels that anything happens six days later. Why? The last time we found something happening on the six day was the creation of humans, which the Bible calls good; in fact, very good. Peter likewise calls it "good" to be on the mountaintop. The sixth day of creation was good, but it was not the ultimate day; the 7th was and is. In the same way, the transfiguration is a good day. Moses and Elijah, the law and the prophets, are great and to be celebrated. By they are not the ultimate; Jesus is. Likewise, turning bright as light is good and to be celebrated. But it is not the crucifixion and resurrection Jesus for which has come.
εξ ("six" -- there is a rough breathing mark over the e, so this word is read "hex" like "hexagon"; 17:1): This is the only event that occurs "six" days after something in any of the Gospels or in the whole Bible. The last event is a series of teachings in which Jesus foretells his death and resurrection and Peter rebukes him. So why six? In the Bible six often refers to incomplete (yet not entirely bad!) things, chiefly creation. Transfiguration is good. But not ultimate.
αναφερω ("took up" or "sacrificed"; 17:1): This word literally means "take up," but is often used to describe the action of the priest in sacrifice. It is also used for Abraham taking up Isaac to Mount Moriah. Is Jesus taking up his disciples for a sacrifice? Is he sacrificing them? I think in this case, the verb probably just means "took up" but an interesting connection.
μεταμορφομαι ("transfigured" or "metamophisized"; 17:2): The Latin "transfigured" is not as "cool", imho, as the Greek "metamorphisized." This word is fairly rare in the NT. It also occurs in Romans 12:2 (Do not let your minds be confirmed to this world, but transformed by the renewing of your minds...) and 2 Corinthians 3:18, which reads something like "Shine, Jesus, Shine." Transfigured sounds so churchy. Try "transformation" or "metamorphosed" as see what reaction you get. "Transformer Sunday"
φος ("light"; 17:2 see also 5:14). Jesus called his disciples to be the light of the world; a city on a hill cannot be hidden. In this passage we again have light on the hill, but this time it is Jesus himself. The NRSV covers up the literal phrase, "white as light," which is too bad because it is one of the few times, outside of John, that Jesus is referred to as light. Even the angel at the resurrection (28:3) will not be bright as light!
αγαπητος ("beloved"; 17:5; 12:18; 3:17): This phrase harkens back to Jesus baptism. It also reaches back to the prophet Isaiah and the love song for the beloved. (A few times God calls Israel his beloved). Most significantly, it leads us back to Abraham and his near sacrifice of Isaac, his beloved son. Baptism, sacrifice, a mountaintop, God's promises to Abraham. Something Lutheran stirs in these waters...
αψαμενος ("touch"; aorist participle of απτω; 17:7): I find it interesting that Jesus touches them. I had missed that before. I think it greatly softens Jesus words. He touches them. Tells them to arise and not be afraid. We often remember his words at the end of the story, not to tell anyone, but this is a powerful gesture by Jesus: to uplift with his touch and his words.
οραμα ("vision," 17:9): The NIV probably gets this right by translating it "what you have seen" instead of vision, because vision for most of us sounds like something made up. Freiburg Lexicon says, (1) literally what is seen, appearance, spectacle; (2) in the NT a supernatural vision, given as a means of divine communication, to be distinguished from a dream (οναρ)
εγερθη ("stand up" or "resurrect"; aorist passive of εγειρω; 17:7 &9) Jesus uses the same verb for talking about his resurrection as he does to tell the disciples to "stand up." Jesus tells them to stand up. And then he tells them he will "stand up." Jesus resurrection leads to our own resurrection too.
Grammar: The quick and easy circumstantial participle
A number of verses in this section have easy circumstantial participles. 17:7 for example, puts one right in the middle of the sentence (after the και)
και αψαμενος αυτων ειπεν
first step: plug in English words in "untranslated format." I will put an * by the part that we need to clarify in order to translate.
and touch* of them he said
It turns out that the "he said" is the main part of the sentence. The αψαμενος αυτων is the participle
The participle is in the aorist, which means it happened before the other verb. So
"touched of them, he said."
We need to clean up the word "touched" but two things are tricky. First, the verb is in the middle voice. Don't worry about that. He did not touch himself; what languages consider "middle voice" varies. In this case, we can translate this as an active voice, "touch." Second, αυτων is in the genitive simply because this verb takes a genitive object. So
"and touched them, he said."
Now we figure out who is doing the action
Here it should be obvious that Jesus touched them. You could also check that the participle is in the nominative, which means the subject of the rest of the sentence is doing the action...who is Jesus.
Then we add in the circumstance
"and after he touched them, he said"
Consider also 17:9
επαραντες δε τους οφθαλμους αυτων ειδον...ει μη...
Here again we have a circumstantial participle. Step one, fill in English that you know
look up* and the eyes of them they saw...
Once you figure out that ειδον = they saw = the main verb of the sentence, you should be able to move quickly through this participle. Indeed, your brain can probably figure out the actual reading:
"After lifting their eyes they saw..."
You could work through this in sequential steps:
Fix tense: "lifted their eyes they saw"
Fix voice...already done
Figure out who -- the disciples! (Again, you can check the case and number, but disciples makes sense!)
Then add circumstance. Since it is aorist, it happened first...
"After lifting their eyes they saw..."