Quadratic Functions
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.
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