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When I was creating my Bard with a standard array, I considered making constitution my lowest.
I know that starting HP = the highest number of your class' hit dice + CON mod (so 8 + CON for a Bard)
But I'm wondering if that formula still applies if the modifier is negative. My friend says it doesn't, and I've tried looking it up, but all I get is if it affects your HP leveling up, not your staring HP
The rules on "Hit Points and Hit Dice" state:
[...] At 1st level, your character has 1 Hit Die, and the die type is determined by your class. You start with hit points equal to the highest roll of that die, as indicated in your class description. (You also add your Constitution modifier, which you’ll determine in step 3.) This is also your hit point maximum. [...]
Nowhere does it make any exception for negative Constitution modifiers. In fact, I don't know if there is any feature where you add a modifier only if it is positive.
Instead of ignoring negative modifiers, feature often impose minimums on the total such as when gaining health at level other than 1:
Each time you gain a level, you gain 1 additional Hit Die. Roll that Hit Die, add your Constitution modifier to the roll, and add the total (minimum of 1) to your hit point maximum.
Notably, a negative Constitution modifier still applies it just also gets effectively ignored if the modifier is sufficiently negative to change the total to a non-positive number.
Applying this sort of minimum to your starting hit points would be awkward to phrase, getting you something like "minimum of the maximum of your hit die", which clearly isn't the best wording (though that doesn't particularly mean much).
Because nothing explicitly states that a negative Constitution modifier doesn't apply and nothing states a minimum on your starting hit points, your Constitution modifier can actually decrease your starting hit points.
There might actually at least one feature that specifies a minimum on the modifier to a roll. The Artillerist Artificer's Protector Eldritch Canon states:
[...] 1d8 + your Intelligence modifier (minimum of +1)
It looks like this minimum applies to the modifier and not the total, though perhaps this feature is just weirdly worded and has a "+" to denote positivity for some reason... Regardless, such a wording does not appear in the section on starting hit points.
When a spell or ability overrides the normal value derived from your ability modifier it will say so. For example, Reckless Abandon for Barbarians:
Beginning at 6th level, when you use Reckless Attack while raging, you also gain temporary hit points equal to your Constitution modifier (minimum of 1). They vanish when your rage ends.
Unless there is wording to suggest otherwise, your modifier, even if negative, is fully applied.
The rules concerning starting hp state:
You start with hit points equal to the highest roll of that die, as indicated in your class description. (You also add your Constitution modifier, which you’ll determine in step 3).
When we gain a level, as you have observed, the negative CON modifier still applies, but is floored at 1 hp gained:
Each time you gain a level, you gain 1 additional Hit Die. Roll that Hit Die, add your Constitution modifier to the roll, and add the total (minimum of 1) to your hit point maximum.
In the end, the rules are consistent across all levels. The negative CON modifier always applies, and just as you cannot gain less than 1 hp at a level up, you cannot start the game at less than 1 max hp.
Like you said, your starting HP is max hit die number plus your Con modifier. If you have a negative Con modifier, that still applies.
So a first level bard with an 8 Constitution (-1 modifier) has 8 + -1 = 7 hit points.
