Dragon's breath is a second-level transmutation spell from Xanathar's Guide to Everything (p. 154) which allows a creature to spew magical energy from its mouth (sort of like a dragonborn's breath weapon). And School of Evocation wizards gain the Empowered Evocation feature at 10th level, allowing them to add their Intelligence modifier to one damage roll of every evocation spell they cast.
From the transmutation label it seems clear that this bonus wouldn't apply to the spell dragon's breath; however, one of my players was adamant about the attack-part being "quite obviously evocation-based", and I felt she made a pretty good case at the table when she went into an epic rant about how:
Sure, the concentration part of the spell is definitely transmutation, but the attack isn't transmuting anything! It's just a cone of pure elemental magic. So unless you want to be silly and say that blasting flames on a bunch of kobolds falls under the school of transmutation because they were 'transformed' into a smoking pile of ashes - we should acknowledge that the cone of energy created by dragon's breath is really no different from the cone produced by burning hands.
Now, I couldn't back this up with the rules at the table, and I'm actually still having a hard time formulating a good justification using RAW. but since she made such a strong case logically, I let her do it. To my mind, even if I was in the wrong, what's the big deal really? Five damage points? Anyway, now I'm in trouble with a guy in my group who sent me an email accusing me of favoritism.
So, could you guys help me find appropriate page numbers which describe evocation spells as being the kind where you blast your foes with energy?