168 CHAPTER 3 Linear and Quadratic Functions 76. The graph of the function f x ax bx c 2 ( ) = + + has vertex at 1, 4 ( ) and passes through the point 1, 8. ( ) − − Find a, b, and c. 75. The graph of the function f x ax bx c 2 ( ) = + + has vertex at 0, 2 ( ) and passes through the point 1, 8 ( ). Find a, b, and c. In Problems 61–66, determine the quadratic function whose graph is given. 61. x y –3 –2 –1 1 1 2 –2 Vertex: (–1, –2) (0, –1) 62. x y –1 2 3 4 5 1 8 4 2 (0, 5) Vertex: (2, 1) 63. x y –6 –3 1 2 –8 4 6 (0, –4) Vertex: (–3, 5) 64. (0, –1) x y –1 3 5 1 2 4 –4 Vertex: (2, 3) 65. (3, 5) x y –1 3 6 8 –2 –4 2 4 Vertex: (1, –3) 66. (–4, –2) x y –3 –1 1 2 4 6 –4 –2 Vertex: (–2, 6) In Problems 67–74, determine, without graphing, whether the given quadratic function has a maximum value or a minimum value, and then find the value. 67. f x x x 3 24 2 ( ) = + 68. f x x x 2 12 2 ( ) = − + 69. f x x x 2 12 3 2 ( ) = + − 70. f x x x 4 8 3 2 ( ) = − + 71. f x x x6 1 2 ( ) = − + − 72. f x x x 2 8 3 2 ( ) = − + + 73. f x x x 5 20 3 2 ( ) = − + + 74. f x x x 4 4 2 ( ) = − Applications and Extensions In Problems 77–82, for the given functions f and g: (a) Graph f and g on the same Cartesian plane. (b) Solve f x g x . ( ) ( ) = (c) Use the result of part (b) to label the points of intersection of the graphs of f and g. (d) Shade the region for which f x g x ( ) ( ) > ; that is, the region below f and above g. 77. f x x g x x 2 1; 4 2 ( ) ( ) = − = − 78. f x x g x x 2 1; 9 2 ( ) ( ) = − − = − 79. f x x g x x 4; 2 1 2 ( ) ( ) = − + = − + 80. f x x g x x 9; 2 1 2 ( ) ( ) = − + = + 81. f x x x g x x x 5 ; 3 4 2 2 ( ) ( ) = − + = + − 82. f x x x g x x x 7 6; 6 2 2 ( ) ( ) = − + − = + − For Problems 83 and 84, use the fact that a quadratic function of the form f x ax bx c 2 ( ) = + + with b ac 4 0 2 − > may also be written intheformf x a x r x r 1 2 ( )( ) ( ) = − − , where r1 and r2 are the x-intercepts of the graph of the quadratic function. 85. Suppose that f x x x4 21. 2 ( ) = + − (a) What is the vertex of f? (b) What are the x-intercepts of the graph of f? (c) Solve f x 21 ( ) = − for x. What points are on the graph of f? (d) Use the information obtained in parts (a)–(c) to graph f x x x4 21. 2 ( ) = + − 86. Suppose that f x x x2 8. 2 ( ) = + − (a) What is the vertex of f? (b) What are the x-intercepts of the graph of f? (c) Solve f x 8 ( ) = − for x. What points are on the graph of f? (d) Use the information obtained in parts (a)–(c) to graph f x x x2 8. 2 ( ) = + − 83. (a) Find quadratic functions whose x-intercepts are 3− and 1 with a a a a 1; 2; 2; 5. = = = − = (b) How does the value of a affect the intercepts? (c) How does the value of a affect the axis of symmetry? (d) How does the value of a affect the vertex? (e) Compare the x-coordinate of the vertex with the midpoint of the x-intercepts. What might you conclude? 84. (a) Find quadratic functions whose x-intercepts are 5− and 3 with a a a a 1; 2; 2; 5. = = = − = (b) How does the value of a affect the intercepts? (c) How does the value of a affect the axis of symmetry? (d) How does the value of a affect the vertex? (e) Compare the x-coordinate of the vertex with the midpoint of the x-intercepts. What might you conclude?
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