Warm – up #4 1. Find the exact value of 2

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Warm – up #4 1. Find the exact value of 2 𝑐𝑜𝑠 2 𝜋 4 −1 2 2 2 2 −1 2 2 4 −1 1 – 1 = 0

Homework Log Fri 3/11 Lesson 7 – 8 Learning Objective: To graph sine & cosine Hw: #720 Pg. 452 #1 – 4 all, 13 – 20 all

3/11/16 Lesson 7 – 8 Sinusoidal Graph Advanced Math/Trig

Learning Objective To graph sine & cosine

Graph of y = sinx Period = 2𝜋 Amplitude = 𝑚𝑎𝑥−𝑚𝑖𝑛 2 = 1−(−1) 2 =1 x y 1 2 2 2 ≈0.7 3 2 ≈0.87 1 – 1 𝜋 6 𝜋 4 𝜋 3 𝜋 2 𝜋 3𝜋 2 2𝜋 1 –1 Period = 2𝜋 Amplitude = 𝑚𝑎𝑥−𝑚𝑖𝑛 2 = 1−(−1) 2 =1

General Sine Curve 𝑦= asin 𝑏 𝑥−𝑐 +𝑑 Amp = 𝑎 Period = 2𝜋 𝑏 We’ll do c & d tomorrow Period = 2𝜋 𝑏 If a is (+), typical sine curve If a is (–), reflect through x–axis

1. Graph y = 3sinx 𝑦= asin 𝑏 𝑥−𝑐 +𝑑 b = 1 c = 0 a = 3 d Amp = 3 𝜋 2 𝜋 3𝜋 2 2𝜋 –3 Amp = 3 Period = 2𝜋 Increment length = Period∙ 1 4 = 2π∙ 1 4 = 𝜋 2

2. Graph y = sin2x 𝑦= asin 𝑏 𝑥−𝑐 +𝑑 b = 2 c = 0 a = 1 d Amp = 1 Per: 𝜋 4 𝜋 2 3𝜋 4 𝜋 –1 Amp = 1 Per: IL: = 2𝜋 𝑏 = 2𝜋 2 = 𝜋 = Period∙ 1 4 = π∙ 1 4 = 𝜋 4

3. Graph y = 2sin3x 𝑦= asin 𝑏 𝑥−𝑐 +𝑑 b = 3 c = 0 a = 2 d Amp = 2 Per: 𝜋 6 𝜋 3 𝜋 2 2𝜋 3 –2 Amp = 2 Per: IL: = 2𝜋 𝑏 = 2𝜋 3 = Period∙ 1 4 = 2𝜋 3 ∙ 1 4 = 𝜋 6

4. Graph y = –sin4x 𝑦= asin 𝑏 𝑥−𝑐 +𝑑 b = 4 c = 0 a = –1 d Amp = 1 Per: 𝜋 8 𝜋 4 3𝜋 8 𝜋 2 –1 Amp = 1 Per: IL: = 2𝜋 𝑏 = 2𝜋 4 = 𝜋 2 a = (–) so reflect over x – axis! = Period∙ 1 4 = 𝜋 2 ∙ 1 4 = 𝜋 8

5. Graph y = –3sinx 𝑦= asin 𝑏 𝑥−𝑐 +𝑑 b = 1 c = 0 a = –3 d Amp = 3 Per: 𝜋 2 𝜋 3𝜋 2 2𝜋 –3 Amp = 3 Per: IL: = 2𝜋 𝑏 = 2𝜋 1 = 2𝜋 a = (–) so reflect over x – axis! = Period∙ 1 4 = 2𝜋∙ 1 4 = 𝜋 2

Graph of y = cosx Period = 2𝜋 Amplitude = 𝑚𝑎𝑥−𝑚𝑖𝑛 2 = 1−(−1) 2 =1 x y 3 2 ≈0.87 2 2 ≈0.7 1 2 – 1 𝜋 6 𝜋 4 𝜋 3 𝜋 2 𝜋 3𝜋 2 2𝜋 1 –1 Period = 2𝜋 Amplitude = 𝑚𝑎𝑥−𝑚𝑖𝑛 2 = 1−(−1) 2 =1

General Cosine Curve 𝑦= a𝑐𝑜𝑠 𝑏 𝑥−𝑐 +𝑑 Amp = 𝑎 Period = 2𝜋 𝑏 We’ll do c & d tomorrow Period = 2𝜋 𝑏 If a is (+), typical cosine curve If a is (–), reflect through x–axis

6. Graph y= 1 3 cos(2𝑥) 𝑦= a𝑐𝑜𝑠 𝑏 𝑥−𝑐 +𝑑 = 2 b = 0 c = 1/3 a d 𝜋 4 𝜋 2 3𝜋 4 𝜋 − 1 3 Amp = 1/3 Per: IL: = 2𝜋 𝑏 = 2𝜋 2 = 𝜋 = Period∙ 1 4 = π∙ 1 4 = 𝜋 4

7. Graph y=2cos(5𝑥) 𝑦= a𝑐𝑜𝑠 𝑏 𝑥−𝑐 +𝑑 = 5 b = 0 c = 2 a d Amp = 2 Per: 𝜋 10 𝜋 5 3𝜋 10 2𝜋 5 −2 Amp = 2 Per: IL: = 2𝜋 𝑏 = 2𝜋 5 = Period∙ 1 4 = 2𝜋 5 ∙ 1 4 = 𝜋 10

8. Graph y=−2cos( 1 2 𝑥) 𝑦= a𝑐𝑜𝑠 𝑏 𝑥−𝑐 +𝑑 = 1/2 b = 0 c = –2 a d 𝜋 2𝜋 3𝜋 4𝜋 Amp = 2 Per: IL: –2 = 2𝜋 𝑏 = 2𝜋 1/2 = 4𝜋 a = (–) so reflect over x – axis! = Period∙ 1 4 = 4π∙ 1 4 = 𝜋

9. Graph y=− 1 4 cos( 𝑥 5 ) 𝑦= a𝑐𝑜𝑠 𝑏 𝑥−𝑐 +𝑑 = 1/5 b = 0 c = –1/4 a d 5𝜋 2 5𝜋 15𝜋 2 10𝜋 Amp = 1 4 Per: IL: − 1 4 = 2𝜋 𝑏 = 2𝜋 1/5 = 10𝜋 a = (–) so reflect over x – axis! = Period∙ 1 4 = 10π∙ 1 4 = 5𝜋 2

Ticket Out the Door Graph y = – cos 𝑥 4

Homework #720 Pg. 452 #1 – 4 all, 13 – 20 all