Graphing Form of Sine and Cosine Functions

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Graphing Form of Sine and Cosine Functions
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Period

• The length of one cycle of a graph.

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Initial Trigonometric Graphing Form
Sine
Do not write these on your worksheet yet. We
still need to add one more parameter.
Cosine
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Requirements for a Sine/Cosine Graph
x-intercept
2
Arrows (to show that there infinite cycles)
3
1
5
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At least one Period (in other words, at least 5
consecutive critical points accurately plotted)
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The Amplitude and the Effect of a
Amplitude Half of the distance between the
maximum and minimum values of the range of a
periodic function with a bounded range.
a lt 0
a gt 1
0ltalt1
a 1
Amplitude
1
3
0.5
1
The amplitude is the absolute value of a! It is a
positive distance.
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Example Sine
Transformation Flip the parent graph and
translate it 3Pi/2 units to the left.
Transformation
New Equation
y 0
Period
x -3p/2
You need at least 5 consecutive critical points.
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Example Cosine
Transformation Translate the parent graph Pi/2
units to the left and 1 unit down.
Transformation
New Equation
Period
y -1
x -p/2
You need at least 5 consecutive critical points.
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Sine v Cosine
Sine
Cosine
(Press the Graph)
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Example Sine or Cosine?
Transformation
Amplitude -
2
Graph -
Translation -
3 units up and
Orientation -
Period -
2p
New Equation
Since the Sine and Cosine graphs are periodic and
translations of each other, there are infinite
equations that represent the same curve. Here
are two examples.
y 3
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Example Sine or Cosine?
Transformation
Amplitude -
2
Graph -
Sine
3 units up and 3p/4 to the left
Translation -
3 units up and
Orientation -
Positive
Period -
2p
New Equation
y 3
x -3p/4
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OR
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Example Sine or Cosine?
Transformation
Amplitude -
2
Graph -
Cosine
3 units up and p/4 to the left
Translation -
3 units up and
Orientation -
Positive
Period -
2p
New Equation
y 3
x -p/4
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Changing the Period

• Find the period for each graph and generalize the
  result.

1 cycle in 2p
1/4 cycle in 2p
Period 2p
Period 8p
2 cycles in 2p
4 cycles in 2p
Period p
Period 0.5p
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Determining the Period of Sine/Cosine Graph

• If or , the period
  (the length of one cycle) is determined by
• Ex What is the period of
  ?

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Changing the Period w/o Affecting (h,k)
The key point (h,k) is a point on the sine graph.
Also, multiplying x by a constant changes the
period. Below are two different ways to write a
transformation. In order for the equation to be
useful, it must directly change the graph in a
specific manner. Which equation changes the
period and contains the point (-3,4)?
or
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Graphing Form for Sine
k
h
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Graphing Form for Cosine
k
h
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Notation Trigonometric Functions
Correct way for the calculator!
is equivalent to
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Example Sine
Transformation Change the amplitude to 0.5 and
the period to p. Then translate it p/2 units to
the right and 1 unit down.
Transformation
Not in Graphing form
New Equation
Period
y -1
x p/2
You need at least 5 consecutive critical points.
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Example Cosine
Transformation Change the period to 4p and
translate the parent graph 1 unit up.
Transformation
New Equation
y 1
Period
x 0
You need at least 5 consecutive critical points.
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Example Sine or Cosine?
Transformation
Amplitude -
1.5
Graph -
Translation -
2 units down and
Orientation -
Period -
p/2
New Equation
Since the Sine and Cosine graphs are periodic and
translations of each other, there are infinite
equations that represent the same curve. Here
are two examples.
Period
y -2
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Example Sine or Cosine?
Transformation
Amplitude -
Graph -
Cosine
1.5
2 units down
Translation -
2 units down and
Orientation -
Positive
Period -
p/2
x 0
New Equation
Period
y -2
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OR
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Example Sine or Cosine?
Transformation
Amplitude -
Graph -
Sine
1.5
2 units down and 5p/8 to the right
Translation -
2 units down and
Orientation -
Negative
Period -
p/2
x 5p/8
New Equation
Period
y -2