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x - intercepts

y. x - intercepts. x. y. x. x - intercepts. y. x. y. x. y. x. y. x. Draw one full period of 2sin(3 x– π /2). Period = 2 π /|b| = 2 π /3. 2. Amplitude = |a| = 2. Phase shift = -c/b = π /6. 2 π /3. π /2. 5 π /6. π /6. π /3. -2.

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x - intercepts

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  1. y x- intercepts x

  2. y x x- intercepts

  3. y x

  4. y x

  5. y x

  6. y x

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

  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 Section 5.7 Question 43 Vertical translation ½ units down 1 y = sin(x–π /2) 1/2 –1 y = sin(x) –3/2 y = sin(x–π /2) –1/2 Phase shift π/2 units right

  9. Graph one full period of 2sin(3x–π /2) +1 y 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 3 2 1 π/2 2π/3 π/6 π/3 x –1 –2 This is the graph of 2sin(3x). Now click to see the phase shift , vertical translation and to get 2sin(3x–π/2)+1

  10. Graph one full period of sin(x+π /6) y a =1, b =1,c =π/6 Amplitude= |a| =1 Period= 2π/b = 2π Phase shift= – c/b = –π/6 Section 5.7 Question 18 1 3π/2 2π –π/6 π/2 π x –1 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) y Section 5.7 Question 20 a =1, b = 2,c = – π/3 Amplitude= |a| =1 Period= 2π/b = π Phase shift= – c/b =π/6 1 π/2 π/6 π/4 3π/4 π 7π/6 x –1 This is the graph of cos(2x). Now click to see the phase shift and to get cos(2x-π/3)

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

  13. y x

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

  15. y x

  16. y x

  17. y x

  18. y x

  19. Draw one full period of y= 2tan(x/2) y a = 2 and b = 1/2 , 4b = 2 Asymptotes: x = ±2π/4b = ± 2π/2 = ± π Lets draw asymptotes 2 Mark 2 and –2 on the y-axis and ±π/4b = ±π/2 on the x-axis x –2 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 = 2π/3 3/2 π/2 π/6 π/3 2π/3 –3/2 Section 5.6 Question 34

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

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

  23. Graph one full period of -3sec(2x/3) y a = –3 , b = 2/3 Period = 2π/b = 3π 3 3π/4 3π/2 9π/4 3π x –3 Section 5.6 Question 36

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

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

  26. Graph one full period of 3/2sin(x /4+3π /4) y a =3/2, b = 1/4,c = 3π/4 Amplitude= |a| = 3/2 Period= 2π/b = 8π Phase shift= – c/b = –3π 3/2 6π 8π x –3π 4π 2π –3/2 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 y sec(x − π/2 )+1 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 sec(x) sec(x − π/2 ) 2 1 cos(x) π | | | | 3π/2 2π x π/2 –1 Click to shift π/2 unit to right Click to shift 1 unit up Section 5.7 Question 50

  28. Graph one full period of csc(x/3-π/12)+4 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 csc(x/3-π/12)+4 5 csc(x/3−π/12) 3 1 x sin(x/3) 9π/2 | | | | 3π 6π 3π/2 π/4 –1 csc(x/3) Section 5.7 Question 48

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

  30. Sketch the graph of y = cos2(x) 1 1/2 -π -π/4 π 3π/2 π/2 -3π/2 -π/2 π/4 Section 5.5 Question 65

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

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