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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?

From SRD

Evocation spells manipulate magical energy to produce a desired effect. Some call up blasts of fire or lightning. Others channel positive energy to heal wounds.

Your spell does not manipulate magical energy to produce a desired effect, but:

Transmutation spells change the properties of a creature, object, or environment. They might turn an enemy into a harmless creature, bolster the strength of an ally, make an object move at the caster’s command, or enhance a creature’s innate healing abilities to rapidly recover from injury.

The Dragon's Breath spell change the affected creature inability to use breathe weapon (as in dragonborn's racial trait) to be able to do so. Although the destructive effect similar to what some evocation spell does, breath weapon is not evocation spell.

Using the same logic, breath weapon granted by Dragon's Breath spell is only similar to what some evocation spell does, but it is actually not. Thus, the breath weapon does not benefit from Empowered Evocation.

Note that dragon's and dragonborn's breath weapon is not even considered magical. It is natural for them to be able to breathe destructive energy. The breath weapon given by Dragon's Breath spell grants you the ability to breathe destructive energy, albeit magically, not naturally (thus surpressed in an anti-magic field, unlike dragon's and dragonborn's breath attack).

Note to self: narrate the spell as giving them a dragon's gland to breathe fire/acid/whatever.

Empowered Evocation would not apply

Empowered Evocation only takes effect when you cast an Evocation spell (emphasis mine):

to one damage roll of any wizard evocation spell you cast.

Even if the spell effect of the Transmutation spell dragon's breath could be described as an evocative effect, it would not be an Evocation spell. The ability to use the exhale action is a consequence of the spell, not a spell itself.

For another example, take the dragonborn's Dragon Breath racial trait. This is a mostly congruent effect that is certainly not an Evocation spell even though the effect can be highly similar to spells like burning hands.

"The Schools of Magic" appears as a sidebar in the Spells chapter of the Basic Rules or the Player's Handbook. It describes evocation spells as:

spells [that] manipulate magical energy to produce a desired effect. Some call up blasts of fire or lightning. Others channel positive energy to heal wounds.

The dragon's breath spell says:

imbue it with the power to spew magical energy from its mouth...

... but it is unclear if this magical energy is actually being "manipulated" by the creature, the spell, or is just a virtue of the transformation.

As such, it could be argued that the dragon's breath spell does not even qualify as evocative in nature It could just be changing the physiology of the target to be capable of performing an action similar to a dragon's breath weapon which is certainly fantastical, but not magical in a game-terms sense.

Since the character cannot normally breath fire, it is a transmutation spell. The spell is not "evocating" itself, it transmutes the target to be able to breath fire, hence the 1 minute duration.

It seems you have a rules lawyer, except even worse, as they do not seem to be using the rules. Remember, you can't combine real world and in-game rules, otherwise you get stuff like the Peasant Railgun.