202 CHAPTER 4 Polynomial and Rational Functions (iii) Set the slider for a to 1 and c to 1. Now use the slider to change b from 1 to 2 to 3 to 4.As you do this, notice the behavior of the graph of the polynomial function at x 1. = − (iv) Based on the results from parts (i)–(iii), if the zero of a polynomial function is of even multiplicity, the graph of the function will (cross/touch/avoid) the x-axis at the zero. If the zero of a polynomial function is of odd multiplicity, the graph of the function will (cross/touch/avoid) the x-axis at the zero. (c) Use the slider to set a to 1 and b to 2. Use the slider to change c from 2− to 1− to 1 to 2. Note the behavior of the graph near each x-intercept. True or False The value of c affects the behavior of the graph near each x-intercept. 7. Interactive Figure Exercise Exploring Turning Points Open the “Multiplicity” interactive figure, which is available in the Video & Resource Library of MyLab Math (under Sullivan Interactive Figures). (a) Set the value of c to 1. Set the value of a to 1 and b to 1. What is the degree of the polynomial? How many turning points does the graph have? (b) Keep the value of c at 1. Set the value of a to 2 and b to 1. What is the degree of the polynomial? How many turning points does the graph have? (c) Keep the value of c at 1. Set the value of a to 2 and b to 2. What is the degree of the polynomial? How many turning points does the graph have? (d) Keep the value of c at 1. Set the value of a to 3 and b to 1. What is the degree of the polynomial? How many turning points does the graph have? (e) Keep the value of c at 1. Set the value of a to 3 and b to 3. What is the degree of the polynomial? How many turning points does the graph have? (f) If f is a polynomial of degree n, then f has at most turning points. (g) If f is a polynomial of degree 5, then f could have 4, , or 0 turning points. (h) True or False If f is a polynomial of degree 4, then f could have 2 turning points. 8. The graph of every polynomial function is both and . 9. Multiple Choice If r is a real zero of even multiplicity of a polynomial function f, then the graph of f the x-axis at r. (a) crosses (b) touches 10. The graphs of power functions of the form f x x ,n ( ) = where n is an even integer, always contain the points , , and . 11. If r is a real solution of the equation f x 0, ( ) = list three equivalent statements regarding f and r. 12. The points at which a graph changes direction (from increasing to decreasing or decreasing to increasing) are called . 13. The graph of the function ( ) = − + − − f x x x x x 3 5 2 7 4 3 2 resembles the graph of for large values of x . 14. If f x x x x 2 5 7, 5 3 2 ( ) = − + − + then f x( ) → as x , →−∞ and f x( ) → as x . →∞ 15. Multiple Choice The of a real zero is the number of times its corresponding factor occurs. (a) degree (b) multiplicity (c) turning point (d) limit 16. Multiple Choice The graph of y x x x 5 3 2 9 6 4 = − + − has at most how many turning points? (a) 9− (b) 14 (c) 6 (d) 5 Skill Building In Problems 17–28, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, state why not. Write each polynomial in standard form. Then identify the leading term and the constant term. 17. f x x x 4 3 ( ) = + 18. f x x x 5 4 2 4 ( ) = + 19. g x x 2 3 5 2 ( ) = + 20. h x x 3 1 2 ( ) = − 21. f x x 1 1 ( ) = − 22. f x x x 1 ( ) ( ) = − 23. g x x x 2 2/3 1/3 ( ) = − + 24. h x x x 1 ( ) ( ) = − 25. F x x x 5 1 2 4 3 π ( ) = − + 26. F x x x 5 2 3 ( ) = − 27. G x x x 2 1 1 2 2 ( ) ( ) ( ) = − + 28. G x x x 3 2 2 3 ( ) ( ) = − + In Problems 29–42, use transformations of the graph of = = y x or y x 4 5 to graph each function. 29. f x x 1 4 ( ) ( ) = + 30. f x x 2 5 ( ) ( ) = − 31. f x x 3 5 ( ) = − 32. f x x 2 4 ( ) = + 33. f x x 1 2 4 ( ) = 34. f x x3 5 ( ) = 35. f x x5 ( ) = − 36. f x x4 ( ) = − 37. f x x 1 2 5 ( ) ( ) = − + 38. f x x 2 3 4 ( ) ( ) = + − 39. f x x 2 1 1 4 ( ) ( ) = + + 40. f x x 1 2 1 2 5 ( ) ( ) = − − 41. f x x 4 2 5 ( ) ( ) = − − 42. f x x 3 2 4 ( ) ( ) = − + In Problems 43–50, find a polynomial function whose real zeros and degree are given. Answers will vary depending on the choice of the leading coefficient. 43. Zeros: 1, − 1, 3; degree 3 44. Zeros: 2, − 2, 3; degree 3 45. Zeros: 5, − 0, 6; degree 3 46. Zeros: 4, − 0, 2; degree 3 47. Zeros: 5, 2, − − 3, 5; degree 4 48. Zeros: 3, 1, − − 2, 5; degree 4 49. Zeros: 1, − multiplicity 1; 3, multiplicity 2; degree 3 50. Zeros: 2, − multiplicity 2; 4, multiplicity 1; degree 3
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