Cambridge O Level Additional Mathematics · Syllabus 4037 · Quadratic Functions
Quadratic Function
What is Quadratic Function?
A function of a single variable whose highest power is two, written f(x) = ax^2 + bx + c with a not equal to zero. Its graph is a parabola, symmetric about a vertical axis through a single turning point. The same function can be written in expanded form, in factorised form a(x - p)(x - q) which displays its roots, or in completed-square form a(x - h)^2 + k which displays its turning point (h, k); all three describe one curve and are equal for every value of x.
This definition is part of the Quadratic Functions chapter in Cambridge O Level Additional Mathematics.
Quadratic Function in context
A quadratic function is any function that can be written \(f(x)=ax^2+bx+c\) with \(a\neq0\); its graph is a parabola with one turning point and a vertical axis of symmetry. The same function has three algebraic forms that are equal at every \(x\): expanded form shows the \(y\)-intercept \(c\), factorised form \(a(x-p)(x-q)\) shows the roots \(x=p\) and \(x=q\), and completed-square form \(a(x-h)^2+k\) shows the turning point \((h,k)\). Choosing the form that makes the answer visible is the central skill of this chapter.