section3_4 - Section 3.4 Horizontal and Vertical Shifts of...

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Unformatted text preview: Section 3.4 Horizontal and Vertical Shifts of Graphs Page 1 of 32 Section 3.4 Horizontal and Vertical Shifts of Graphs Page 2 of 32 Example 1: Sketch the graph of the function f ( x ) = x + 1. Do not plot points, but instead apply transformations to the graph of a standard function. Solution to Example 1: Example 2: Sketch the graph of the function f ( x ) = x − 1 . Do not plot points, but instead apply transformations to the graph of a standard function. Solution to Example 2: Section 3.4 Horizontal and Vertical Shifts of Graphs Page 3 of 32 Example 3: Solution to Example 3: Section 3.4 Horizontal and Vertical Shifts of Graphs Page 4 of 32 Example 4: Solution to Example 4: Section 3.4 Horizontal and Vertical Shifts of Graphs Page 5 of 32 Section 3.4 Horizontal and Vertical Shifts of Graphs Page 6 of 32 Example 5: Solution to Example 5: Section 3.4 Horizontal and Vertical Shifts of Graphs Page 7 of 32 Example 6: Solution to Example 6: Section 3.4 Horizontal and Vertical Shifts of Graphs Page 8 of 32 Section 3.4 Horizontal and Vertical Shifts of Graphs Page 9 of 32 Example 7: Solution to Example 7: Section 3.4 Horizontal and Vertical Shifts of Graphs Page 10 of 32 Section 3.4 Horizontal and Vertical Shifts of Graphs Reflecting, Stretching, and Shrinking of Graphs Page 11 of 32 Section 3.4 Reflecting, Stretching, and Shrinking of Graphs Page 12 of 32 Section 3.4 Reflecting, Stretching, and Shrinking of Graphs Page 13 of 32 Example 1: Sketch the graph of the function ( ) . Do not plot points, but instead apply transformations to the g8.(o)-nTp6(i)-.(o8ard un)-.ncat io.d 5 4 w 36 5 53 Sket ch t he gr of the funet i aph ( Do not ot points , but instead ) . pl Section 3.4 Reflecting, Stretching, and Shrinking of Graphs Page 14 of 32 Example 3: Solution to Example 3: Section 3.4 Reflecting, Stretching, and Shrinking of Graphs Page 15 of 32 Example 4: Section 3.4 Reflecting, Stretching, and Shrinking of Graphs Page 16 of 32 Solution to Example 4: Example 5: Solution to Example 5: Section 3.4 Reflecting, Stretching, and Shrinking of Graphs Page 17 of 32 Section 3.4 Reflecting, Stretching, and Shrinking of Graphs Page 18 of 32 Example 6: Solution to Example 6: Section 3.4 Reflecting, Stretching, and Shrinking of Graphs Page 19 of 32 Section 3.4 Reflecting, Stretching, and Shrinking of Graphs Page 20 of 32 Example 7: Solution to Example 7: Section 3.4 Reflecting, Stretching, and Shrinking of Graphs Page 21 of 32 Section 3.4 Reflecting, Stretching, and Shrinking of Graphs Even and Odd Functions Page 22 of 32 Section 3.4 Even and Odd Functions Page 23 of 32 Example 1: Solution to Example 1: Section 3.4 Even and Odd Functions Page 24 of 32 Therefore, the function is neither even nor odd. Example 2: Solution to Example 2: Section 3.4 Even and Odd Functions Page 25 of 32 Section 3.4 Even and Odd Functions Page 26 of 32 Example 3: Solution to Example 3: Example 4: Solution to Example 4: Section 3.4 Even and Odd Functions Page 27 of 32 Example 5: Solution to Example 5: Example 6: Solution to Example 6: Section 3.4 Even and Odd Functions Page 28 of 32 Example 7: Solution to Example 7: Section 3.4 Even and Odd Functions Page 29 of 32 Exercise Set 3.4: Transforming Functions Matching. The left-hand column contains equations that represent transformations of f(x) = x2. Match the equations on the left with the description on the right of how to obtain the graph of g from the graph of f. 1. 2. 3. 4. 5. 6. 7. 8. 9. g ( x) = ( x − 4) 2 g ( x) = x 2 − 4 g ( x) = x 2 + 4 g ( x) = ( x + 4) 2 g ( x) = − x 2 g ( x) = ( − x ) 2 g ( x) = 4x 2 Describe how the graphs of each of the following functions can be obtained from the graph of y = f(x). 19. y = f ( x) + 1 20. y = f ( x − 7 ) 21. y = f ( − x) + 3 22. y = − f ( x + 3) − 8 23. y = − 14 f ( x − 2) − 5 A. Rnithcelf B. S4hnt,esui t f e lh i t c ne il h f r C. Rnithcelf d ro n a w t h f s i e 4us.nti D. Shgit4.stinu rf E. Shgit3,stinu rf h e n t c lf r i f t h ne i s , xa r4.nsuti duwap F. Sd4.nsuti prua w ht f i G. Rnithcelf H. S4hnt,esui t f e lh i d3un.sti rauwphstfi I. S4.nsuti t f he l i S4 d r a wo n th f i u n .sti 4. f oa cr t J. K. Syblacvihtre x-axis. y.xsa-i x-axis, x- 24. y = −5 f ( − x) + 1 25. y = f ( 7 − x) + 2 26. y = f ( − x − 5 ) − 7 y-axis. Standard Functions: Sketch the graph of each of the following functions. (Plot points if necessary, but then memorize the general shape of each graph.) 27. f ( x) = x 2 28. f ( x) = x 29. f ( x) = x 30. f ( x) = x3 31. f ( x) = 1 x g ( x ) = 14 x 2 g ( x) = − x 2 − 4 10. g ( x) = ( x + 4) 2 + 3 11. g ( x) = −( x − 3) 2 + 4 12. g ( x) = ( − x + 4) 2 L. Syblacvitkrneh 4. f oa cr t Describe how the graph of g is obtained from the graph of f. (Do not sketch the graph.) 13. f ( x) = x , 14. f ( x) = x3, 15. f ( x) = x , 16. f ( x) = x 2, 17. f ( x) = , x 1 g ( x) = − x − 2 32. f ( x) = 3 x Sketch the graph of each of the following functions. Do not plot points, but instead apply transformations to the graph of a standard function. 33. f ( x) = x 2 + 3 34. f ( x) = ( x − 5 ) 2 35. f ( x) = 6 − x 2 36. f ( x) = 2− ( x − 1 )2 37. f ( x) = −3( x − 4) 2 − 2 g ( x) = −2( x + 5 )3 g ( x) = −5 x − 2 + 1 g ( x) = 16 ( x + 3) 2 − 7 3 g ( x) = +2 x+8 g ( x) = 3 − x + 4 18. f (x ) = 3 x , Exercise Set 3.4: Transforming Functions 38. f ( x) = ( x + 5 )2 + 3 39. f ( x) = 6 − x + 2 40. f ( x) = 12 − x + 1 41. f ( x) = − x + 4 + 2 42. f ( x) = 5 − x − 1 43. f ( x) = 2 x + 5 − 3 44. f ( x) = − x − 2 + 4 45. f ( x) = −( x − 4)3 + 1 46. f ( x) = − x3 − 5 47. f ( x) = 1 x −3 +6 Determine whether each of the following functions is even, odd, both or neither. 55. f ( x) = x3 − 5 x 56. f ( x) = x 2 + 3x 57. f ( x) = x 4 + 2x 2 58. f ( x) = x5 + 2x3 59. f ( x) = 2x3 + x 2 − 5 x + 1 60. f ( x) = 3x 6 + x2 2 Answer the following. 61. Tohfspgrea Defprowhagetbics w . o b n l e h rs a ho f t re a g p n m d w bi s y f ( x ) = 4 − x 2 dna g ( x) = 4 − x 2 g f. 6 4 2 x 48. f ( x) = − 2 x+4 f ( x) = 4 − x 2 g ( x) = 4 − x 2 y 6 4 2 x 49. f ( x) = − + 3 50. f ( x) = x − 6 51. f ( x) = 3 − x + 2 52. f ( x) = − 3 x + 1 − 5 Answer the following. 53. (a) Icirteymshnod,ufa _ . h t e o pr c s w i ( x-s,xia (b) Iceirtymshn,voufa _ . h t e o pr c s w i ( x-s,xia 54. (a) Iheotpcsriwymnufa y-_.nsouihfetc,xa (O?) r h no t e , i b v d (b) Iheotpcsriwymnufa ._ n s o u hi g f et , c r (O?) h n ter i o b ,v d 3 4 x −6 −4 −2 −2 −4 2 4 6 −6 −4 −2 −2 −4 2 4 6 62. Tohfspgrea Defprowhagetbics w . o b n l e h rs a ho f t re a g p n m d w bi s y f ( x ) = x 3 − 1 dna g ( x) = x3 − 1 g f. y f ( x) = x3 − 1 g ( x) = x3 − 1 2 x 2 x y-?) g i n ro x , a s −2 2 −2 2 −2 −2 y-?) g i n ro x , a s Sketch the graphs of the following functions: 63. (a) f ( x) = x 2 − 9 64. (a) f ( x) = 1 x (b) g ( x) = x 2 − 9 (b) g ( x) = 1 x Exercise Set 3.4: Transforming Functions The graph of y = f(x) is given below. Sketch the graph of each of the following functions. 8 6 4 2 −4 −2 2 4 6 8 x 10 y y = f(x) −2 −4 −6 65. y = f ( x + 2) 66. y = f ( x) − 3 67. y = f ( x − 2) − 1 68. y = f ( x + 1 ) + 5 69. y = f ( − x) 70. y = − f ( x) 71. y = 2f ( x) 72. y = 12 f ( x) 73. y = −2f ( x + 1 ) 74. y = f ( − x) − 4 Answer the following. 75. Shatosepu ( −3, 6 ) ofhpgreantsi y = f ( x ) h t a dn o i c su f fFirohnteda . y = f ( x ) d n o i a fc ut s e v a . h gp r e n o t i 76. Shatosepu ht a .p r n ha g o et i ( 2, − 7 ) ohpfargtneis f.Find ...
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