198 CHAPTER 4 Polynomial and Rational Functions Identifying the Graph of a Polynomial Function Which of the graphs in Figure 10 could be the graph of a polynomial function? For those that could, list the real zeros and state the least degree the polynomial function can have. For those that could not, say why not. EXAMPLE 7 Figure 10 x y –2 2 2 –2 x y –2 2 2 –2 x y –2 2 2 –2 x y –3 3 3 –3 (a) (b) (c) (d) Based on the first bullet of the theorem, a polynomial function of degree 5 will have at most 5 1 4 − = turning points. Based on the second bullet of the theorem, if a polynomial function has 3 turning points, then its degree must be at least 4. Solution (a) The graph in Figure 10(a) cannot be the graph of a polynomial function because of the gap that occurs at x 1. = − Remember, the graph of a polynomial function is continuous–no gaps or holes. (See Figure 1 on p. 191.) (b) The graph in Figure 10(b) could be the graph of a polynomial function because the graph is smooth and continuous. It has three real zeros, at 2, − 1, and 2. Since the graph has two turning points, the degree of the polynomial function must be at least 3. (c) The graph in Figure 10(c) cannot be the graph of a polynomial function because of the cusp at x 1. = Remember, the graph of a polynomial function is smooth. (d) The graph in Figure 10(d) could be the graph of a polynomial function. It has two real zeros, at 2− and 1. Since the graph has three turning points, the degree of the polynomial function is at least 4. Now Work PROBLEMS 61(c) AND 73 End Behavior One last remark about Figure 8 on page 196. For very large values of x, either positive or negative, the graph of f x x x 1 2 2 ( ) ( ) ( ) = + − looks like the graph of y x .3 = To see why, write f in the form f x x x x x x x x 1 2 3 2 1 3 2 2 3 3 2 3 ( ) ( ) ( ) ( ) = + − = − − = − − For large values of x, either positive or negative, the terms x 3 2 and x 2 3 are close to 0. Do you see why? Evaluate x 3 2 and x 2 3 for x 10, 100, 1000 = and for x 10, 100, 1000. = − − − What happens to the value of each expression for large values of x ? So, for large values of x , f x x x x x x x 3 2 1 3 2 3 3 2 3 3 ( ) ( ) = − − = − − ≈ The behavior of the graph of a function for large values of x, either positive or negative, is referred to as its end behavior.
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