Quadratic Function

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) $$y = a{x^2}$$, $$a < 0$$