Graphs of Composite Trig Functions

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Graphs of Composite Trig Functions

• Objective
• Be able to combine trigonometric and algebraic
  functions together.
• TS Demonstrating understanding of concepts.
• Warm-Up Graph each of the below on your
  calculator. Which seem to be periodic?

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How can we verify if something is periodic?

• If we believe some function f(x) has a period
  of a, then to verify we need to show
• f(x a) f(x).
• Example Verify y(sin x)2 is periodic.

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You Try

• Is y (sin3x)(cosx) periodic? Use your
  calculator to figure out what the period is.

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Graph the following functions one at a time in
the window -2p x 2p and -6 y 6
Which appear to be sinusoids? What
relationship between the sine and cosine
functions ensures their sum or difference is a
sinusoid?
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Sums that are Sinusoid Functions

• Given the two functions f(x) a1sin(bxc1)
  and
• g(x) a2cos(bxc2) both with the same b
  value then the sum (fg)(x) a1sin(bxc1)
  a2cos(bxc2) is a sinusoid with period 2p/b

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Examples

• Determine whether each of the following
  functions is or is not a sinusoid.

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Putting the two together

• Show that g(x) sin(2x) cos(3x) is periodic
  but not a sinusoid.

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What if I just want to graph some crazy trig
functions? (dont roll your eyes, you know you
want to graph crazy trig functions)
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• Functions involving the absolute values of Trig
  functions
• The key is to remember absolute values
  create all positive values.
• Examples
• f(x) tanx b) g(x) sinx

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• Functions involving the absolute values of Trig
  functions
• Examples
• b) g(x) sinx

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• Functions involving a sinusoid and a linear
  function
• The key is to remember sine and cosine can
  be at most 1 and at least -1.
• Examples
• f(x) 3x cosx b) g(x) ½x cosx

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• Functions involving a sinusoid and a linear
  function
• Examples
• b) g(x) ½x cosx

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• Dampened Trig Functions (Trig functions muliplied
  by a algebraic function)
• The key is to remember sine and cosine can
  be at most 1 and at least -1.
• Example
• f(x) (2x)cosx