21. π = 3csc ⎛ 2 − ⎞ y x ⎜ 4 ⎟ ⎝ ⎠ Begin with the graph of = csc and apply the y x following transformations: π π 1) Shift right units ⎡ = csc ⎛ − ⎞⎤ y x 4 ⎢ ⎜ 4 ⎟⎥ ⎝ ⎠ 2) Horizontally compress by a factor of 1 2 π ⎡ ⎛ ⎞⎤ = − csc 2 y x ⎢ ⎜ 4 ⎟⎥ π ⎛ ⎞ ⎝ ⎠ 23. = − cot ⎜ 2x + ⎟ y 3) Vertically stretch by a factor of 3 ⎝ 2 ⎠ Begin with the graph of π ⎡ ⎛ ⎞⎤ following transformations: = 3csc 2 − y x ⎢ ⎜ ⎟⎥ 4 ⎝ ⎠ π π 1) Shift left units ⎡ = cot ⎛ + ⎞⎤ y x ⎢ ⎜ 2 ⎟⎥ 2 ⎝ ⎠ 2) Horizontally compress by a factor of 1 2 π ⎡ ⎛ ⎞⎤ = cot 2 + y x ⎢ ⎜ 2 ⎟⎥ ⎝ ⎠ 3) Reflect about the x-axis ⎡ ⎛ π ⎞⎤ = − cot 2x y + ⎢ ⎜ 2 ⎟⎥ ⎝ ⎠ (3x ) π 1 sec − 22. = sec and apply the y x 2 Begin with the graph of = y following transformations: ( )⎤ π π 1) Shift right units ⎡ = sec − y x 2) Horizontally compress by a factor of 1 3 (3x )⎤ π ⎡ = sec − y
21. y = 3csc ⎛ 2 x − π ⎞ ⎜ 4 ⎟ ⎝ ⎠ Begin with the graph of y = csc x and apply the following transformations: 1) Shift right π units ⎡ y = csc ⎛ x − π ⎞⎤ 4 ⎢ ⎜ 4 ⎟⎥ ⎝ ⎠ 2) Horizontally compress by a factor of 1 2 ⎡ ⎛ π ⎞⎤ y = csc 2 x − ⎢ ⎜ 4 ⎟⎥ ⎛ π ⎞ ⎝ ⎠ 23. y = − cot ⎜ 2x + ⎟ 3) Vertically stretch by a factor of 3 ⎝ 2 ⎠ Begin with the graph of ⎡ ⎛ π ⎞⎤ y = 3csc 2 x − following transformations: ⎢ ⎜ 4 ⎟⎥ ⎝ ⎠ 1) Shift left π units ⎡ y = cot ⎛ x + π ⎞⎤ 2 ⎢ ⎜ 2 ⎟⎥ ⎝ ⎠ 2) Horizontally compress by a factor of 1 2 ⎡ ⎛ π ⎞⎤ y = cot 2 x + ⎢ ⎜ 2 ⎟⎥ ⎝ ⎠ 3) Reflect about the x-axis ⎡ ⎛ π ⎞⎤ y = − cot 2x + ⎢ ⎜ 2 ⎟⎥ ⎝ ⎠ 22. y = 1 sec (3x − π ) 2 Begin with the graph of y = sec x and apply the following transformations: 1) Shift right π units ⎡ y = sec ( x − π )⎤ 2) Horizontally compress by a factor of 1 3 ⎡ y = sec (3x − π )⎤
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