The shape and its mirror line
A [[quadratic-function|quadratic function]] f(x) = a x^2 + b x + c graphs as a [[parabola|parabola]], a smooth U-shaped curve. If a > 0 the U opens upward; if a < 0 it opens downward. The parabola is perfectly symmetric about a vertical line called the [[axis-of-symmetry|axis of symmetry]], and the single point where the curve turns around — its lowest or highest point — is the [[vertex|vertex]], which sits right on that axis.
Axis of symmetry of f(x) = a x^2 + b x + c:
x = -b / (2a)
Find the vertex of f(x) = x^2 - 4x + 1:
x = -(-4) / (2·1) = 4/2 = 2 (axis: x = 2)
f(2) = 2^2 - 4·2 + 1 = 4 - 8 + 1 = -3
Vertex = (2, -3), opens upward (a = 1 > 0)
so the minimum value of f is -3.Vertex form reveals the vertex at a glance
The same parabola can be written in [[vertex-form|vertex form]] f(x) = a(x - h)^2 + k, where the vertex is simply (h, k). You get there from standard form by completing the square. Vertex form is the honest x-ray of a parabola: h shifts it left or right, k shifts it up or down, and a still controls how wide and which way it opens.
Rewrite f(x) = x^2 - 4x + 1 in vertex form:
f(x) = (x^2 - 4x) + 1
= (x^2 - 4x + 4) - 4 + 1 (add & subtract (4/2)^2 = 4)
= (x - 2)^2 - 3
Vertex form: f(x) = (x - 2)^2 - 3
Read directly: vertex (h, k) = (2, -3).
Same vertex we found from x = -b/(2a) — they must agree.