# 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. 13. The increase and decrease of
14. 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

15. Extrema
16. a) when n doubles  when n is odd — no

b) no

c) when  when — no

17. Asymptotes
18. a) no

b) c) while — not when 19. Convexity and points of inflection
20. a) when n doubles — bulge down when n is odd,  ;  ; 0 — inflection point

b) when n doubles —  ;   when n is odd —  ;  c) when  ;   when 21. A special case
22. If , (when ) ## Graphs of power functions • n is an even natural number
•  • n — an odd natural number
•  • n is an odd number VDA
•  • n is an even number VDA
•  • n is not positive integer
• (green) • n is not an integer WMA
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