After the small rework, Jax now can deal a lot of dmg when you go AP.
This build will utilize all the strongest sides of Jax - you burst down an enemy in a sec - you are ****ing tanky - you dominate the early game - you are too annoying with your E.

My starting idea is that Jax is a perfect counter to the melee initiator meta. You don't have to care about Nautilus, Blitz, Thresh, Camille anymore. Just be Jax and play it simple.
After300+ games playing Jax Support among Plat/Diamond/Grandmaster elo, I finally gave you this guide to help you master a real weapon xD

You shoulda team-up with your jungler/toplaner at level 1 and secure the first blood before go back to your lane. You are strong, don't be scared!

In the early game, try to make some short trades with the enemy botlane. Dominate the lane and start roaming mid/secure Dragon or even invade enemy jungle with your teammates. Your E is a ****ing crazy tool to win any early fight, try to ultilize it as much as possible.

Also, don't forget to control the early game vision. Always check for dragonside hermit to prepare for a countergank in mind. You are the main initiator, but you are also a support. Please don't forget your position

Now it's time to be ****ing annoying!

After Zeke, you can start team-up to get some kills. You can also 1v1 any champion if you want to cuz you are ****ing tanky and you deal crazy dmg.

Try to gain as much vision as you can. You are the initiator, so that vision is the best way to choose whether to initiate a fight or not. Make a way for your teammates to follow up and Zhonya if needed. Always keep in mind one thing: It's "SURVIVAL OF THE FITTEST!" Every decision you make will result in the late game.

Well, I rarely make it longer than 35 mins so I'm not sure whether this may work or not.

In the late game, your main point is no longer initiate the fight but controlling it. You have the vision, you have the information of every aspects in your game. Now it's your decision. Once you go wrong, you lose the game.

Try to think carefully before you act. Always prepare for the worst to come when you make a wrong choice.

Just simply E1-Q-AA-W-AA-R-AA-E2-AA
Remember that your Q also works on trinket, ward, any targetable object (except for turret), ultilize it to escape, to close the gap or to make a suprise attack
One more thing, your W will reset your AA, this will help you deal lots of dmg in the early game xD