# Quadratic Function

A function of the form , where and are constants and , 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 , whose graph is a curve. The graph of is obtained from the graph of by vertical stretching if , or flattening if. Furthermore, the graph of for negative values of is obtained by reflecting the graph across the x-axis. Figure 1 shows the graph of for various values of .

Figure 1 , where , and |

The graph of the equation of the form is , and are examples of curves called **parabolas**. These parabolas are systemic about the y-axis; they **open upward** and have a lowest point at if (Figure 2 (a)), and they **open downward **and have a highest point at (0, 0) if (Figure 2 (b)). The highest or lowest point of the graph of is called theĀ **vertex** of the parabola, and its line symmetry is called the **axis of symmetry** or simply the **axis** of the parabola.

Figure 2 (a) , |
Figure 2 (b) , |