Graphs of Sine and Cosine Functions

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Graphs of Sine and Cosine Functions

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Summary

• Basic sine and cosine curves
• Sinusoidal curve y A sin(B(x-C)) D
• Amplitude A
• Period 2p / B
• Phase shift C
• Vertical shift D

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y sin x
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y cos x
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y sin x y cos x
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Amplitude

• The amplitude is the maximum height above the
  center line.
• The standard sine curve y sin x has amplitude 1
• y A sin x has amplitude A

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Amplitude adjustment

• y sin x
• (Amp 1)
• y 2 sin x
• (Amp 2)
• y 3 sin x
• (Amp 3)

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Negative values of A

• y sin x
• (Amp 1)
• y -2 sin x
• (Amp 2)

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More fun with amplitude

• y sin x
• Amp 1
• y ½ sin x
• Amp ½
• y -½ sin x
• Amp ½

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Period

• The period is the distance between two peaks or
  valleys.
• y sin x has period 2p
• y sin(Bx) has
• period 2p/B

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Speeding up

• y cos x
• Period 2p
• y cos 2x
• Period p

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Slowing down

• y sin x
• Period 2p
• y sin(x/2)
• Period 4p
• y sin(x/3)
• Period 6p
• y sin(x/4)
• Period 8p

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Amplitude Period

• y sin x
• Period 2p
• Amp 1
• y 2 sin 3x
• Period 2p/3
• Amp 2

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Period Amplitude

• y sin x
• Period 2p
• Amp 1
• y -3 sin x/2
• Period 4p
• Amp 3

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Are you on my wavelength?
y sin x Period 2p Amp 1 y -2 cos 3x Period
2p/3 Amp 2
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Phase Shift

• Replacing x with x-c shifts the graph to the
  right by c units.
• Replacing x with xc shifts the graph to the left
  by c units.
• Example If the graph of y sin 2x is shifted to
  the right by 3 units, then the equation of the
  new graph is y sin 2(x-3)

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Its just a jump to the left

• y sin x
• y sin (x1)
• y sin (x2)
• y sin (x3)

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and then a step to the right

• y cos x
• y cos (x-1)
• y cos (x-2)
• y cos (x-3)

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Period phase shift

• y cos x/2
• y cos (x-1)/2
• y cos (x-2)/2
• y cos (x-3)/2

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Everything changes

• y -10 sin 2x
• y -10 sin 2(x-½)
• y -10 sin 2(x-1)

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Transformations

• y sin x
• y 3 sin x
• y 3 sin (2x)
• y 3 sin (2(x½))

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Vertical shifts

• If y f(x) is changed to y f(x) c, then the
  graph is shifted up by c units.
• If y f(x) is changed to y f(x) - c, then the
  graph is shifted down by c units.

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Vertical shifts - Example

• y (sin x) 1
• y (sin x)
• y (sin x) 1

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How to graph a sine wave (1)

• y A sin(B(x-C)) D
• Identify the amplitude, period, phase shift, and
  vertical shift
• Amplitude A
• Period 2p / B
• Phase shift C
• Vertical shift D

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How to graph a sine wave (2)

• The graph is centered on the line yD, and it
  oscillates between yDA and yD-A. Mark D,
  DA, D-A on the y-axis.
• A complete cycle runs between C and C2p/B. Mark
  C and C2p/B on the x-axis.
• Divide the interval from C to C2p/B into four
  equal parts
• Sketch the graph

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y 2 4 sin(2(x-p/6))

• Amplitude4. Graph oscillates between 24 and
  2-4. Mark -2,4,6 on the y-axis
• Periodp and phase shiftp/6. A complete cycle
  runs from p/6 to p/6p7p/6
• Divide p/6, 7p/6 into four equal parts (each of
  length p/4)
• Sketch the graph

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How to graph a sine wave (3)