4-4 Graphing Sine and Cosine Functions
State the amplitude, period, frequency, phase
shift, and vertical shift of each function. Then
graph two periods of the function.
15. SOLUTION: 14. In this function, a = 1, b = , c = , and d = 0.
Because d = 0, there is no vertical shift.
SOLUTION: In this function, a = 3, b = 1, c = −
, and d = 0.
Because d = 0, there is no vertical shift.
Graph y = 3 sin x shifted
units to the right.
ANSWER: ,
ANSWER: amplitude = 3; period = 2π; frequency =
shift =
; phase
; vertical shift = 0
16. y = 0.25 cos x + 3
SOLUTION: 15. In this function, a =
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In this function, a = 1, b = , c = , and d = 0.
Because d = 0, there is no vertical shift.
, b = 1, c = 0, and d = 3.
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4-4 Graphing Sine and Cosine Functions
16. y = 0.25 cos x + 3
17. y = sin 3x – 2
SOLUTION: In this function, a =
SOLUTION: In this function, a = 1, b = 3, c = 0, and d = –2.
, b = 1, c = 0, and d = 3.
Graph y = sin 3x shifted 2 units down.
Graph y =
cos x shifted 3 units up.
ANSWER: amplitude = 1; period =
; frequency =
; phase
shift = 0; vertical shift = −2
ANSWER: amplitude =
; period = 2π; frequency =
; phase
shift = 0; vertical shift = 3
18. SOLUTION: 17. y = sin 3x – 2
SOLUTION: In this function, a = 1, b = 1, c = .
, and d = –1.
In this function, a = 1, b = 3, c = 0, and d = –2.
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4-4 Graphing Sine and Cosine Functions
19. 18. SOLUTION: SOLUTION: In this function, a = 1, b = 1, c = .
, and d = –1.
In this function, a = 1, b = 1, c =
.
, and d = 4.
Graph y = sin x shifted
Graph y = cos x shifted
unit down.
units to the right and 1 ANSWER: ANSWER: amplitude = 1; period = 2π; frequency =
shift = units to the left and 4 uni
; phase
; vertical shift = −1
19. SOLUTION: In this function, a = 1, b = 1, c =
.
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, and d = 4.
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