Today in Precalculus

1
Today in Precalculus

• Need a calculator
• Notes
• Power Functions
• Go over homework
• Homework

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Power Functions

• Definition Any function that can be written in
  the form f(x)kxa, where k and a are nonzero
  constants.
• The constant a is the power, and k is the
  constant of variation or constant of proportion.
• f(x) varies as the ath power of x or is
  proportional to the ath power of x.
• If the power is positive, its direct variation
• If the power is negative, its inverse variation.

3
Power Functions

• Which five of the ten basic functions are power
  functions?
• Many familiar formulas from geometry and science
  are power functions
• Ex C 2pr
• power is 1, constant of variation is 2p
• Ex A s2
• power is 2, constant of variation is 1

4
Power Functions

• Which of the following are power functions?
  State the power and the constant of variation.
  If it isnt a power function, explain why
• a)
• b) g(x) 43x
• c) A pr2

Power function Power -4 Constant of variation 2
Not a power function, power isnt constant
Power function Independent variable r Power
2 Constant of variation p
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Writing Power Functions

• Express the following as power function
  equations
• a) In physics, Hookes Law for a spring states
  that the distance a spring is stretched (or
  compressed) varies directly as the force on the
  spring. (let d distance spring is stretched, F
  force, k constant)
• b) The distance a ball rolls down an inclined
  plane is directly proportional to the square of
  the time it rolls. (let d distance, t time,
  and k constant)

d kF
d kt2
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Even Functions

• Graph x2, x4, x6 and compare/contrast the graphs.

• All have shape similar to x2.
• Same domain and range
• All have the same end behavior.
• All are bounded below b0.
• All have even symmetry.
• All increase/decrease on same interval.
• All go through (-1,1), (0,0), (1,1)

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Odd Functions

• Graph x, x3, x5 and compare/contrast the graphs.

• All have the same end behavior.
• All are unbounded
• All have odd symmetry.
• All increase on (-8,8)
• Domain and Range all reals
• All go through (-1,-1), (0,0), (1,1)

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All functions

• Graph x, x2, x3,x4, x5, x6 with window 0,1
  by0,1

• Lower the power the higher the graph with x
  values between 0 and 1.

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All functions

• Graph x, x2, x3,x4, x5, x6 with window 0,2
  by0,2

• The higher the power the higher the function with
  x values greater than 1.

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All functions

• We learned a power function has the form
  f(x)kxa, so how does k change these graphs?
• If k gt 1, there is a vertical stretch of k
• If klt 1, there is a vertical shrink of k
• If k is negative there is a reflection over the
  x-axis.

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Graphing

• Graph f(x) 4x6 and f(x)

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Homework

• Pg. 196 1-10, 17-22, 31-42