AN16 Answers: Chapter 3 51. (a) Vertex: ( ) −1, 0 ; Axis of symmetry: = − x 1; Concave up (b) y-intercept: 1; x-intercept: −1 (c) y x 5 5 (0, 1) ( 2, 1) ( 1, 0) x 1 (d) Domain: ( ) −∞∞, ; Range: 0, [ )∞ (e) Decreasing: , 1 ( ] −∞ − ; Increasing: 1, [ ) − ∞ (f) f x x x 0 on , 1 1, or for 1, 1; ( ) ( ) ( ) > −∞− ∪−∞ <− >− ( ) f x is never negative. 55. (a) Vertex: ( ) − 1 2 , 5 2 ; Axis of symmetry: = x 1 2 ; Concave down (b) y-intercept: −3; x-intercept: None (c) y x 2 5 x 1 2 1 , 2 5 2 (0, 3) (1, 3) (d) Domain: ( ) −∞∞, ; Range: (−∞ − ⎤ ⎦ ⎥ , 5 2 (e) Increasing: , 1 2 (−∞ ⎤ ⎦ ⎥ ; Decreasing: 1 2 , ) ⎡ ∞ ⎣ ⎢ (f) ( ) f x is never positive; ( ) < f x 0 on ( ) −∞∞, or for all real numbers 57. (a) Vertex: ( ) − − 1, 1 ; Axis of symmetry: = − x 1; Concave up (b) y-intercept: 2; x-intercepts: − − − + 3 3 3 , 3 3 3 (c) (d) Domain: ( ) −∞∞, ; Range: 1, [ ) − ∞ (e) Decreasing: ( ] −∞ −, 1 ; Increasing: [ ) − ∞ 1, (f) ( ) > f x 0 on , 3 3 3 3 3 3 , −∞ ⎛ − − ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ ∪ − + ∞ ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ or for < − − > − + x x 3 3 3 , 3 3 3 ; ( ) < f x 0 on ⎛− − − + ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ 3 3 3 , 3 3 3 or for − − < < − + x 3 3 3 3 3 3 59. (a) Vertex: 3 4 , 17 4 ; ( ) − Axis of symmetry: = − x 3 4 ; Concave down (b) y-intercept: 2; x-intercepts: − − − + 3 17 4 , 3 17 4 (c) y x 5 2.5 (0, 2) (0.28, 0) ( 1.78, 0) 3 4 17 4 , x 3 4 (d) Domain: ( ) −∞∞, ; Range: , 17 4 (−∞ ⎤ ⎦ ⎥ (e) Increasing: , 3 4 (−∞ − ⎤ ⎦ ⎥ ; Decreasing: 3 4 , ) ⎡ − ∞ ⎣ ⎢ (f) ( ) > ⎛− − − + ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ f x 0 on 3 17 4 , 3 17 4 or for − − < < − + x 3 17 4 3 17 4 ; ( ) < −∞ ⎛ − − ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ ∪ − + ∞ ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ f x 0 on , 3 17 4 3 17 4 , or for < − − > − + x x 3 17 4 , 3 17 4 y 8 x 4 ( 1.58, 0) ( 0.42, 0) (1, 1) (0, 2) x 1 53. (a) Vertex: ( ) 1 4 , 15 8 ; Axis of symmetry: = x 1 4 ; Concave up (b) y-intercept: 2; x-intercept: None (c) y x 5 3 (0, 2) 1 2 , 2 x 1 4 1 4 15 8 , (d) Domain: ( ) −∞∞, ; Range: 15 8 , ) ⎡ ∞ ⎣ ⎢ (e) Decreasing: (−∞ ⎤ ⎦ ⎥ , 1 4 ; Increasing: 1 4 , ) ⎡ ∞ ⎣ ⎢ (f) ( ) > f x 0 on ( ) −∞∞, or for all real numbers; ( ) f x is never negative. 61. ( ) ( ) = + − = + − f x x x x 1 2 2 1 2 2 63. ( ) ( ) = − + + = − − − f x x x x 3 5 6 4 2 2 65. ( ) ( ) = − − = − − f x x x x 2 1 3 2 4 1 2 2 67. Minimum value; −48 69. Minimum value; −21 71. Maximum value; 8 73. Maximum value; 23 75. = = = a b c 6, 0, 2 77. (a), (c), (d) y x 6 5 (1, 3) (3, 5) (b) { } −1, 3 79. (a), (c), (d) y x 1 5 ( 1, 3) (3, 5) (b) { } −1, 3 81. (a), (c), (d) y x 10 10 (2, 6) (1, 6) (b) { } −1, 2 83. (a) ( ) ( )( ) = = + − = + − a f x x x x x 1: 3 1 2 3 2 ( ) ( )( ) = = + − = + − a f x x x x x 2: 2 3 1 2 4 6 2 ( ) ( )( ) =− =−+ −=−−+ a f x x x x x 2: 2 3 1 2 4 6 2 ( ) ( )( ) = = + − = + − a f x x x x x 5: 5 3 1 5 10 15 2 (b) The value of a does not affect the x-intercepts, but it changes the y-intercept by a factor of a. (c) The value of a does not affect the axis of symmetry. It is = − x 1 for all values of a. (d) The value of a does not affect the x-coordinate of the vertex. However, the y-coordinate of the vertex is multiplied by a. (e) The mean of the x-intercepts is the x-coordinate of the vertex. 85. (a) ( ) − − 2, 25 (b) −7, 3 (c) ( ) ( ) − − − − 4,0; 4, 21 , 0, 21 (d) y x 12 10 (3, 0) ( 7, 0) (0, 21) (4, 21) (2, 25)
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