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Based on the following statement about spells then they should be able to Know a spell with Metamagic already applied to it.
"These new spells can be common spells chosen from the sorcerer/wizard spell list, or they can be unusual spells that the sorcerer has gained some understanding of through study."
Magic Missile is a spell.
Empowered Magic Missile is a spell modified by a meta-magic feat.
Your spells known is appropriate for the former, but not the later.
As to your question about 'unusual' spells, that is up to your GM. He could allow you to research a new spell, we'll call it 'Magic Rocket' that has mechanical effects exactly like an Empowered Magic Missile.
Personally, as a GM I would approve that, but your GM will have to make the call. If it is PFS, it wouldn't be allowed.
"Unusual Spells" is addressed in the section on Sorcerer Repertoire. It covers new, unique spells learned at GM discretion, not spell+metamagic. Of course, with GM permission anything's possible.
The feat Spontaneous Metafocus allows normal metamagic casting times.
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Based on the following statement about spells then they should be able to Know a spell with Metamagic already applied to it.
"These new spells can be common spells chosen from the sorcerer/wizard spell list, or they can be unusual spells that the sorcerer has gained some understanding of through study."
If allowed by a GM, you can create a variant spell, like Empowered Burning Hands, maybe call it Cinder's Fan, but using up one of your known 3rd level spell slots. As a new spell, if approved, it wouldn't require you to possess the feat Empower spell, but a DM may exclude it from being further Empowered. Remember though, you are using one of your known spell slots which you can change out a single spell from every time you level up (one level lower than your max allowed spell level), not at every rest like prepared casters.
With Great Power comes Great Limitations!
Thank you this will allow what I am wanting to do.
Not by the basic sorcerer; empowered magic missile is not a separate spell, though you might research a spell like it.
There's at least one feat for a sorcerer to prepare a spell with metamagic in one or more spell slots to speed casting (not spells known slots) and an arcanist can do this. A sorcerer with the arcane bloodline can ignore the extended casting time a few times a day from 3rd level. Quickened spells ignore extended casting time from any metamagic.
The CRB is rather restrictive about it:
Spells: A sorcerer casts arcane spells drawn primarily from the sorcerer/wizard spell list presented in Chapter 10.
Now this list was expanded by other books, and there is the "unusual spells that the sorcerer has gained some understanding of through study" loophole. A GM can allow you to learn a new spell that's basically empowered magic missile.
However, it's usually not a good deal. Sorcerers have a very limited amount of known spells already. With spending a precious known spell on something that could be done via metamagic you cut down your versatility, in favor of a (tiny?) advantage in action economy. You could have learned fireball, haste or fly instead, to stick with the 3rd-level example...
So what's your intention? Do you want to stick with magic missile as your primary spell?
all effects of spells are done at the end of casting it. not the start .(unless said otherwise in the spell itself.) such as which creature to summon. what energy to resist. targets etc.
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Another feat is Spontaneous Metafocus.