SECTION 4.1 Polynomial Functions 199 For example, if f x x x x 2 5 4, 3 2 ( ) = − + + − then the graph of f will behave like the graph of y x2 3 = − for very large values of x, either positive or negative. We can see that the graphs of f and y x2 3 = − “behave” the same by considering Table 4 and Figure 11. In Words The end behavior of a polynomial function resembles that of its leading term. THEOREM End Behavior of the Graph of a Polynomial Function The end behavior of the graph of the polynomial function f x a x a x a x a a 0 n n n n n 1 1 1 0 ( ) = + + + + ≠ − − is the same as that of the graph of the power function y a xn n = x ( ) f x 2 3 = − y x 10 1494 − 2000 − 100 1,949,904 − 2,000,000 − 500 248,749,504 − 250,000,000 − 1000 1,994,999,004 − 2,000,000,000 − Table 4 Figure 11 Y2 5 22x3 1 5x2 1 x 2 4 Y1 5 22x3 25 2175 175 5 Notice that, as x becomes a larger and larger positive number, the values of f become larger and larger negative numbers. When this happens, we say that f is unbounded in the negative direction. Rather than using words to describe the behavior of the graph of the function, we explain its behavior using notation. We symbolize “the value of f becomes a larger and larger negative number as x becomes a larger and larger positive number” by writing f x( ) →−∞ as x →∞ (read “the values of f approach negative infinity as x approaches infinity”). In calculus, limits are used to convey these ideas. There we use the symbolism f x lim , x ( ) = −∞ →∞ read “the limit of f x( ) as x approaches infinity equals negative infinity,” to mean that f x( ) →−∞ as x . →∞ When the value of a limit equals infinity (or negative infinity), we mean that the values of the function are unbounded in the positive (or negative) direction and call the limit an infinite limit . When we discuss limits as x becomes unbounded in the negative direction or unbounded in the positive direction, we are discussing limits at infinity. Look back at Figures 2 and 3 (pp. 192 and 193). Based on the preceding theorem and the previous discussion on power functions, the end behavior of a polynomial function can be of only four types. See Figure 12. NOTE Infinity ( )∞ and negative infinity ( ) −∞ are not numbers. Rather, they are symbols that represent unboundedness. j Figure 12 End behavior of f x a x a x a x a n n n n 1 1 1 0 ( ) = + + + + − − y x y x y x y x n ≥ 2 even; an > 0 (a) n ≥ 2 even; an < 0 ` as x 2` f(x) ` as x ` f(x) (b) n ≥ 3 odd; an > 0 (c) n ≥ 3 odd; an < 0 (d) ` as x ` f(x) ` as x 2` f(x) 2` as x 2` f(x) 2` as x 2` f(x) 2` as x ` f(x) 2` as x ` f(x)
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