A function of the form \(Q:\mathbb{R} \longrightarrow \mathbb{R}\), where&nbsp;\(Q(x) = ax^2+bx+c\) 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. \(a &gt; 0\) 
Since&nbsp;\( a &gt; 0 \), 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. \(a &lt; 0\)
Since&nbsp;\( a &lt; 0 \), 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 \[
\left( -\frac{b}{2a}, c - \frac{b^2}{4a} \right).
\]

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
\[
x = -\frac{b}{2a}.
\]

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
\[
\left( -\frac{b}{2a}, c - \frac{b^2}{4a} \right).
\]

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
\[
\left( -\frac{b}{2a}, c - \frac{b^2}{4a} \right).
\]

Functions

International Baccalaureate Higher Level Mathematics

Quadratic functions

Quadratic functions