A Language of Dragons by S. F. Williamson

Vivien Featherswallow is willing to do anything to accomplish her goal of attending university to become a dragon translator, even if it means betraying her best friend. 

When her family is arrested all her plans fall apart. Vivien in her first act of rebellion ever, releases a dragon and starts a war.

Now she must work for the Prime Minister in order to save her family.

Teenage misfits with nothing to lose are recruited to work in Bletchley Park.

The secret assignments are in codebreaking, aviation and zoology in hopes of winning the war against the dragon rebels.

Will Vivien do what is right or what is being asked of her to save her family?

This is a great start to a YA fantasy series with lots of dragons sure to appeal to teens and adults alike.