1. 1. Graph y=5cos( 1 4 𝑥) 𝑦= a𝑐𝑜𝑠 𝑏 𝑥−𝑐 +𝑑 = 1/4 b = 0 c = 5 a d Amp = 52𝜋 4𝜋 6𝜋 8𝜋 −5
Amp = 5
Per:
IL:
= 2𝜋 𝑏
= 2𝜋 1/4
= 8𝜋
= Period∙ 1 4
= 8𝜋∙ 1 4
= 2𝜋

2. Homework Log Mon 3/14 Lesson 7 – 8 Learning Objective:
To graph sine & cosine with phase shift
Hw: #721 Pg – 11 odd, 29 – 35 odd Add ‘d’ values:
#29: + 1, #31: – 1,
#33: + 2, #35: – 2

3. 3/14/16 Lesson 7 – 8 Sinusoidal Graph with Phase Shift
Advanced Math/Trig

4. Learning Objective
To graph sine & cosine with phase shift

5. General Sine & Cosine Curves
𝑦= asin 𝑏 𝑥−𝑐 +𝑑
𝑦= a𝑐𝑜𝑠 𝑏 𝑥−𝑐 +𝑑
Amp = 𝑎
𝑐=𝐻𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑆ℎ𝑖𝑓𝑡
Period = 2𝜋 𝑏
𝑑=𝑉𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑆ℎ𝑖𝑓𝑡
first point of graph

6. 1. Graph y=sin 𝑥+ 𝜋 4 𝑦= asin 𝑏 𝑥−𝑐 +𝑑 b = 1 c = −𝜋/4 a d = 0 Per = 2𝜋
Check Period: 1
7𝜋 4 − − 𝜋 4 =2𝜋
𝜋 4
3𝜋 4
5𝜋 4
− 𝜋 4
7𝜋 4
Per = 2𝜋
IL:
Period∙ 1 4
= 2π∙ 1 4
= 2𝜋 4 –1
Check a point:
(easier to match c)
𝜋 4 , 1
− 𝜋 4 + 2𝜋 4 = 𝜋 4
+ 2𝜋 4 = 3𝜋 4
+ 2𝜋 4 = 5𝜋 4
+ 2𝜋 4 = 7𝜋 4

7. 2. Graph y=3cos 𝑥− 𝜋 6 b = 1 c =𝜋/6 a = 3 d = 0 Per = IL:
Check Period: b c a d
= 1
=𝜋/6
= 3
= 0
13𝜋 6 − 𝜋 6 =2𝜋 3
Per =
IL:
2𝜋 𝑏
= 2𝜋 1
=2𝜋
𝜋 6
2𝜋 3
7𝜋 6
5𝜋 3
13𝜋 6 –3
= 2𝜋∙ 1 4
= 𝜋 2
Check a point:
7𝜋 6 , 3
𝜋 6 + 3𝜋 6 = 4𝜋 6
+ 3𝜋 6 = 7𝜋 6
+ 3𝜋 6 = 10𝜋 6
+ 3𝜋 6 = 13𝜋 6

8. 3. Graph y= 1 2 sin 3𝑥− 𝜋 2 b c a d = 3 =𝜋/6 = 1/2 = 0
Factor out b 
Check Period:
1 2
5𝜋 6 − 𝜋 6 = 4𝜋 6 = 2𝜋 3
Per =
IL:
2𝜋 𝑏
= 2𝜋 3
𝜋 6
𝜋 3
𝜋 2
2𝜋 3
5𝜋 6
= 2𝜋 3 ∙ 1 4
= 𝜋 6
− 1 2
Check a point:
𝜋 3 , 1 2
𝜋 6 + 𝜋 6 = 2𝜋 6
+ 𝜋 6 = 3𝜋 6
+ 𝜋 6 = 4𝜋 6
+ 𝜋 6 = 5𝜋 6

9. 4. Graph y=− 1 3 cos 4𝑥+𝜋 𝑦=− 1 3 cos 4 𝑥+ 𝜋 4 b = 4 Factor out b  cd
= 4
=−𝜋/4
= –1/3
= 0
Factor out b 
1 3
Per =
IL:
2𝜋 𝑏
= 2𝜋 4
= 𝜋 2
− 𝜋 4
− 𝜋 8
𝜋 4
𝜋 8
Check Period:
− 1 3
= 𝜋 2 ∙ 1 4
= 𝜋 8
𝜋 4 − − 𝜋 4 = 𝜋 2
Check a point:
− 2𝜋 8 + 𝜋 8 =− 𝜋 8
+ 𝜋 8 = 0𝜋 8
+ 𝜋 8 = 𝜋 8
+ 𝜋 8 = 2𝜋 8
𝜋 4 , − 1 3

10. 5. Graph y=4cos 1 2 𝑥+ 𝜋 4 −2 𝑦=4 cos 1 2 𝑥+ 𝜋 2 −2 b = 1/2d
= 1/2
=−𝜋/2
= 4
= –2
Factor out b: 4 2
Per =
IL:
2𝜋 𝑏
= 2𝜋 1/2
𝜋 2
=4𝜋
− 𝜋 2
3𝜋 2
5𝜋 2
7𝜋 2 –2
Check Period: –4
= 4𝜋∙ 1 4
= 𝜋
7𝜋 2 − − 𝜋 2 =4𝜋 –6
Check a point:
− 𝜋 2 + 2𝜋 2 = 𝜋 2
+ 2𝜋 2 = 3𝜋 2
+ 2𝜋 2 = 5𝜋 2
+ 2𝜋 2 = 7𝜋 2
𝜋 2 , −2

11. 6. Graph y=2sin 1 3 𝑥− 𝜋 12 +3 𝑦=2 sin 1 3 𝑥− 𝜋 4 +3 b = 1/3c a d
= 1/3
=𝜋/4
= 2
= 3 5
Factor out b: 4
Check Period: 3
25𝜋 4 − 𝜋 4 =6𝜋 2
= 2𝜋 1/3
Per =
IL: 1
2𝜋 𝑏
=6𝜋
19𝜋 4
𝜋 4
7𝜋 4
13𝜋 4
25𝜋 4
= 6𝜋∙ 1 4
= 6𝜋 4 –2
Check a point:
𝜋 4 + 6𝜋 4 = 7𝜋 4
+ 6𝜋 4 = 13𝜋 4
+ 6𝜋 4 = 19𝜋 4
+ 6𝜋 4 = 25𝜋 4
13𝜋 4 , 3

12. 7. Graph y=−sin 1 2 𝑥+3𝜋 −2 𝑦= −sin 1 2 𝑥+6𝜋 −2 b = 1/2 Factor out b:d
= 1/2
=−6𝜋
= –1
= –2
Factor out b: 1
−6𝜋
−5𝜋
−4𝜋
−3𝜋
−2𝜋 –1
= 2𝜋 1/2
Per =
IL:
2𝜋 𝑏
=4𝜋
Check Period:
−6𝜋− −2𝜋 =4𝜋 –2
= 4𝜋∙ 1 4
= 𝜋
Check a point:
13𝜋 4 , 3 –3
−6𝜋+𝜋=−5𝜋
+π=−4𝜋
+𝜋=−3𝜋
+𝜋=−2𝜋

13. Ticket Out the Door
Graph y=2sin 3x+4𝜋 −5

14. Homework
#721 Pg – 11 odd, 29 – 35 odd Add ‘d’ values: #29: + 1 #31: – 1 #33: + 2 #35: – 2