New Functions from Old: Stretches and Shifts (1/28/09)

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New Functions from Old Stretches and Shifts
(1/28/09)

• Stretches If you multiply y (the output) by a
  positive constant c, it stretches the graph
  vertically (if c gt 1) or compresses it (if c lt
  1). If c is negative, it also turns the graph
  upside down!
• Multiplying x (the input) by c gt 1 compresses the
  graph horizontally, etc.
• Shifts Replacing y by y k shifts the graph
  upward if k is positive and downward if k is
  negative.
• Replacing x by x k shifts to the right if k
  gt0, etc.

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Clicker Question 1

• The graph of f (x) (x2)2 3 can be gotten
  from the graph of f (x) x 2 by shifting it
• A. left 2 and down 3.
• B. right 2 and up 3.
• C. left 2 and up 3.
• D. left 3 and up 2.
• E. right 3 and down 2.

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New Functions from Old

• Obviously, one way to get a new function from old
  ones is to add, subtract, multiply, or divide
  them.
• For example, a polynomial is obtained by adding
  or subtracting certain monomials.
• For example, a rational function is obtained from
  dividing two polynomials.

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New Functions from Old Composite Functions

• Composite functions If you apply one function,
  and then apply another function to the output of
  the first function, this is called a composite.
• Example f (x) (3x 4)4 results from first
  applying the linear function 3x 4 and then
  applying the raise to the 4th power function.
• The composite of f followed by g is commonly
  denoted g ? f (note the order!).
• More examples

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Clicker Question 2

• The function f (t) sin2(3t 2) is a composite
  of
• A. a linear function followed by a trig function
  followed by a power function.
• B. linear followed by power followed by trig.
• C. trig followed by power followed by linear.
• D. trig followed by exponential followed by
  linear.
• E. linear followed by trig followed by
  exponential.

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Trigonometric Functions

• Sin, cos, tan, and so on, are called
  trigonometric because their origins were in the
  study of right triangles.
• However, what they really should be called are
  Circular Functions or Periodic Functions, since
  their definitions are in terms of circles, and
  because they repeat themselves.

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Radians

• Start with a unit circle (i.e., a circle whose
  radius is 1 unit) whose center is at the origin.
• An angle measured counterclockwise from the
  x-axis has measure t radians if its arc on the
  circle is t units long.
• Since the circumference is 2 ?, there are 2 ?
  radians in a whole circle, so 2 ? radians 360

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Sin and Cos

• If we have an angle of t radians, the sin(t )
  is defined to be the y-coordinate on the unit
  circle. The cos(t ) is the x-coordinate.
• Hence both sin and cos repeat themselves every 2
  ? radians (i.e., they are periodic functions).
• Also, sin2(t ) cos2(t ) 1

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Clicker Question 3

• What is sin(?/4)?
• A. 1
• B. ?2
• C. ?2 / 2
• D. ½
• E. ?3 / 2

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Amplitude and Period

• The amplitude of any periodic function of time is
  half the distance between its highest and lowest
  points.
• The period is the shortest time before the
  function begins to repeat.
• Hence the function y A sin(Bt ) has amplitude
  A and period 2 ? /B .

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Tangent

• The tangent function is defined as tan(t )
  sin(t )/cos(t ).
• Note that this is simply the slope of the line
  which the angle t makes.
• Hence tan(t ) is periodic with period ? rather
  than 2 ?.

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Clicker Question 4

• What is the tan(?/4)?
• A. 0
• B. ½
• C. 1
• D. ?2
• E. undefined

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Secant and Cosecant

• Though less commonly used, we need to know that
• The secant (denoted sec) is the reciprocal of the
  cosine, and
• The cosecant (denoted csc) is the reciprocal of
  the sine.
• Hence, again, both these functions are period of
  period 2?.

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Assignment for Friday

• Read Section 1.3.
• In Sec 1.3, do Exercises 1, 3, 9-17 odd, 23, 29,
  31, 33, and 41.
• Read Appendix D as needed (many of the formulas
  will not be of use to us).
• In Appendix D, do Exercises 1, 3, 9, 13, 23, 29,
  37, 65, 67, and 77.