Section 4.1 AN19 Cumulative Review (page 187) 1. ( ) 5 2; 3 2 , 1 2 2. ( ) − − 2, 1 and ( ) 2, 3 are on the graph. 3. { } ) ≥− ⎡ − ∞ ⎣ ⎢ x x 3 5 ; 3 5 , 3 5 4. = − + y x2 2 y x (2, 2) ( 1, 4) 5 5 5. = − + y x 1 2 13 2 y x (3, 5) 5 9 6. ( ) ( ) − + + = x y 2 4 25 2 2 y x (7, 4) (2, 4) (2, 9) (3, 4) (2, 1) 10 10 7. Yes 8. (a) −3 (b) − − x x4 2 2 (c) + + x x4 1 2 (d) − + − x x4 1 2 (e) − x 3 2 (f) + − x h 2 4 9. { } ≠ z z 7 6 10. Yes 11. (a) No (b) ( ) − − − 1; 2, 1 is on the graph. (c) ( ) − − 8; 8, 2 is on the graph. 12. Neither 13. Local maximum value is 5.30 and occurs at = − x 1.29. Local minimum value is −3.30 and occurs at = x 1.29. Increasing: 4, 1.29 [ ] − − and [ ] 1.29, 4 ; Decreasing: [ ] −1.29, 1.29 14. (a) −4 (b) x x 4 or 4, { } ( ) >− − ∞ 15. (a) Domain: x x 4 4 ; { } − ≤ ≤ Range: y y 1 3 { } − ≤ ≤ (b) ( ) ( ) − − 1,0, 0, 1, ( ) 1, 0 (c) y-axis (d) 1 (e) −4 and 4 (f) x x 1 1 { } − < < (g) y x (4, 5) (2, 3) (1, 2) ( 1, 2) ( 2, 3) ( 4, 5) (0, 1) 5 5 (h) y (1, 0) (0, 1) ( 1, 0) ( 2, 1) ( 4, 3) (2, 1) (4, 3) x 5 5 (i) y x (1, 0) (0, 2) ( 1, 0) ( 2, 2) ( 4, 6) (2, 2) (4, 6) 5 10 ( j) Even (k) 0, 4 [ ] CHAPTER 4 Polynomial and Rational Functions 4.1 Assess Your Understanding (page 201) 6. (a) −1, 3 (b) touch; cross (c) F 7. (a) 2; 1 (b) 3; 2 (c) 4; 3 (d) 4; 1 (e) 6; 1 (f) −n 1 (g) 2 (h) F 8. smooth; continuous 9. b 10. ( ) ( ) ( ) −1, 1 ; 0, 0 ; 1, 1 11. r is a real zero of f ; r is an x-intercept of the graph of f ; −x r is a factor of f . 12. turning points 13. = y x3 4 14. ∞ −∞ ; 15. b 16. d 17. Polynomial function; degree 3; ( ) = + f x x x4 ; 3 leading term: x ;3 constant term: 0 19. Polynomial function; degree 2; ( ) = + g x x 3 5 2 5 ; 2 leading term: x 3 5 ;2 constant term: 2 5 21. Not a polynomial function; x is raised to the −1 power. 23. Not a polynomial function; x is raised to non-integer powers. 25. Polynomial function; degree 4; π ( ) = − + F x x x 5 1 2 ; 4 3 leading term: x5 ;4 constant term: 1 2 27. Polynomial function; degree 4; ( ) = − + − + G x x x x x 2 4 4 4 2; 4 3 2 leading term: x2 ;4 constant term: 2 29. y x 5 5 (0, 1) (22, 1) (21, 0) 31. y 5 x 5 (1, 22) (21, 24) (0, 23) 33. y 5 x 5 (0, 0) 1, 1 2 21, 1 2 35. y 5 x 5 (21, 1) (1, 21) (0, 0) 37. y 5 x 5 (2, 3) (1, 2) (0, 1) 39. y 5 x 5 (21, 1) (22, 3) (0, 3) 41. y 6 x 5 (3, 3) (2, 4) (1, 5) 43. ( ) ( )( )( ) = + − − f x x x x 1 1 3 for = a 1 45. ( ) ( )( ) = + − f x x x x 5 6 for = a 1 47. ( ) ( )( )( )( ) = + + − − f x x x x x 5 2 3 5 for = a 1 49. ( ) ( )( ) = + − f x x x 1 3 2 for = a 1 51. ( ) ( )( )( ) = + − − f x x x x 3 2 3 5 53. ( ) ( )( )( ) = + − − f x x x x x 16 2 1 3 55. ( ) ( )( )( ) = + − − f x x x x 3 3 1 4 57. ( ) ( ) ( ) = + − f x x x 5 1 1 2 2 59. ( ) ( ) ( )( ) = − + − − f x x x x 2 5 2 4 2 61. (a) 7, multiplicity 1; −3, multiplicity 2 (b) Graph touches the x-axis at −3 and crosses it at 7. (c) 2 (d) = y x3 3 63. (a) 5, multiplicity 3 (b) Graph crosses x-axis at 5. (c) 6 (d) = y x7 7 65. (a) − 1 2 , multiplicity 2; −4, multiplicity 3 (b) Graph touches the x-axis at − 1 2 and crosses at −4. (c) 4 (d) = − y x2 5 67. (a) 5, multiplicity 3; −4, multiplicity 2 (b) Graph touches the x-axis at −4 and crosses it at 5. (c) 4 (d) = y x5 69. (a) No real zeros (b) Graph neither crosses nor touches the x-axis. (c) 5 (d) = y x2 6 71. (a) 0, multiplicity 2; − 2, 2, multiplicity 1 (b) Graph touches the x-axis at 0 and crosses at − 2 and 2. (c) 3 (d) = − y x2 4 73. Could be; zeros: −1, 1, 2; Least degree is 3. 75. Cannot be the graph of a polynomial; gap at = − x 1 77. ( ) ( )( ) = − − f x x x x 1 2 79. ( ) ( )( ) ( ) = − + − − f x x x x 1 2 1 1 2 2 81. ( ) ( )( ) ( ) = + + − f x x x x 0.2 4 1 3 2 83. ( ) ( ) ( ) = − + − f x x x x 3 3 2 2 85. (a) −3, 2 (b) − − 6, 1 87. (a) 2 (b) 210 25 5 10 (c) = y x2 3 (d) The end behavior of the graph resembles = y x2 3 and two turning points are visible, the maximum possible. (e) ( ] [ ) −∞ ∞ , 0 and 1, (f) Increasing on ( ] −∞, 0 and [ )∞1, 89. −3, multiplicity 2; −1, multiplicity 3; 1, multiplicity 1 91. No, every polynomial function is defined at 0 so has a y-intercept; yes, the graph of a polynomial function can be completely above or below the x-axis (e.g., = + y x 1 2 ) 93. a, b, c, d 99. = − − y x 2 5 11 5 100. { } ≠ − x x 5 101. − − − + 2 7 2 , 2 7 2 102. { } − 4 5 , 2 103. Decreasing 104. ( ) = + + y x 2 5 2 105. − 1 3 106. ( ) − 8, 14 107. Quotient: −x4 7; remainder: −x4 2 108. ( ) ( ) −∞ ∪ ∞ , 0 1,
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