Thank you for giving me the opportunity to assist you. I will give the best answer that I can with the information provided.
If you did not include your stock trades on your 2009, the IRS will just look at the amount cashed in and tax you on that amount.
Whenever you sell stock, you must include the sales on your return, and show the date acquired, date sold, amount of cost basis, and the amount sold for. This needs to be reported on Schedule D.
Since you co-owned the stock, your basis will be a little tricky, but not impossible to figure out. Your father's basis in the stock will be what he purchased it for (or possibly different if he co-owned with your mother). When he added you as a co-owner in 2005, you assumed 1/2 of the ownership and your basis in that 1/2 is your Dad's basis. Then, when he passed away, the basis in the inherited half is the Fair market value when he passed away.
For example, He buys the stock for 20,000 in 2000. Then he adds you in 2005. Your basis in your half is $10,000. When he passes away, the stock is worth 28,000. So you inherit his half at the value of $14,000. So your basis in the whole lot is $24,000. When you sell for $31,000 your capital gain is a long term $6000. You could even have a capital loss if the stock declined in value.
You need to tell the IRS that you will file an amended 2009 return ASAP. Include the Schedule D showing the stock sales. You will need to pay any balance due with the amended return. I recommend sending any correspondence/amended returns certified mail.
you should also amend your NJ state return.
Please let me know if you need additional assistance.