Function y = ax² + bx + c. Properties, study, range

General form and domain

A function

y=ax^2+bx+c

a\neq0

, is called quadratic. Its graph is a parabola. The domain in the school course is usually

D(f)=\mathbb{R}

— you may substitute any real

x

. The range

E(f)

depends only on the vertex and the direction of the branches: if

a>0

the "bottom of the bowl" is the minimum point; if

a<0

the "roof" is the maximum point.

Vertex

(x_v;y_v)

x_v=-\dfrac{b}{2a}

, find

y_v

by substitution or via the discriminant

Zeros — solutions of

ax^2+bx+c=0

(if any)

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