Careful, you are mixing apples and oranges. The max(min) will be the value of Y which is bigger(smaller) than every other value of Y for some region around your max(min) value.

Not X like you said, X is the input variable that determines your Y.

You should try graphing ax^2 + bx + c for various values of a, b, and c to verify if it has "sinusoidal waves."

that also might give you some intuition into the the max(min) of a parabola

Theorem: If E ⊂ R and f: E → R, and f has a maximum or minimum at x ∈ E, then one of the following three is true:
(1) x is a boundary point of E,
(2) f'(x) = 0, or
(3) f is not differentiable at x.

In your case, f(x) = ax^{2} + bx + c and E is the interval [x_{1}, x_{2}]. Then the only possibilities are these: (1) x is one of the boundary points x_{1} or x_{2} of E, or (2) f'(x) = 2ax + b = 0, so x = -b/2a. Look at the values of f at those three points; the largest one is the maximum, and the smallest one is the minimum.