形状与它的镜像线
[[quadratic-function|二次函数]] f(x) = a x^2 + b x + c 的图像是一条[[parabola|抛物线]]，一条光滑的 U 形曲线。若 a > 0，U 形开口向上；若 a < 0，开口向下。抛物线关于一条竖直线完全对称，这条线叫做[[axis-of-symmetry|对称轴]]；而曲线转向的那个唯一的点——最低点或最高点——就是[[vertex|顶点]]，它恰好落在对称轴上。
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.顶点式一眼看出顶点
同一条抛物线也可以写成[[vertex-form|顶点式]] f(x) = a(x - h)^2 + k，其中顶点就是 (h, k)。你可以通过配方法从标准形式得到它。顶点式是抛物线最直白的 X 光片：h 把它左右平移，k 把它上下平移，而 a 仍然控制开口的宽窄和朝向。
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.