A function $f$ of the form $f\left( x \right) = a{x^2} + bx + c$, where $a,\,b$ and $c$ are constants and $a \ne 0$, is called a quadratic function. Such functions often arise in applied mathematics. For instance, the height of a projectile is a quadratic function of time, the velocity of blood flow is a quadratic function of the distance from the center of the blood vessel, and the force exerted by the wind on the blades of a wind-powered generator is a quadratic function of the wind speed.
The simplest quadratic function is the square function $f\left( x \right) = {x^2}$, whose graph is a curve. The graph of $f\left( x \right) = a{x^2}$ is obtained from the graph of $f\left( x \right) = {x^2}$ by vertical stretching if $a > 1$, or flattening if$0 < a < 1$. Furthermore, the graph of $f\left( x \right) = a{x^2}$ for negative values of $a$ is obtained by reflecting the graph $f\left( x \right) = \left| a \right|{x^2}$ across the x-axis. Figure 1 shows the graph of $f\left( x \right) = a{x^2}$ for various values of $a$.
 Figure 1 $f\left( x \right) = a{x^2}$, where $a = \pm 2$, $a = \pm 1$ and $a = \pm 1/2$
The graph of the equation of the form is $f\left( x \right) = a{x^2}$, and are examples of curves called parabolas. These parabolas are systemic about the y-axis; they open upward and have a lowest point at $\left( {0,0} \right)$ if $a > 0$ (Figure 2 (a)), and they open downward and have a highest point at (0, 0) if $a < 0$ (Figure 2 (b)). The highest or lowest point of the graph of $f\left( x \right) = a{x^2}$ is called the vertex of the parabola, and its line symmetry is called the axis of symmetry or simply the axis of the parabola.
 Figure 2 (a) $y = a{x^2}$, $a > 0$ Figure 2 (b) $y = a{x^2}$, $a < 0$