1. y x- intercepts x

2. y x x- intercepts

3. x y

4. x y

5. x y

6. x y

7. Period = 2π/|b| = 2π/3 2 -2 π/6 5π/6 π/2 Draw one full period of 2sin(3x–π/2) 2π/3 This is the graph of 2sin(3x). Now click to see the phase shift and to get 2sin(3x–π/2) π/3 Amplitude = |a| = 2 Phase shift = -c/b = π/6

8. Graph one full period of sin(x–π /2) –1/2 a = 1, b = 1,c = – π/2 and d = –1/2 Amplitude = |a| = 1 Period = 2π/b = 2π Phase shift = – c/b = π/2 Vertical translation: 1/2 units down y = sin(x) y = sin(x–π /2) y = sin(x–π /2) –1/2 1 1/2 –1 –3/2 Phase shift π/2 units right Vertical translation ½ units down Section 5.7 Question 43

9. Graph one full period of 2sin(3x–π /2) +1 a = 2, b = 3,c = – π/2 and d = 1 Amplitude = |a| = 2 Period = 2π/b = 2π/3 Phase shift = – c/b = π/6 Vertical translation: 1 unit up 2 3 –1 –2 π/6 π/22π/3 x y π/3 This is the graph of 2sin(3x). Now click to see the phase shift, vertical translation and to get 2sin(3x–π/2)+1 1

10. Graph one full period of sin(x+π /6) a = 1, b = 1,c = π/6 Amplitude = |a| = 1 Period = 2π/b = 2π Phase shift = – c/b = –π/6 1 –1 π/2π 3π/22π2π x y –π/6 Section 5.7 Question 18 This is the graph of sin(x). Now click to see the phase shift and to get sin(x+π/6)

11. Graph one full period of cos(2x–π/3) a = 1, b = 2,c = – π/3 Amplitude = |a| = 1 Period = 2π/b = π Phase shift = – c/b = π/6 1 –1 x y π/2 ππ/6π/43π/4 Section 5.7 Question 20 This is the graph of cos(2x). Now click to see the phase shift and to get cos(2x-π/3) 7π/6

12. Graph one full period of (1/2)sin(πx/3) a = 1/2, b = π/3 Amplitude = |a| = 1/2 Period = 2π/b = 6 1/2 –1/2 3/2 9/2 6 x y 3 Section 5.5 Question 38

13. x y

14. 2 –2–2 1 –1 In [0, π], In [π, 2π], 0≤ sinx ≤ 2sinx 2sinx ≤ sinx ≤ 0 Graph one full period of 2sinx and sinx

15. x y

16. x y

17. x y

18. x y

19. x y 2 –2 –2 Draw one full period of y = 2tan(x/2) a = 2 and b = 1/2, 4b = 2 Asymptotes: Lets draw asymptotes Mark 2 and –2 on the y-axis and ±π/4b = ±π/2 on the x- axis x = ±2π/4b = ± 2π/2 = ± π Now we can draw the graph Section 5.6 Question 29

20. Graph one full period of (3/2)csc(3x) a = 3/2, b = 3 Period = 2π/b = 2 π/3 π/6π/3 π/2 2π/3 Section 5.6 Question 34 3/2 –3/2

21. Graph one full period of (1/3)tanx a = 1/3, b = 1→4b = 4 Period = π/|b| = π π/4 –π/4 1/3 –1/3 Section 5.6 Question 22 –π/2π/2

22. Graph one full period of 2cscx x y a =2, b = 1 Period = 2π/b = 2π 2 –2–2 π/2 π 3π/2 2π2π Section 5.6 Question 28

23. Graph one full period of -3sec(2x/3) x y a = –3, b = 2/3 Period = 2π/b = 3π 3 –3 3π/43π/29π/43π3π Section 5.6 Question 36

24. x y 3 –3 –3 Draw one full period of y = –3tan(3x) a = –3 and b = 3, 4b = 12 Asymptotes: Lets draw asymptotes Mark 3 and –3 on the y-axis and ±π/4b = ±π/12 on the x-axis x = ±2π/4b = ± 2π/12 = ± π/6 Period = π/b = π/2 Now we can draw the graph Section 5.6 Question 30

25. x y 1/2 –1/2 Draw one full period of y = (1/2)cot(2x) a = 1/2 and b = 2, 4b = 8 Asymptotes: Lets draw asymptotes Mark 1/2 and –1/2 on the y-axis and π/8, 2π/8, 3π/8 and 4π/8 on the x-axis x = π/b = π/2 and x = 0(y-axis) Period = π/b = π/2 Now we can draw the graph Section 5.6 Question 32

26. Graph one full period of 3/2sin(x /4+3π /4) a =3/2, b = 1/4,c = 3π/4 Amplitude = |a| = 3/2 Period = 2π/b = 8π Phase shift = – c/b = –3π 2π 6π6π 8π8π 3/2 –3/2 –3π x y 4π4π This is the graph of 3/2sin(x/4). Now click to see the phase shift and to get 3/2sin(x/4+3π/4)

27. Graph one full period of sec(x − π/2 )+1 x y a = 1, b = 1, c = −π/2, d = 1 Period = 2π/|b| = 2π Phase shift = −c/b = π/2 Vertical translation : (d =) 1 unit up 1 –1 π 3π/22π2π Section 5.7 Question 50 π/2 2 cos(x) sec(x) sec(x − π/2 ) Click to shift π/2 unit to right Click to shift 1 unit up sec(x − π/2 )+1 | |||

28. Graph one full period of csc(x/3-π/12)+4 x y a = 1, b = 1/3, c = π/3, d = 4 Period = 2π/|b| = 6π Phase shift = -c/b = π/4 Vertical translation: 4 unit up 1 –1–1 3π/2 3π3π 9π/2 6π6π Section 5.7 Question 48 || | | π/4 5 sin(x/3) csc(x/3−π/12) csc(x/3) csc(x/3-π/12)+4 3

29. 1/2sin(3x) ≥ 0 1/2sin(3x) ≤ 0 Sketch the graph of y = |(1/2)sin(3x)| Click to see y = |(1/2)sin(3x)| π/3 2π/3 -π/3 -2π/3 -1/2 1/2 Range of 1/2sin(3x) Range of |1/2sin(3x)| 0 One period of 1/2sin(3x) One period of |1/2sin(3x)| Section 5.5 Question 48

30. Sketch the graph of y = cos 2 (x) 1 π/2 -π/2 π -π-π π/4 3π/2 -3π/2 -π/4 Section 5.5 Question 65 1/2

31. Sketch the graph of y = sin|x| π 2π2π -π-π -2π 1 0 Section 5.5 Question 68 y = sin|x| = sin(x) if x ≥ 0 − sin(x) if x ≤ 0