359 3.4 Polynomial Functions: Graphs, Applications, and Models 3.4 Polynomial Functions: Graphs, Applications, and Models Graphs of f 1x2 =ax n We can now graph polynomial functions of degree 3 or greater with real number coefficients and domains (because the graphs are in the real number plane). We begin by inspecting the graphs of several functions of the form ƒ1x2 =axn, with a = 1. The identity function ƒ1x2 = x, the squaring function ƒ1x2 = x2, and the cubing function ƒ1x2 = x3 were graphed earlier using a general point-plotting method. Each function in Figure 20 has odd degree and is an odd function exhibiting symmetry about the origin. Each has domain 1-∞, ∞2 and range 1-∞, ∞2 and is continuous on its entire domain 1-∞, ∞2. Additionally, these odd functions are increasing on their entire domain 1-∞, ∞2, appearing as though they fall to the left and rise to the right. ■ Graphs of f 1x2 =ax n ■ Graphs of General Polynomial Functions ■ Behavior at Zeros ■ Turning Points and End Behavior ■ Graphing Techniques ■ Intermediate Value and Boundedness Theorems ■ Approximations of Real Zeros ■ Polynomial Models x y f(x) = x 1 0 –1 –1 1 2 –2 –2 2 x y f(x) = x3 1 0 –1 –1 1 2 –2 –2 2 Falls to the left x y f(x) = x 5 Rises to the right 1 0 –1 –1 1 2 –2 –2 2 Figure 20 Each function in Figure 21 has even degree and is an even function exhibiting symmetry about the y-axis. Each has domain 1-∞, ∞2 but restricted range 30, ∞2. These even functions are also continuous on their entire domain 1-∞, ∞2. However, they are decreasing on 1-∞, 02 and increasing on 10, ∞2, appearing as though they rise both to the left and to the right. x y f(x) = x2 1 0 –1 –1 1 2 –2 –2 2 x y f(x) = x4 1 0 –1 –1 1 2 –2 –2 2 x y f(x) = x 6 1 0 –1 –1 1 2 –2 –2 2 Rises to the right Rises to the left Figure 21 The behaviors in the graphs of these basic polynomial functions as x increases (decreases) without bound also apply to more complicated polynomial functions. Graphs of General Polynomial Functions As with quadratic functions, the absolute value of a in ƒ1x2 =axn determines the width of the graph. • When a 71, the graph is stretched vertically, making it narrower. • When 0 6 a 61, the graph is shrunk or compressed vertically, making it wider.
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