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 if0 < 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.

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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 theĀ vertex of the parabola, and its line symmetry is called the axis of symmetry or simply the axis of the parabola.

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Figure 2 (a) y = a{x^2}, a > 0
Figure 2 (b) y = a{x^2}, a < 0