Hey, i'm stefanko, s13 1327LP rank 1 EUNE and 1023LP challenger EUW. I've been rank 1 Jax world on league of graphs multiple times. I'm known for my unrivaled Jax pick, and i recently started streaming at https://www.twitch.tv/stefankolol I offer coaching(feel free to add me on discord stefanko#5246 if interested), and have over 1000 hours of coaching done and probably close to 10M mastery points on Jax, and i'd be happy to answer any of your questions on my stream.

When playing Jax, you should look to sidelane, get as much resources on the map as you can, be the highest level in the game, and play objective focused gameplay, fighting when enemies aren't close to their towers, so you have a lot of space to get good flanks and all ins in.

Pros:

-Huge 1v9 potential
-Great scaling
-Good mobility
-Great against autoattackers
-Great duelist, perhaps the best in the game

Cons:

-Has very hard matchups
-Mana issues
-Low sustain
-Needs to farm melee range
-Very bad in lane if frozen
-Needs resources
-Bad at sieging
-Weak if counterstrike gets kited

Jax early game warries a lot since there are many matchups you should approach differently. Generally you don't want to stay in the lane for too long as Jax is mana hungry and has no sustain. You usually want to have the wave on your side of the lane so you can all in the enemy. Jax has hard time last hitting if wave is pushing to him, so try to get prio level 1 if matchups allow you to so you are not forced to lose cs. Once you crashed the wave you can look to:
-Stay in the lane if you have a favorable state(are stronger than enemy, healthy and have mana)
-Make a play with your jungler if he is pathing to top
-Look for cheater recall if you are weaksided or low on resources.
Try to pay attention to your jungler's pathing and look to have good wave state and have your gold spent when he is about to make a play top side.

Once the lane is over you mostly want to sidelane and get resources on the map. Try to reset(spend your gold and get healthy) around 1 minute and 20 seconds before objective spawns so you are strong on the map when you need to pressure the map. You want to try to pressure sidelanes opposite of objectives around 30 seconds before objective spawns, and then get your team 5v4 by rotating or keep pushing sidelane if nobody comes. The way you wanna snowball on Jax is by taking resources on the map on cooldown. Only time you wanna group is if enemy overextends. It's really important to be strong on the map when objectives are about to spawn, so you can secure your team an objective or be ready to trade something else on the map if your team is not in a good state to contest the objective. Avoid sieging with your team as Jax, as it makes Jax(and other bruisers) very useless.

Jax works the best with junglers that don't do full clears, so he can take their jungle camps and receive ganks from them if lane isn't in a good state. Supportive junglers are very good to have, but Jax can work well with some full clear junglers asw, such as Eve or Karthus since they give you easy resets and force enemies to build mr. Having any enchanter on bot lane is also scaling really well into mid and late game with Jax.

There are few different scenarios when it comes to teamfighting:

-If enemy has ranged champs, look to find a good flanking spot where you can approach enemy team from close range, and wait for enemy to get out of position by going on your team, or getting into your face.
-If enemy has assasisns, look to stick to your team's squishy champs, as assasins need to go melee range on them.
-If your team has few enchanters, look to stick to them, avoid flanking where they can't follow up, as it's gonna make them useless.
-If your team has tanks or champs that can buy a lot of time, let them engage first, as Jax is really good at cleaning up.
-If you need to be the one that engages, look to buy stopwatch to buy time for your team.

Ultimate mechanics:

-You can cancel your ult animation with your e if you are not in leap strike animation
-You can cast ultimate and flash after to give enemies less time to react to your ultimate
-You can use your ultimate while your q is casting
-When you engage on someone and want to go for all in you usually want to q>e>aa>w>r as you should use your ultimate when your aa is on cooldown


Trading:

-When you are chasing someone who has flash, use your flash first and then q after he flashes for maximum q range and more optimal damage
-Try to e in the wave so your e damage gets ramped up by dodging minions autos when trading
-When someone gets out of your e range and you can't close the distance, fake walking back and then go forward again, it's gonna make enemy think you are backing off and they are gonna get baited into your e.
-Q on ranged champs when they are stuck in aa animation to make it easier to stick to them.
-Once you are level 6 always make sure to stack up 3rd aa ult passive before trading.
-W extends your aa range, so try to w enemy when they go for last hit on a minion.
-When you q on someone look to autoattack and then w+aa for maximum dps instead of using w instantly
-Look to auto attack minions to ramp up attack speed from your passive before you go for a trade or an all in
-Most of the matchups you can camp third lane bush and come behind enemy and look for extended trade while enemy cant kite your e, sometimes you can even deny enemy xp from 3 melees


Teamfighting:

-A lot of times in teamfights you can q on enemy frontline and cast your e. It's gonna make enemy backline think you are committing on frontline so they stop being scared from you and walk up, and once you stack up damage on your e, you can flash on enemy backline and look to oneshot them.

I went on runes and matchups in detail in build options and threats, so you can check them out there, i added notes to explain my decisions if you have any confusions.

Thanks for reading the guide, i hope it helped you. I look to update the guide frequently, but as i said you can always ask me anything you want on my stream at https://www.twitch.tv/stefankolol or on discord stefanko#5246
If you liked the guide, i would highly appreciate if you could give it a thumbs up :)