plotting a parabola

From: j0mbolar (j0mbolar_at_engineer.com)
Date: 01/18/05 

Date: 17 Jan 2005 17:23:44 -0800

My book states that to find the base point from a quadratic
equation of the form y = ax^2 + bx + c
you use x_sub0 = -b/(2a) and y_sub0 = (c - b^2)/(4a)

we have the Axis of symmetry and x-Coordinate of the Vertex
along with the y-Coordinate of the vertex respectively.

my question is this, how were these equations derived
from the quadratic equation y = ax^2 + bx + c?

my book does not state this. It just states the equations.
can anyone tell me how we can logically conclude that
-b/(2a) can be deduced from y = ax^2 + bx + c for
the x-Coordinate and likewise for the y-Coordinate?

Also, I wonder similarly for how we can deduce equations
that give us the sum of interior angles of a non-convex
polygon? How did people before us deduce these things?
I feel to truly understand what we use, we must know how
what we use was determined. i.e., the logical process of
deducing these equations.

How did we conclude that 180n - 360(in degrees) gives
us the sum of all interior angles of a general polygon?
or PIn - 2PI if you prefer radians. Where n represents the
sides of our polygon.

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