# 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 )

## 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
• (green)

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