324 CHAPTER 3 Polynomial and Rational Functions In this chapter we primarily consider polynomial functions having real coefficients. When analyzing a polynomial function, the degree n and the leading coefficient an are important. These are both given in the dominating termanx n. 3.1 Quadratic Functions and Models ■ Polynomial Functions ■ Quadratic Functions ■ Graphing Techniques ■ Completing the Square ■ The Vertex Formula ■ Quadratic Models Polynomial Functions Polynomial Function A polynomial functionƒ of degree n, where n is a nonnegative integer, is a function of the form ƒ1x2 =anxn +a n−1xn−1 + P+a1x +a0, where an, an-1, c , a1, and a0 are complex numbers, with an ≠0. Polynomial Function Function Type Degree n Leading Coefficient an ƒ1x2 = 2 Constant 0 2 ƒ1x2 = 5x - 1 Linear 1 5 ƒ1x2 = 4x2 - x + 1 Quadratic 2 4 ƒ1x2 = 2x3 - 1 2x + 5 Cubic 3 2 ƒ1x2 = x4 + 22x3 - 3x2 Quartic 4 1 The function ƒ1x2 = 0 is the zero polynomial and has no degree. LOOKING AHEAD TO CALCULUS In calculus, polynomial functions are used to approximate more complicated functions. For example, the trigonometric function sin x is approximated by the polynomial x - x3 6 + x5 120 - x7 5040 . Quadratic Functions Polynomial functions of degree 2 are quadratic functions. Again, we are most often concerned with real coefficients. Quadratic Function A quadratic functionƒ is a function of the form ƒ1x2 =ax2 +bx +c, where a, b, and c are complex numbers, with a≠0. The simplest quadratic function is ƒ1x2 =x2. Squaring function See Figure 1. This graph is a parabola. Every quadratic function with real coefficients defined over the real numbers has a graph that is a parabola. The domain of ƒ1x2 = x2 is 1-∞, ∞2, and the range is 30, ∞2. The lowest point on the graph occurs at the origin 10, 02. Thus, the function decreases on the open interval y x (–2, 4) (–1, 1) (1, 1) (0, 0) Domain (–∞, ∞) Range [0, ∞) f(x) = x2 (2, 4) Figure 1 x ƒ1x2 -2 4 -1 1 0 0 1 1 2 4 1-∞, 02 and increases on the open interval 10, ∞2. (Remember that these intervals indicate x-values.)
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