Graphs of Sine and Cosine Five Point Method

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Graphs of Sine and Cosine Five Point Method
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Plan for the Day

• Review Homework
• 4.5 P 307 3-21 odd, 23-26 all
• The effects of b and c together in the
  equationsy a (cos (bx c)) dy a (sin
  (bx c)) d
• Graphing of Sine and Cosine Functions using the 5
  key points
• Homework
• Quiz next time

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Cosine Function
Graph of the Cosine Function
To sketch the graph of y cos x first locate the
key points.These are the maximum points, the
minimum points, and the intercepts.
Then, connect the points on the graph with a
smooth curve that extends in both directions
beyond the five points. A single cycle is called
a period.
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Sine Function
Graph of the Sine Function
To sketch the graph of y sin x first locate the
key points.These are the maximum points, the
minimum points, and the intercepts.
Then, connect the points on the graph with a
smooth curve that extends in both directions
beyond the five points. A single cycle is called
a period.
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Properties of Sine and Cosine Functions
Properties of Sine and Cosine Functions
The graphs of y sin x and y cos x have
similar properties
1. The domain is the set of real numbers.
3. The maximum value is 1 and the minimum value
is 1.
4. The graph is a smooth curve.
6. The cycle repeats itself indefinitely in both
directions of the x-axis.
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Summarizing

• Standard form of the equations y a (cos (bx
  c)) dy a (sin (bx c)) d
• a - a is called the amplitude, like our other
  functions it is like a stretch it affects y or
  the output
• If a lt 0 it also causes a reflection across the
  x- axis
• d vertical shift, it affects y or the
  output
• c horizontal shift, it affects x or ? or
  the input
• b period change (squishes or stretches
  out the graph horizontal stretch or shrink) to
  find the new period, 2p/b
• The combination of b and c has another effect
  

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Dealing with (bx c)

• The c causes a shift (opposite the sign) left
  and right, b it changes the frequency of the
  graph (2p/b is the new period), it is a
  horizontal stretch or shrink.
• When they are together, you apply the frequency
  change and then the shift
• There is a method to complete this

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You begin by adjusting the reference period

• Start with the standard key points
• Determine where the new reference period begins
  and end
• Set new intervals for the maximums, minimums, and
  zeros.
• Adjust the x values based upon this
  information.
• Adjust the y values with the amplitude and
  vertical shift.
• Plot your new points and graph!

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You begin by adjusting the reference period

• The original reference period is 2p and regular
  intervals of p/2. If there is a b or c (or both)
  that can change.
• The parent has a reference period that begins
  at zero. You need to find the new beginning of
  the reference period.Find the new beginning, (bx
  c 0), solve for x. x is the new beginning.
• The original reference period ends at 2p, find
  the new end (bx c 2p), solve for x. x is the
  new end.
• The original reference period is 2p and has 4
  equal periods of p/2. Find the new period (2p/b
  ), and divide the new period into 4 equal parts
  to create the new intervals.
• Use this information to find new x values in key
  points
• Adjust the y values of the key points by applying
  the amplitude (with sign or a) and the vertical
  shift (d)

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Example

• Graph
• Begin with our key points.
• Where do they come from?

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Example

• Find the new beginning bx c 0,solve for x.
  x is the new beginning.
• Find the new end bx c 2p, solve for x. x
  is the new end.
• Find the new period 2p/b

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Example

• Find the new beginning p/8
• Find the new end 9p/8
• Find the new period p
• Break the new period into 4 equal intervals p/4

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Example

• Beginning p/8 End 9p/8
• New intervals p/4

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Example
3x114
3x011
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Example
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Calculator Issues

• Window settings
• Using your reference period to set your window
• Setting scale based upon your new intervals

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Summarizing

• How do you put it all together?
• Identify the key points of your basic graph
• Find the new period (2p/b)
• Find the new beginning (bx - c 0)
• Find the new end (bx - c 2p)
• Divide the new period into 4 equal parts to
  create new interval to find x values of the key
  points
• Adjust the y values of the key points by applying
  the amplitude (with sign or a) and the vertical
  shift (d)

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Homework 25

• Page 308 41, 42, 44, 49, 51