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EARLY-GAME
Usually the support stays with the AD Carry at bottom lane.
They are there because a lot of reasons
They don't kill minions, letting the AD Carry farm faster and Gear up
They make the support for the ad carry, so the AD Carry can be more
offensive. (Slow the opponent, heal the ally, buff the ally, etc)
As support, they usually take full responsibility for warding their
lane. Letting the AD Carry gold for gear only.
MID-GAME
They buy more wards than their teammates. So they can ward more of the map (How do I ward properly in League of Legends?), also they usually buy Oracles so they can hunt down enemy wards (Jungles also usually buy oracles, because for a Jungler it's easier to walk around looking for enemy wards).
END-GAME
At this point your function is to keep your group alive as much as you can. Make the opponent lose their focus and stop attacking your allies (knocking up, silence, stun, etc). Keep Warding and if possible keep an Oracle activated.
NOTES:
To be a good support, you need to:
Learn to ward as much as you can (also, where you should be warding)
Keep a good sync with your AD Carry. Some people are more aggressive
others are more defensive, try to learn which one is your partner, and
try to follow him.
Stay tuned in with your team, especially your lane partner and your
jungler.
You don't need this or that summoner spell (People tend to think
that CV or Heal is a must have for a good support, and that is not the
real deal). Try and learn, CV is amazing tool for Jungle invading, so
if you and your jungle have a good sync, you can get this, and make a
living hell for the enemy jungler. Heal is usually taken by the Carry,
but some prefer two heals (carry + support) or even none (I usually
play lulu with exhaustion, for better chasing)
If you fall behind, don't blame yourself too much, a good support is
not so good without a good partner (jungle and/or your lane partner)
Remember, the jungler and you must have a good synergy, he will be
warding a lot, so try not to make you or him waste a ward (double ward
happens sometimes, be careful)
You don't have to be Janna, Soraka, Taric or Sona to be a good
support. Lulu, Leona and a lot more (like my favorite, Galio) are
great supports, if they are played and mastered.
Gold Mastery and Gold Runes are very important. Remember you are not
farming minions, and you still need to buy your gear, oracles and wards.
