sin ⎛ − 2π ⎞ = − 5 sin ⎛ 2π ⎞ x x 27. (ω = 4 cos to y x = cos y A 43. Comparing ) = 5 y , we x 3 ⎜ 3 ⎟ ⎜ ⎟ 3 3 ⎝ ⎠ ⎝ ⎠ ω find = 4 and = 1 . Therefore, the amplitude A = 4 and the period is 2π 2π . Because This is in the form 1 = − 5 A is 4 = 3 = sin(ω where y A x) ω and = 2π . Thus, the amplitude = 4 cos will the amplitude is 4, the graph of y x is 3 −4 lie between and 4 on the Because the y-axis. = − 5 A period is 2π , one cycle will begin at = 0 and x [0, ] end at = 2π . We divide the interval 2π x = 5 and the period is 3 3 π into four subintervals, each of length 2π = = = = 2π 2π 3 T by . finding the following values: ω 2π π π 3 0, , , 3π , and 2π 4 2 28. = 9 cos ⎛ − 3π ⎞ = 9 cos ⎛ 3π ⎞ 2 2 y x x 5 ⎜ 2 ⎟ 5 These values of determine the of ⎜ 2 ⎟ x x-coordinates the five key points on the graph. To obtain the y- ⎝ ⎠ ⎝ ⎠ of the five key points for This is in the form = 4 cos , we multiply the of the y x y-coordinates = 9 A 5 = cos(ω where y A x) 4 . The five ω = and 3π . Thus, the amplitude = cos y x five key points for is 2 key points are = by A = 9 A ( ) π = 9 and the period is ⎛ ⎛ 3π ⎞ ⎞ (π ⎜ ⎟ −4) ⎜ 5 5 0, 4 , 2 ,0 , , , 2 (2π ,0 ⎟ , , 4) ⎝ ⎠ ⎝ ⎠ = 2π = 2π = T We plot these five points and fill in the graph of 4 . ω 33. H 31. A 3 3π 29. F 32. 2 30. E I the curve. We then extend the graph in either
27. y = 5 sin ⎛ − 2π x ⎞ = − 5 sin ⎛ 2π x ⎞ 43. Comparing y = 4 cos x to y = A cos (ω x ) , we 3 ⎜ 3 ⎟ 3 ⎜ 3 ⎟ ⎝ ⎠ ⎝ ⎠ find A = 4 and ω = 1 . Therefore, the amplitude This is in the form y = A sin(ω x) where A = − 5 3 is 4 = 4 and the period is 2π = 2π . Because 1 and ω = 2π . Thus, the amplitude is the amplitude is 4, the graph of y = 4 cos x will 3 A = − 5 = 5 and the period is lie between −4 and 4 on the y-axis. Because the period is 2π , one cycle will begin at x = 0 and end at x = 2π . We divide the interval [0, 2π ] 3 3 T = 2π = 2π = 3 . into four subintervals, each of length 2π = π by ω 2π 3 4 2 finding the following values: 0, π , π , 3π , and 2π 28. y = 9 cos ⎛ − 3π x ⎞ = 9 cos ⎛ 3π x ⎞ 2 2 5 ⎜ 2 ⎟ 5 ⎜ 2 ⎟ These values of x determine the x-coordinates of ⎝ ⎠ ⎝ ⎠ the five key points on the graph. To obtain the y- This is in the form y = A cos(ω x) where A = 9 5 of the five key points for y = 4 cos x , we multiply the y-coordinates of the and ω = 3π . Thus, the amplitude is 2 five key points for key points are y = cos x by A = 4 . The five A = 9 = 9 and the period is ( ) ⎛ π ⎞ ⎛ 3π ⎞ 5 5 0, 4 , ⎜ 2 ,0 ⎟ , (π , −4) , ⎜ ,0 ⎟ , (2π , 4) 2 ⎝ ⎠ ⎝ ⎠ T = 2π = 2π = 4 . We plot these five points and fill in the graph of ω 29. F 30. E 31. A 32. I 33. H 3π 3 2 the curve. We then extend the graph in either
62514-0132318806_ISM_02-1.docx