(4 ) π 2 tan − x 15. = π; T 19. φ = y = A 1 2; = ω 2 Begin with the graph of = tan and apply the y x ω = 2π = 2π = following transformations: φ = φ = 2 1 ω π φ T = 1 ( )⎤ π π 1) Shift right units ⎡ = tan − y x 2 2 Assuming is positive, we have that A φ = sin(ω − ) = 2 sin(2 y A x x 2) Horizontally compress by a factor of 1 −1) 4 = 2 sin ⎡2 ⎛ − 1 (4x )⎤ x π ⎡ = tan − y ⎞⎤ ⎢ ⎜ 2 ⎟⎥ ⎣ ⎦ stretch by a factor of 2 ⎣ ⎝ ⎠⎦ (4 )⎦ = π T ⎣ = − π 2 tan y x = 3; A 16. 3) ; φ = 2 Vertical 2 ω ω = = = 2π 2π 4 φ φ = = 2 π T ω 4 φ = 2 8 Assuming is positive, we have that A = − φ = − sin(ω ) 3sin(4x 8) y A x ( = ⎡⎣4 − 3sin 2)⎤⎦ x φ = − 1 = 3; = 3π; A T 17. ω 2 y = = cot and apply the φ φ y x = = − 1 Begin with the graph of ω = = = 2π 2π 2 (2 1 x − 3 ) π 20. 3 3 9 3π 3 T ( )⎤ π ⎡ = − π 1) Shift right units cot y x following ω 2 3 1 transformations: 3 φ 1 ⋅ 2 = − 2 Assuming is positive, we have that A 2) Horizontally compress by a factor of 2 = − φ = ⎛ + ⎞ sin(ω ) 3 sin 2 2 y A x x (2 )⎤ π ⎝ ⎠ ⎡ = cot − y x 1 = 3sin ⎡ 2 ⎛ + 1 x ⎞⎤ 3) Vertically compress by a factor of ⎢ 3 ⎜ ⎟⎥ 3 2
15. A = 2; T = π; φ = 1 19. y = 2 tan (4 x − π ) ω 2 Begin with the graph of y = tan x and apply the ω = 2π = 2π = 2 φ = φ = 1 following transformations: T π ω 2 2 φ = 1 1) Shift right π units ⎡ y = tan ( x − π )⎤ Assuming A is positive, we have that y = A sin(ω x − φ ) = 2 sin(2 x −1) = 2 sin ⎡2 ⎛ x − 1 ⎞⎤ 2) Horizontally compress by a factor of 1 4 ⎡ y = tan (4x − π )⎤ ⎢ ⎜ 2 ⎟⎥ ⎣ ⎦ 16. A = 3; T = π ; 2 ⎣ φ = 2 ω ⎝ ⎠⎦ 3) stretch by a factor of 2 ⎣ y = 2 tan (4 x − π )⎦ ω = 2π = 2π = 4 φ = φ = 2 T π ω 2 4 φ = 8 Assuming A is positive, we have that y = A sin(ω x − φ ) = 3sin(4x − 8) = 3sin ⎡⎣4 ( x − 2)⎤⎦ 17. A = 3; T = 3π; φ = − 1 ω ω = 2π = 2π = 2 3 φ = φ = − 1 20. y = 1 (2 x − π ) 2 Begin with the graph of y = cot x and apply the T 3π 3 ω 2 3 3 φ 1 ⋅ 2 = − 2 3 3 9 following transformations: 1) Shift right π units ⎡ y = cot ( x − π )⎤ 1 Assuming A is positive, we have that y = A sin(ω x − φ ) = 3 sin ⎛ 2 x + 2 ⎞ ⎝ ⎠ 2) Horizontally compress by a factor of 2 ⎡ y = cot (2 x − π )⎤ 1 = 3sin ⎡ 2 ⎛ x + 1 ⎞⎤ 3) Vertically compress by a factor of ⎢ 3 ⎜ 3 ⎟⎥ 2 Vertical
62514-0132318806_ISM_02-1.docx