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How do you find the formula for a quadratic equation?

I know how do to do linear equations and how to figure out the equation when you have two points.

But how do you know what the formula for a quadratic equation?

For example: If I graphed this -

how would I find its formula? I do not seek an answer, I want to know how you arrive at that answer.



Kimi wa ne tashika ni ano toki watashi no soba ni ita

Itsudatte itsudatte itsudatte

Sugu yoko de waratteita

Nakushitemo torimodosu kimi wo

I will never leave you