SECTION 3.3 Quadratic Functions and Their Properties 167 14. True or False If the discriminant b ac 4 0, 2 − = the graph of f x ax bx c a , 0, 2 ( ) = + + ≠ touches the x-axis at its vertex. 15. Multiple Choice If b ac 4 0, 2 − > which conclusion can be made about the graph of f x ax bx c a , 0? 2 ( ) = + + ≠ (a) The graph has two distinct x-intercepts. (b) The graph has no x-intercepts. (c) The graph has three distinct x-intercepts. (d) The graph has one x-intercept. 16. Multiple Choice If the graph of f x ax bx c a , 0, 2 ( )= + + ≠ has a maximum value at its vertex, which condition must be true? (a) − > b a2 0 (b) − < b a2 0 (c) a 0 > (d) a 0 < (f) Set the value of a to 2, − b to 2, and c to 1. − Determine the value of the discriminant. (g) Suppose the value of the discriminant of a quadratic function is 8. How many x-intercepts will the graph of the quadratic function have? 9. The graph of a quadratic function is called a(n) . 10. The vertical line passing through the vertex of a parabola is called the . 11. The x-coordinate of the vertex of f x ax bx c a , 0, 2 ( ) = + + ≠ is . 12. True or False The graph of f x x x 2 3 4 2 ( ) = + − is concave up. 13. True or False The y-coordinate of the vertex of f x x x4 5 2 ( ) = − + + is f 2( ). Skill Building In Problems 17–24, match each graph to one the following functions. 17. f x x 1 2 ( ) = − 18. f x x 1 2 ( ) = − − 19. f x x x2 1 2 ( ) = − + 20. f x x x2 1 2 ( ) = + + 21. f x x x2 2 2 ( ) = − + 22. f x x x2 2 ( ) = + 23. f x x x2 2 ( ) = − 24. f x x x2 2 2 ( ) = + + A. x y 3 2 22 21 (21, 0) B. (21, 21) x y 2 2 22 C. (0, 21) x y 2 2 22 22 D. (21, 1) x y 3 2 22 21 E. (0, 21) x y 1 2 22 23 F. (1, 0) x y 21 3 22 G. (1, 1) x y 3 21 3 21 H. (1, 21) x y 2 22 22 In Problems 25–32, (a) find the vertex and axis of symmetry of each quadratic function. (b) Determine whether the graph is concave up or concave down. (c) Graph the quadratic function. 25. f x x 3 2 2 ( ) ( ) = − − 26. f x x 4 1 2 ( ) ( ) = − + − 27. f x x 2 3 5 2 ( ) ( ) = − − + 28. f x x 3 1 4 2 ( ) ( ) = + − 29. f x x 2 6 3 2 ( ) ( ) = − + 30. f x x 1 2 1 3 2 ( ) ( ) = + − 31. f x x 1 3 1 2 7 6 2 ( ) ( ) = − − − 32. f x x 5 2 ( ) ( ) = − + In Problems 33–44, graph the function f by starting with the graph of y x2 = and using transformations (shifting, compressing, stretching, and/or reflecting). [Hint: If necessary, write f in the form f x a x h k. 2 ( ) ( ) = − + ] 33. f x x 1 4 2 ( ) = 34. f x x2 4 2 ( ) = + 35. f x x 2 2 2 ( ) ( ) = + − 36. f x x 3 10 2 ( ) ( ) = − − 37. f x x x4 2 2 ( ) = + + 38. f x x x6 1 2 ( ) = − − 39. f x x x 2 4 1 2 ( ) = − + 40. f x x x 3 6 2 ( ) = + 41. f x x x2 2 ( ) = − − 42. f x x x 2 6 2 2 ( ) = − + + 43. f x x x 1 2 1 2 ( ) = + − 44. ( ) = + − f x x x 2 3 4 3 1 2 In Problems 45–60, (a) find the vertex and the axis of symmetry of each quadratic function, and determine whether the graph is concave up or concave down. (b) Find the y-intercept and the x-intercepts, if any. (c) Use parts (a) and (b) to graph the function. (d) Find the domain and the range of the quadratic function. (e) Determine where the quadratic function is increasing and where it is decreasing. (f) Determine where f x 0 ( ) > and where f x 0. ( ) < 45. f x x x2 2 ( ) = + 46. f x x x4 2 ( ) = − 47. f x x x6 2 ( ) = − − 48. f x x x4 2 ( ) = − + 49. f x x x2 8 2 ( ) = + − 50. f x x x2 3 2 ( ) = − − 51. f x x x2 1 2 ( ) = + + 52. f x x x6 9 2 ( ) = + + 53. f x x x 2 2 2 ( ) = − + 54. f x x x 4 2 1 2 ( ) = − + 55. f x x x 2 2 3 2 ( ) = − + − 56. f x x x 3 3 2 2 ( ) = − + − 57. f x x x 3 6 2 2 ( ) = + + 58. f x x x 2 5 3 2 ( ) = + + 59. f x x x 4 6 2 2 ( ) = − − + 60. f x x x 3 8 2 2 ( ) = − +
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