section4_3 - Section 4.1 Polynomial Functions and Basic...

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Unformatted text preview: Section 4.1 Polynomial Functions and Basic Graphs Page 1 of 23 Basic Graphs: Section 4.1 Polynomial Functions and Basic Graphs Page 2 of 23 Example 1: Sketch the graph of the polynomial function P ( x) = ( x + 1)3 by transforming the graph of the function y = x3 . Solution to Example 1: Section 4.1 Polynomial Functions and Basic Graphs Page 3 of 23 Example 2: Sketch the graph of the polynomial function P ( x) = x 4 + 1 by transforming the graph of the function y = x 4 . Solution to Example 2: Section 4.1 Polynomial Functions and Basic Graphs Page 4 of 23 Example 3: Solution to Example 3: Section 4.1 Polynomial Functions and Basic Graphs Page 5 of 23 Example 4: Solution to Example 4: Section 4.1 Polynomial Functions and Basic Graphs Page 6 of 23 Example 5: Solution to Example 5: Section 4.1 Polynomial Functions and Basic Graphs Page 7 of 23 Section 4.1 Polynomial Functions and Basic Graphs Page 8 of 23 Example 6: Solution to Example 6: Section 4.1 Polynomial Functions and Basic Graphs Page 9 of 23 Section 4.1 Polynomial Functions and Basic Graphs A Strategy for Graphing Polynomial Functions Page 10 of 23 Section 4.1 A Strategy for Graphing Polynomial Functions Page 11 of 23 Section 4.1 A Strategy for Graphing Polynomial Functions Page 12 of 23 Section 4.1 A Strategy for Graphing Polynomial Functions Page 13 of 23 Example 1: Use the strategy for graphing polynomial functions to sketch the graph of P( x) = x3 − x 2 − 2 x. Solution to Example 1: Section 4.1 A Strategy for Graphing Polynomial Functions Page 14 of 23 Example 2: Solution to Example 2: Section 4.1 A Strategy for Graphing Polynomial Functions Page 15 of 23 Example 3: Solution to Example 3: Section 4.1 A Strategy for Graphing Polynomial Functions Page 16 of 23 Example 4: Solution to Example 4: Section 4.1 A Strategy for Graphing Polynomial Functions Page 17 of 23 Example 5: Solution to Example 5: Section 4.1 A Strategy for Graphing Polynomial Functions Page 18 of 23 Section 4.1 A Strategy for Graphing Polynomial Functions Page 19 of 23 Example 6: Solution to Example 6: Section 4.1 A Strategy for Graphing Polynomial Functions Page 20 of 23 Section 4.1 A Strategy for Graphing Polynomial Functions Page 21 of 23 Exercise Set 4.1: Polynomial Functions Sketch a graph of each of the following functions. 1. 2. 3. 4. 5. 6. P ( x) = x3 P ( x) = x 4 P ( x) = x 6 P ( x) = x5 P ( x) = x n , where n is odd and n > 0. P ( x) = x n , where n is even and n > 0. Match each of the polynomial functions below with its graph. 9. P ( x) = ( x − 2)( x + 1)( x + 4) 10. Q( x) = −( x + 2)( x − 1)( x − 4) 11. R ( x) = −( x − 2)( x + 1)2 ( x + 4)2 12. S ( x) = ( x − 2) 2 ( x + 1)( x + 4) 13. U ( x) = ( x + 2)2 ( x − 1)3 ( x − 4) 14. V ( x) = −( x + 2)3 ( x − 1)3 ( x − 4)2 Answer the following. 7. The graph of P ( x) = ( x − 1)( x − 2)3 ( x + 4)2 has xintercepts at x = 1, x = 2, and x = −4. (a) At and immediately surrounding the point x = 2 , the graph resembles the graph of what familiar function? (Choose one.) y=x Choices for 9-14: A. 40 y = x2 y = x3 y B. 80 y (b) At and immediately surrounding the point x = −4 , the graph resembles the graph of what familiar function? (Choose one.) y=x x −4 −2 2 4 −4 −2 40 x 2 40 y = x2 y = x3 −40 (c) If P ( x) were to be multiplied out completely, the leading term of the polynomial would be: (Choose one; do not actually multiply out the polynomial.) x3 ; − x3 ; x 4 ; − x 4 ; x5 ; − x5 ; x 6 ; − x 6 C. −2 400 y x 2 −400 4 D. y 10 x −4 −2 2 4 6 8 8. The graph of Q( x) = −( x + 3) ( x − 5) has xintercepts at x = −3 and x = 5. (a) At and immediately surrounding the point x = −3 , the graph resembles the graph of what familiar function? (Choose one.) y=x 2 3 −800 −10 1200 y = x2 y = x3 E. y 10 F. 200 y (b) At and immediately surrounding the point x = 5 , the graph resembles the graph of what familiar function? (Choose one.) y=x x −6 −4 −2 2 4 6 −2 2 4 x y = x2 y = x3 −200 (c) If P ( x) were to be multiplied out completely, the leading term of the polynomial would be: (Choose one; do not actually multiply out the polynomial.) x3 ; − x3 ; x 4 ; − x 4 ; x5 ; − x5 ; x 6 ; − x 6 −10 Exercise Set 4.1: Polynomial Functions For each of the functions below: (a) Find the x- and y-intercepts. (b) Sketch the graph of the function. Be sure to show all x- and y-intercepts, along with the proper behavior at each x-intercept, as well as the proper end behavior. 15. P( x) = ( x − 5)( x + 3) 16. P( x) = ( x − 2)( x − 6) 17. P( x) = − x( x + 4) 18. P( x) = −( x − 3)( x + 1) 19. P( x) = ( x + 3)2 20. P( x) = −( x − 6)2 21. P( x) = ( x − 5)( x + 2)( x + 6) 22. P( x) = 3x( x − 4)( x − 7) 23. P ( x) = − 1 ( x − 4)( x − 1)( x + 3) 2 24. P( x) = −( x + 6)( x − 2)( x − 5) 25. P( x) = ( x + 2)2 ( x − 4) 26. P( x) = (5 − x)( x + 3)2 27. P( x) = (3x − 2)( x + 4)( x − 5)( x + 1) 28. P ( x) = − 1 ( x + 5)( x + 1)( x + 3)( x − 2) 3 42. P( x) = x3 − 2 x 2 − 15 x 43. P( x) = 25 x − x3 44. P( x) = −3x3 − 5 x 2 + 2 x 45. P( x) = − x 4 + x3 + 12 x 2 46. P( x) = x 4 − 16 x 2 47. P( x) = x5 − 9 x3 48. P( x) = − x5 − 3x 4 + 18 x3 49. P( x) = x3 + 4 x 2 − x − 4 50. P( x) = x3 − 5 x 2 − 4 x + 20 51. P( x) = x 4 − 13x 2 + 36 52. P( x) = x 4 − 17 x 2 + 16 Use transformations (the concepts of shifting, reflecting, stretching, and shrinking) to sketch each of the following graphs. 53. P( x) = x3 + 5 54. P( x) = − x3 − 2 55. P( x) = −( x − 2)3 + 4 56. P( x) = ( x + 5)3 − 1 57. P( x) = 2 x 4 − 3 58. P( x) = −( x − 2) 4 + 5 59. P( x) = −( x + 1)5 − 4 60. P( x) = ( x + 3)5 + 2 29. P( x) = x( x + 2)(4 − x)( x + 6) 30. P( x) = ( x − 1)( x − 3)( x + 2)( x + 5) 31. P( x) = ( x − 3) ( x + 4) 32. P( x) = − x(2 x − 5) 3 3 2 2 33. P( x) = ( x + 5) ( x − 4) 34. P( x) = x 2 ( x − 6)2 35. P( x) = ( x + 3)2 ( x − 4)3 36. P( x) = −2 x(3 − x)3 ( x + 1) 37. P( x) = − x( x − 2)2 ( x + 3) 2 ( x − 4) 38. P( x) = ( x − 5)3 ( x − 2) 2 ( x + 1) 39. P( x) = x8 ( x − 1)6 ( x + 1)7 40. P( x) = − x3 ( x + 1) 4 ( x − 1)7 41. P( x) = x3 − 6 x 2 + 8 x ...
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This document was uploaded on 07/14/2010.

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