Quadratic functions

A function of the form \(Q:\mathbb{R} \longrightarrow \mathbb{R}\), where \(Q(x) = ax^2+bx+c\) for all \(x \in \mathbb{R}\), and \(a,b,c\) are real constants with \(a \neq 0\) is called a quadratic function.

A useful algebraic technique that is often used to determine the range of quadratic functions is the completing the square technique, given below:

\[ \begin{align*} & a x^2 + b x + c \\ = & a \left(x^2 + \frac{b}{a}x\right) + c \\ = & a \left(x + \frac{b}{2a}\right)^2 + \left(c - \frac{b^2}{4a}\right). \end{align*} \]

From the above form, one deduces that the line of symmetry is the vertical line whose equation is given by
\[ x = -\frac{b}{2a}. \]

Case 1. \(a > 0\)

Since \( a > 0 \), it follows that
\[ a \left( x + \frac{b}{2a} \right)^2 + \left( c - \frac{b^2}{4a}\right) \geq c - \frac{b^2}{4a} \]
for all \(x \in \mathbb{R}\).

Thus, the minimum point of the graph of \(y = Q(x)\) is given by
\[ \left( -\frac{b}{2a}, c - \frac{b^2}{4a} \right). \]

Case 2. \(a < 0\)

Since \( a < 0 \), it follows that
\[ a \left( x + \frac{b}{2a} \right)^2 + \left( c - \frac{b^2}{4a}\right) \leq c - \frac{b^2}{4a} \]
for all \(x \in \mathbb{R}\).

Thus, the maximum point of the graph of \(y = Q(x)\) is given by
\[ \left( -\frac{b}{2a}, c - \frac{b^2}{4a} \right). \]