Graphs of Circular Functions PDF

GRAPHS OF CIRCULAR FUNCTIONS Graph: 𝑦 = sin 𝑥 Graph: 𝑦 = cos 𝑥  The domain of both 𝑦 = asin 𝑏(𝑥 − 𝑐) + 𝑑 and 𝑦 = 𝑎 cos 𝑏(𝑥 − 𝑐) + 𝑑 is the set of real numbers or (−∞, ∞). The range of both 𝑦 = asin 𝑏(𝑥 − 𝑐) + 𝑑 and 𝑦 = 𝑎 cos 𝑏(𝑥 − 𝑐) + 𝑑 is 𝑑 − 𝑎 , 𝑑 + 𝑎. The number 𝑎 is the amplitude of...

GRAPHS OF CIRCULAR FUNCTIONS Graph: 𝑦 = sin 𝑥 Graph: 𝑦 = cos 𝑥  The domain of both 𝑦 = asin 𝑏(𝑥 − 𝑐) + 𝑑 and 𝑦 = 𝑎 cos 𝑏(𝑥 − 𝑐) + 𝑑 is the set of real numbers or (−∞, ∞). The range of both 𝑦 = asin 𝑏(𝑥 − 𝑐) + 𝑑 and 𝑦 = 𝑎 cos 𝑏(𝑥 − 𝑐) + 𝑑 is 𝑑 − 𝑎 , 𝑑 + 𝑎. The number 𝑎 is the amplitude of the graphs of 𝑦 = 𝑎 sin 𝑏(𝑥 − 𝑐) + 𝑑 and 𝑦 = 𝑎 cos(𝑏(𝑥 − 𝑐)) + 𝑑. The period of 𝑦 = 𝑎 sin 𝑏(𝑥 − 𝑐) + 𝑑 and 2𝜋 𝑦 = 𝑎 cos 𝑏(𝑥 − 𝑐) + 𝑑 is. 𝑏  Find the domain, range, amplitude, and period of 𝑦 = − 2 sin(3𝑥 + 𝜋).  Solution: Note that 𝑦 = −2 sin 3𝑥 + 𝜋 can be written as 𝑦 = 𝜋 𝜋 − 2 sin 3 𝑥 +. This means that 𝑎 = −2, 𝑏 = 3, 𝑐 = − , and 3 3 𝑑 = 0. Domain: (−∞, ∞) Range: 𝑑 − 𝑎 , 𝑑 + 𝑎 = 0 − −2 , 0 + −2 = −2,2 Amplitude: 𝑎 = −2 = 2 2𝜋 2𝜋 2𝜋 Period: = = 𝑏 3 3 Find the domain, range, amplitude, and period of 𝜋 𝑦 = −2 cos −2𝑥 + −1 4  Solution: 𝜋 Note that 𝑦 = −2 cos −2𝑥 + − 1 can be written as 𝜋 4 𝑦 = −2 cos −2 𝑥 − − 1. This means that 𝑎 = −2, 𝑏 = −2, 𝜋 8 𝑐 = , and 𝑑 = −1. 8 Domain: (−∞, ∞) Range: 𝑑 − 𝑎 , 𝑑 + 𝑎 = −1 − −2 , −1 + −2 = −3,1 Amplitude: 𝑎 = −2 = 2 2𝜋 2𝜋 2𝜋 Period: = = =𝜋 𝑏 −2 2 Graph: 𝑦 = sec 𝑥 Graph: 𝑦 = csc 𝑥  The domain of 𝑦 = 𝑎 csc 𝑏(𝑥 − 𝑐) + 𝑑 is the set of 𝑥 ∈ ℝ: sin 𝑏 𝑥 − 𝑐 ≠ 0 = 𝑘𝜋 𝑥 ∈ ℝ: 𝑥 ≠ 𝑐 + , 𝑘 ∈ ℤ. 𝑏 The domain of 𝑦 = 𝑎 sec 𝑏(𝑥 − 𝑐) + 𝑑 is the set of 𝑥 ∈ ℝ: cos 𝑏 𝑥 − 𝑐 ≠ 0 = 𝑘𝜋 𝑥 ∈ ℝ: 𝑥 ≠ 𝑐 + , 𝑘 is an odd integer. 2𝑏 The range of both 𝑦 = 𝑎 csc 𝑏(𝑥 − 𝑐) + 𝑑 and 𝑦 = 𝑎 sec 𝑏(𝑥 − 𝑐) + 𝑑 is (−∞, 𝑑 − 𝑎 ሿ ∪ 𝑑 + 𝑎 ,∞.  The period of both 𝑦 = 𝑎 csc(𝑏(𝑥 − 2𝜋 𝑐)) + 𝑑 and 𝑦 = 𝑎 sec 𝑏(𝑥 − 𝑐) + 𝑑 is. 𝑏 The asymptotes of 𝑦 = 𝑎 csc 𝑏(𝑥 − 𝑐) + 𝑑 𝑘𝜋 are 𝑥 = 𝑐 + , 𝑘 ∈ ℤ. 𝑏 The asymptotes of 𝑦 = 𝑎 sec 𝑏(𝑥 − 𝑐) + 𝑑 𝑘𝜋 are 𝑥 = 𝑐 + , 𝑘 is an odd integer. 2𝑏 Find the domain, range, amplitude, and period of 𝑦 = −2 sec(3𝑥 + 𝜋)  Solution: Note that 𝑦 = −2 sec(3𝑥 + 𝜋) can be written as 𝜋 𝜋 𝑦 = −2 sec 3 𝑥 +. This means that 𝑎 = −2, 𝑏 = 3, 𝑐 = − , 3 3 and 𝑑 = 0. 𝑘𝜋 Domain: 𝑥 ∈ ℝ: 𝑥 ≠ 𝑐 + 2𝑏 , 𝑘 is an odd integer = 𝜋 𝑘𝜋 𝑥 ∈ ℝ: 𝑥 ≠ − + ,𝑘 is an odd integer 3 6 Range: −∞, 𝑑 − 𝑎 ∪ 𝑑 + 𝑎 , ∞ = −∞, 0 − −2 ∪ ሾ0 + −2 , ∞) = −∞, −2 ∪ 2, ∞  Solution: 2𝜋 2𝜋 2𝜋 Period: = = 𝑏 3 3 𝑘𝜋 Asymptotes: 𝑥 = 𝑐 + 2𝑏 , 𝑘 is an odd integer; thus, the asymptotes 𝜋 𝑘𝜋 are 𝑥 = − + ,𝑘 is an odd integer 3 6 Find the domain, range, amplitude, and period 𝜋 of 𝑦 = −2 csc(−2𝑥 − ) − 1 4  Solution: 𝜋 Note that 𝑦 = −2 csc(−2𝑥 − )− 1 4 𝜋 can be written as 𝑦 = −2 csc −2 𝑥 + − 1. This means that 8 𝜋 𝑎 = −2, 𝑏 = −2, 𝑐 = − , and 𝑑 = −1. 8 𝑘𝜋 𝜋 𝑘𝜋 Domain: 𝑥 ∈ ℝ: 𝑥 ≠ 𝑐 + 𝑏 ,𝑘 ∈ ℤ = 𝑥 ∈ ℝ: 𝑥 ≠ − 8 + 2 ,𝑘 ∈ℤ Range: −∞, 𝑑 − 𝑎 ∪ 𝑑 + 𝑎 , ∞ = −∞, −1 − −2 ∪ ሾ−1 + −2 , ∞) = −∞, −3 ∪ 1, ∞  Solution: 2𝜋 2𝜋 Period: = =𝜋 𝑏 −2 𝑘𝜋 Asymptotes: 𝑥 = 𝑐 + ,𝑘 𝑏 ∈ ℤ; thus, the asymptotes are 𝜋 𝑘𝜋 𝑥= − + ,𝑘 ∈ℤ 8 2 Graph: 𝑦 = tan 𝑥 Graph: 𝑦 = cot 𝑥  The domain of 𝑦 = 𝑎 cot 𝑏(𝑥 − 𝑐) + 𝑑 is the set of 𝑥 ∈ ℝ: sin 𝑏 𝑥 − 𝑐 ≠ 0 = 𝑘𝜋 𝑥 ∈ ℝ: 𝑥 ≠ 𝑐 + , 𝑘 ∈ ℤ. 𝑏 The domain of 𝑦 = 𝑎 tan 𝑏(𝑥 − 𝑐) + 𝑑 is the set of 𝑥 ∈ ℝ: cos 𝑏 𝑥 − 𝑐 ≠ 0 = 𝑘𝜋 𝑥 ∈ ℝ: 𝑥 ≠ 𝑐 + , 𝑘 is an odd integer. 2𝑏 The range of both 𝑦 = 𝑎 cot 𝑏(𝑥 − 𝑐) + 𝑑 and 𝑦 = 𝑎 tan 𝑏(𝑥 − 𝑐) + 𝑑 is (−∞, ∞).  The period of both 𝑦 = 𝑎 cot(𝑏(𝑥 − 𝜋 𝑐)) + 𝑑 and 𝑦 = 𝑎 tan 𝑏(𝑥 − 𝑐) + 𝑑 is. 𝑏 The asymptotes of 𝑦 = 𝑎 cot 𝑏(𝑥 − 𝑐) + 𝑑 𝑘𝜋 are 𝑥 = 𝑐 + , 𝑘 ∈ ℤ. 𝑏 The asymptotes of 𝑦 = 𝑎 tan 𝑏(𝑥 − 𝑐) + 𝑑 𝑘𝜋 are 𝑥 = 𝑐 + , 𝑘 is an odd integer. 2𝑏  1 Graph: 𝑦 = tan 2𝑥 2  Solution: 1 1 In 𝑦 = tan 2𝑥 , 𝑎 = , 𝑏 = 2, 𝑐 = 0, and 𝑑 = 0. 2 2 𝑘𝜋 Domain: 𝑥 ∈ ℝ: 𝑥 ≠ 𝑐 + , 𝑘 is an odd integer = 2𝑏 𝑘𝜋 𝑥 ∈ ℝ: 𝑥 ≠ , 𝑘 is an odd integer 4 Range: −∞, ∞  Solution: 𝜋 𝜋 𝜋 Period: = = 𝑏 2 2 𝑘𝜋 Asymptote: 𝑥 = 𝑐 + , 𝑘 is an odd integer; thus, the 2𝑏 𝑘𝜋 asymptotes are 𝑥 = , 𝑘 is an odd integer 4  𝑥 Graph: 𝑦 = 2 cot 3  Solution: 𝑥 1 In 𝑦 = 2 cot , 𝑎 = 2, 𝑏 = , 𝑐 = 0, and 𝑑 = 0. 3 3 𝑘𝜋 Domain: 𝑥 ∈ ℝ: 𝑥 ≠ 𝑐 + 𝑏 ,𝑘∈ℤ = 𝑥 ∈ ℝ: 𝑥 ≠ 3𝑘𝜋, 𝑘 ∈ ℤ Range: −∞, ∞  Solution: 𝜋 𝜋 Period: = 1 = 3𝜋 𝑏 3 𝑘𝜋 Asymptote: 𝑥 = 𝑐 + ,𝑘∈ ℤ; thus, the asymptotes are 𝑏 𝑥 = 3𝑘𝜋, 𝑘 ∈ ℤ