4.1 Graphs of Sine and Cosine

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4.1 Graphs of Sine and Cosine

• OBJ Graph sine and cosine

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1 DEF ? Sine Graph

• 1
• 0 p p 3p 2p
• -1 2 2
• 0 1 0 -1 0

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1 DEF ? Sine Graph

• 1
• 0 p p 3p 2p
• -1 2 2
• 0 1 0 -1 0

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y d a(trig b (x c))

• a (amplitude) multiply a times (0 1 0 -1 0 1)
• b (period) 2p
• b
• c (starting point)
• d (vertical shift)

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y sin x

• Ref.
• no
• Amp.
• 1
• Per.
• 2p
• ¼ Per.
• p/2
• St. Pt.
• 0
• Vert. Sh.
• none

• 0 1 0 1 0
• 1
• 0
• -1 p/2 3p/2 4p/2

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2 DEF ? Cosine Graph

• 0 p p 3p 2p
• 2 2
• 1 0 -1 0 1

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2 DEF ? Cosine Graph

• - p 0 p p 3p 2p
• 2 2 2
• 1 0 -1 0 1

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DEF ? Periodic function

• A function f with the property f(x) f(xp)
  for every real number x in the domain of f and
  for some real positive number p. The smallest
  possible positive value of p is the period of the
  function f.

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3 EX ? Graph y 2 sin x

• 0 p p 3p 2p
• 2 2
• 0 1 0 -1 0
• 2(0 1 0 -1 0)
• 0 2 0 -2 0

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3 EX ? Graph y 2 sin x

• 0 p p 3p 2p
• 2 2
• 0 2 0 -2 0

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DEF ? Amplitude of Sine and Cosine

• The graph of y a sin x or y a cos x will
  have the same shape as y sin x or y cos x,
  respectively, except with range - ? a ? ? y ? ? a
  ? . The number ? a ? is called the amplitude.

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y d a(trig b (x c))

• a (amplitude) multiply a times (0 1 0 -1 0 1)
• b (period) 2p
• b
• c (starting point)
• d (vertical shift)

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4 y -2 cos x

• 1 0 -1 0 1
• -2(1 0 -1 0 1)
• -2 0 2 0 -2
• 2
• 1
• 0
• -1 p/2 3p/2 4p/2
• -2

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4 y -2 cos x

• Ref.
• yes
• Amp.
• - 2
• Per.
• 2p
• ¼ Per.
• p/2
• St. Pt.
• 0
• Vert. Sh.
• none

• 1 0 -1 0 1
• -2(1 0 -1 0 1)
• -2 0 2 0 -2
• 2
• 1
• 0
• -1 p/2 3p/2 4p/2
• -2

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4 y -2 cos x

• 2
• 1
• 0
• -1 p/2 3p/2 4p/2
• -2

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DEF ? Vertical Translation

• A function of the form y d a sin b x or of
  the form y d a cos b x is shifted vertically
  when compared with y a sin b x or y a cos b x.

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y d a(trig b (x c))

• a (amplitude) multiply a times (0 1 0 -1 0 1)
• b (period) 2p
• b
• c (starting point)
• d (vertical shift)

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5 EX ? Graph y 3 2 sin x

• 0 p p 3p 2p
• 2 2

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1 DEF ? Sine Graph

• 1
• 0 p p 3p 2p
• -1 2 2
• 0 1 0 -1 0

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3 EX ? Graph y 2 sin x

• 0 p p 3p 2p
• 2 2
• 2(0 1 0 -1 0)
• 0 2 0 -2 0

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5 EX ? Graph y 3 2 sin x

• 1
• 0 p p 3p 2p
• -1 2 2
• 2(0 1 0 -1 0)
• 0 2 0 -2 0

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5 EX ? Graph y 3 2 sin x

• 1
• 0 p p 3p 2p
• -1 2 2
• 2(0 1 0 -1 0)
• 0 2 0 -2 0
• -3-3-3 -3 -3

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5 EX ? Graph y 3 2 sin x

• 1
• 0 p p 3p 2p
• -1 2 2
• -3 -1 -3 -5 -3

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DEF ? Phase Shift

• The function y sin (x c) has the shape of
• the basic sine graph y sin x, but with a
• translation ? c ? units to the right if c lt 0
• and to the left if c gt 0. The number c is
• the phase shift of the graph. The cosine
• graph has the same function traits.

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y d a(trig b (x c)

• a (amplitude) multiply a times (0 1 0 -1 0 1)
• b (period) 2p
• b
• c (starting point)
• d (vertical shift)

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EX ? Graph y sin (x p/3)6 EX ? Graph y
4 sin (x p/3)

• 2? 5? 8? 11? 14?
• -1 6 6 6 6 6
• 0 1 0 -1 0

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6 EX ? Graph y 4 sin (x p/3)

• 2? 5? 8? 11? 14?
• -1 6 6 6 6 6
• 0 1 0 -1 0
• -1(0 1 0 -1 0
• 0 -1 0 1 0

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6 EX ? Graph y 4 sin (x p/3)

• 2? 5? 8? 11? 14?
• -1 6 6 6 6 6
• 0 -1 0 1 0
• 4 4 4 4 4
• 4 3 4 5 4

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6 EX ? Graph y 4 sin (x p/3)

• 2? 5? 8? 11? 14?
• -1 6 6 6 6 6
• 4 3 4 5 4

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EX ? Graph y 3cos (x p/4)7 EX Graph
y -3 3cos(xp/4)

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EX ? Graph y 3cos (x p/4)7 EX Graph y
-3 3cos(xp/4)

• -? ? 3? 5? 7?
• 4 4 4 4 4
• 1 0 -1 0 1

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EX ? Graph y 3cos (x p/4)7 EX Graph y
-3 3cos(xp/4)

• -? ? 3? 5? 7?
• 4 4 4 4 4
• 1 0 -1 0 1
• 3(1 0 -1 0 1)
• 3 0 -3 0 3

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7 EX Graph y -3 3cos(xp/4)

• -? ? 3? 5? 7?
• 4 4 4 4 4
• 3 0 -3 0 3
• -3 -3 -3 -3 -3
• 0 -3 -6 -3 0
• __ __ __ __ __ __ __ __ __

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7 EX Graph y -3 3cos(xp/4)

• -? ? 3? 5? 7?
• 4 4 4 4 4
• __ __ __ __ __ __ __ __ __

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1 EX ? Graph y -2 sin x

• Ref, Amp
• No, 1
• Per
• 2 p
• ¼ Per 0 p p 3p 2p
• p/2 2 2
• St.Pt. 0
• Vert. Shift
• 2

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1 EX ? Graph y -2 sin x

• 0 p p 3p 2p
• 2 2
• 0 1 0 -1 0
• -2 -2 -2 -2 -2
• -2 -1 -2 -3 -2

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1 EX ? Graph y -2 sin x

• 0 p p 3p 2p
• 2 2
• 0 1 0 -1 0
• -2 -2 -2 -2 -2
• -2 -1 -2 -3 -2

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2 EX ? Graph y 3 2 cos x

• Ref, Amp
• Yes, -2
• Per
• 2 p
• ¼ Per 0 p p 3p 2p
• p/2 2 2
• St.Pt. 0
• Vert. Shift
• 3

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2 EX ? Graph y 3 2 cos x

• 0 p p 3p 2p
• 2 2
• 1 0 -1 0 1
• -2(1 0 -1 0 1)
• -2 0 2 0 -2

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2 EX ? Graph y 3 2 cos x

• 0 p p 3p 2p
• -2(1 0 -1 0 1) 2
  2
• -2 0 2 0 -2
• 3 3 3 3 3
• 1 3 5 3 1

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2 EX ? Graph y 3 2 cos x

• 0 p p 3p 2p
• -2(1 0 -1 0 1) 2
  2
• -2 0 2 0 -2
• 3 3 3 3 3
• 1 3 5 3 1