Quadratic Functions

2.3

Quadratic Functions

 
\(\blacksquare\) A quadratic function can be expressed in the form \(f(x)=ax^2+bx+c\), where \(a\), \(b\) and \(c\) are constants and \(a \ne 0\).
 

If \(a\gt 0\), graph has the shape \(\LARGE \smile\) which passes through a minimum point.

 

If \(a\lt 0\), graph has the shape \(\LARGE \frown\) which passes through a maximum point.

 

The relationship between the position of the graph \(f(x)=ax^2+bx+c\) on the \(x-\)axis and its type of roots:

 

\(\boldsymbol{b^2-4ac\gt 0}\)

Two different real roots

\(\boldsymbol{b^2-4ac= 0}\)

Two equal real roots

\(\boldsymbol{b^2-4ac\lt0}\)

No real roots

 

A quadratic function can be expressed in the form

\(f(x)=a(x-h)^2+k\)

where \(a\), \(h\) and \(k\) are constants.

 

In \(f(x)=a(x-h)^2+k\),

\(x=h\) is an axis of symmetry and \((h,k)\) is the coordinates of the minimum or maximum point.