The power function

Definition: the Function of the form where n is any real number, is called power function

Properties of the power function (in )

a) n — natural b) n is an integer WMA c) n is not a whole

  1. Scope
  2. a)

    b)

    c) when

    when

  3. Multiple value
  4. a) when n doubles

    when n is odd

    b) when n doubles

    when n is odd

    c) when

    when

  5. Parity, odd parity
  6. a),b) when n doubles — doubles

    when n is odd — odd

    c) neither even nor odd

  7. The frequency
  8. not periodic

  9. The intersection with the coordinate axes
  10. a)

    b) no

    c) when

    when — no

  11. Derivative
  12. The increase and decrease of
  13. a) when n doubles — decreases, increases

    when n odd — is growing

    b) when n doubles — increases — decreases

    when n odd — comes — comes

    c) when — growing

    when it comes

  14. Extrema
  15. a) when n doubles

    when n is odd — no

    b) no

    c) when

    when — no

  16. Asymptotes
  17. a) no

    b)

    c) while — not

    when

  18. Convexity and points of inflection
  19. a) when n doubles — bulge down

    when n is odd, ; ; 0 — inflection point

    b) when n doubles — ;

    when n is odd — ;

    c) when ;

    when

  20. A special case
  21. If , (when )

    Степенева функція

Graphs of power functions

  • n is an even natural number
  • Степенева функція

  • n — an odd natural number
  • Степенева функція

  • n is an odd number VDA
  • Степенева функція

  • n is an even number VDA
  • Степенева функція

  • n is not positive integer
  • (green)

    Степенева функція

  • n is not an integer WMA
  • (green)

    Степенева функція

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