The maths
A quadratic is y = ax² + bx + c, with a not zero. Its graph is a parabola. When a is positive the curve opens upward and has a lowest point.
You can write the same curve three ways. Each one shows something different.
y = a x² + b x + c y = a(x − p)(x − q) y = a(x − h)² + k
The roots p and q are the x values where the curve meets the x-axis. There y = 0. The factored form shows them straight away.
The vertex (h, k) is the turning point. The line x = h is the axis of symmetry, halfway between the roots.
h = −b / 2a k = c − b² / 4a
The discriminant D decides how many real roots there are. It is the part under the square root in the quadratic formula.
D = b² − 4ac x = (−b ± √D) / 2a
If D is positive the curve cuts the axis twice. If D is zero it just touches, and the two roots are equal. If D is negative it misses the axis, so there are no real roots.
The roots also tie back to the coefficients. Adding them gives −b/a and multiplying them gives c/a. These are Vieta's relations.
p + q = −b / a p q = c / a
When a is positive and D is negative, y stays positive for every x. The curve sits wholly above the axis.