Hello my name is Chris.Ive been playing league of legends since season 3 and i've picked up gragas in season 4 which helped me climb the ladder all the way up to high diamond.I've got many name changes so imma just put some of my names in there Coach Gragas , Rockstar Gragas , do u copy over , Toyota supra mk4.
This is my ultimate gragas guide with all the roles that i play him all the runes and item paths i use along with the few hard matchups which i personally find difficulties.

So what is AP gragas and why it is so good in most matchups and why it can carry games 1v9.
The most unique thing about gragas is that he has 4 AoE abilities , all of his abilities are AoE damage which means his damage output in teamfights has the potential to be gigantic if played correctly.
In the early game versus most matchups you need to poke with Q as much as possible and try to dodge every skillshot the enemy laner targets you with because you are a melee champion so in order to farm/cs you need to get close to the wave which puts you in threat so dodging skillshots is really really important , mouse movement must be fast as hell.If you see an opening to hit E without hitting any minions on the way go for the trade you will outtrade your opponent in most cases
Your burst on level 3 with Electrocute(domination rune), ignite and your 3 abilities is around 400 so you can basically kill anyone who gets half hp or so. if the enemy jungler ganks you and he is half or below you can instantly kill him with 3 abilities + ignite and walk away with a buff or two.On level 6 your kill potential jumps so high in the sky you can't imagine you can basically 100 to 0 pretty much every laner who's not running barrier.Just poke with Q once then wait for you Q cooldown and throw a Q on the minion wave then you have 3.5 seconds to drink your W and throw your R in the direction your Q is or just directly throw the R and the W after that.This is around a 900 hp burst which is insane that early in the game.Once you get penetration shoes(sorc shoes) look for roams especially bot lane.
Once you get your core items Protobelt Lichbane you can oneshot almost anyone in the game.
In teamfights try to flank or if you can't flank just wait for crucial abillities from the enemy team and go in with E flash on the person who deals the most damage in their team or catch of their bot lane duo usually squishy champs.

Pros
naturally tanky
good sustain in lane
good poke/trade potential
heavy cc
4 AoE abilities
ALOT OF BURST DAMAGE

Cons
3 skillshot abilites
falls off abit late game due to other champions scaling
melee in lane

If you are the person who likes oneshotting people for less than a second or using skillshots in particular order or you like his drunkass fat*** looks thats the champion for you.