Transformations of Circular Functions

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Transformations of Circular Functions

• Pre-Calculus 5.6
• 2/4/08

2
Reviewing Transformations

• Graph y x2
• Graph y -x2 What does this do?

Multiplying by a negative reflects the graph over
the x-axis.
3
Reviewing Transformations

• Graph y x2 3 What does this do?
• Graph y (x 3)2 What does this do?

Adding or subtracting outside the function shifts
the graph vertically,
Adding or Subtracting within the function, inside
parenthesis, shifts the graph horizontally in the
opposite direction.
4
Reviewing Transformations

• Graph y x2
• Graph y 3x2 What does this do?
• Graph y (3x )2 What does this do?

Multiplying the function by a numbergt1 stretches
the graph vertically.
Multiplying within the function,inside
parenthesis) by a numbergt1 shrinks the graph
horizontally.
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Transformations of Sin and Cos

• Look at the functions in the form
• y A sin (Bx C) D
• y A cos (Bx C) D
• A, B, C, and D are constants, that transform the
  basic graph of ysin x or ycos x

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Constant D

• How would D affect the graphs?

D shifts the graph vertically In your calculator
Graph y cos x y cos x .5 y cos x - 3
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Constant C

• How would C affect the graphs?

C shifts the graph horizontally In your
calculator Graph y cos x y cos (x .5) y cos
(x 3)
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Constant C

• As long as B 1 the graph is shifted C units.
• If B ? 1 then The shift is C/B
• This is called the Phase Shift

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Constant A

• How would A effect the graphs?

• If A gt 1 then the graph gets vertically
  stretched.
• If A lt 1 then the graph gets vertically shrunk.
• This is called the amplitude.
• Graph y sin x, y 2sin x and y .25 sin x.

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Constant B

• How would B effect the graphs?

• If B lt 1 then the graph gets horizontally
  stretched.
• If B gt 1 then the graph gets horizontally
  shrunk.
• If B lt 0 then the graph is also reflected across
  the y-axis.
• This is affects the period. Since the period is
  2? when B 1, then the period is