Translations 4'2

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Translations4.2

• JMerrill, 2009

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Sine Waves

• Recorded sounds, that are periodic, are some kind
  of sine wave. Most of these sine waves are not
  of the parent function type. They are either
  not lying nicely on the x-axis or are a
  combination of waves (dont worrywell get to
  that!)

YEA!!
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Signals Sound

• These terms are used to describe sine waves and
  other waveforms precisely
• Period The period is the time taken for one
  complete cycle of a repeating waveform.
• Frequency This is the number of cycles
  completed per second. The measurement unit for
  frequency is the hertz, Hz. 1 Hz 1
  cycle/second.
• The pitch of a musical note is the same as its
  frequency (which relates to the period).
• The intensity (loudness) of a musical note is the
  same as its amplitude

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Radio Waves

• Radio transmissions are a combination of two
  kinds of waves audio frequency waves that
  represent the sounds being transmitted and radio
  frequency waves that "carry" the audio
  information. All waves have a wavelength, an
  amplitude and a frequency. These properties allow
  the wave to be modified to carry sound
  information.

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AM Radio Waves

• In AM (amplitude modulation) radio transmissions,
  the amplitude of the combined audio frequency and
  radio frequency waves varies to match the audio
  signal. AM radio is subject to problems with
  static interference. Electromagnetic waves (like
  radio waves) are produced by the spark discharges
  in car ignition systems, brushes of electric
  motors and in all sorts of electrical appliances,
  as well as in thunderstorms. Background noise
  changes the amplitude of the radio wave signal
  adds random crackling noises called static.

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FM Radio Waves

• In FM (frequency modulation) radio transmissions,
  the frequency of the combined waves changes to
  reproduce the audio signal. For example, higher
  frequency is associated with the peak amplitude
  in the audio wave. FM waves do not have a problem
  with interference because the background noise
  does not modify the radio wave frequency. In
  addition FM waves give better sound reproduction.

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General Form

• Equations will be of the form
• y k A sin (Bx C), or
• y k A cos (Bx C), where
• k vertical shift
• A amplitude
• B frequencythe way to find the period
• C phase (phase shift)our new origin, found
  by C/B. However

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General Form

• Use my way and the phase shift finds itself
• y k A sin B(x C)

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Amplitude revisited

• If the center of the wave is not at the x-axis,
  then amplitude can be found by
• Amplitude can still be measured by the vertical
  distance between the center of the wave to the
  peak (and/or valley)

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Shifts

• Phase Shift
• When a graph is shifted h units horizontally,
  then x is replaced with (xc)
• Remember that a phase shifts acts in the opposite
  directionjust like all other functions.

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Shifts

• Vertical Shift
• When a graph is shifted vertically, then we add
  that number to the equation.

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Shifts

• When the sine wave is shifted units to the
  left, what is the result?
• A cosine wave!
• So, sine and cosine curves are referred to as
  general sine waves.

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Axis of the Wave

• If the x-axis is not the center of the wave, then
  you need to find the center. The center is the
  average of the peak and the valley points
• Axis of the wave x

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Example

• To find the axis of the wave
• To find the amplitude
• To verify the amplitude, what is the vertical
  distance from the axis of the wave to the peak or
  valley?

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Shifts

• When the sine wave is shifted units to the
  left, what is the result?
• A cosine wave!
• So, sine and cosine curves are referred to as
  general sine waves.

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Reminder Critical points

• To find the critical intervals (max/min,
  intercepts)
• To find the endpoints of any period

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Write the Equation

• Axis of the wave?
• Amplitude?
• Period?
• 4
• So, a 2, b ?

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Write the Equation

• To write the equation, look at the new x/y-axis
  (forget the old). Here, we changed the x-axis,
  but not the y-axis.
• Sine or cosine?
• Cosine
• The vertical shift is the amount we raised the
  x-axis.
• Equation?

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Write the Equation

• Axis of the wave?
• x 2
• Amplitude?
• Period?
• B?
• Sine or cosine?
• Cant tell? Move the y-axis. Sine or cosine?
• Now, use the new set of axes and write the
  equation.

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Write the Equation

• a 1
• Cosine wave
• Equation
• This is how the book does it.

Easier???
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Same Graph - 2nd Equation

• Axis of wave 2
• Move the y-axis to the left ½ unit.
• Now its a sine wave!
• Equation?

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Equations of the Graph

• Since sine and cosine are both general sine
  waves, both equations are correct!

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You Do

• Cosine wave
• Another cosine wave?

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You DoPart Deux

• Sine wave