Concepts Examples The graph of y =c +a sin 3 b1x −d2 4 or y =c +a cos 3 b1x −d2 4 , with b 70, has the following characteristics. 1. amplitude a 2. period 2p b 3. vertical translation up c units if c 70 or down c units if c 60 4. phase shift to the right d units if d 70 or to the left d units if d 60 6.3 Graphs of the Sine and Cosine Functions 6.4 Translations of the Graphs of the Sine and Cosine Functions Sine and Cosine Functions x y 0 –1 1 y = sin x p 2 3p 2p p 2 Domain: 1-∞, ∞2 Range: 3-1, 14 Amplitude: 1 Period: 2p x y 0 –1 1 y = cos x p 2 3p 2p p 2 Domain: 1-∞, ∞2 Range: 3-1, 14 Amplitude: 1 Period: 2p Graph y = 1 + sin 3x. x y 0 –1 1 2 y = 1 + sin 3x 3 4 p p p 3 2p 3 amplitude: 1 domain: 1-∞, ∞2 period: 2p 3 range: 30, 24 vertical translation: up 1 unit Graph y = -2 cos Ax + p 2 B . x y 0 –2 2 y = –2 cos (x + ) P 2 p 2 5p 2 3p 2 7p 2 p 2 – amplitude: 2 domain: 1-∞, ∞2 period: 2p range: 3-2, 24 phase shift: left p 2 units 670 CHAPTER 6 The Circular Functions and Their Graphs Graph y = 2 tan x over a one-period interval. y x 0 y = 2 tan x 2 –2 –p 2 p 2 period: p domain: Ex x ≠12n + 12 p 2 , where n is any integerF range: 1-∞, ∞2 x y 0 y = tan x –p 2 p 2 1 Do main: Ex x ≠12n + 12 p 2 , where n is any integerF Range: 1-∞, ∞2 Period: p x y 0 y = cot x p 2 p 1 Do main: 5x x ≠np, where n is any integer6 Range: 1-∞, ∞2 Period: p Tangent and Cotangent Functions 6.5 Graphs of theTangent and Cotangent Functions
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