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

a)

b)

c) when

when

2. Multiple value

a) when n doubles

when n is odd

b) when n doubles

when n is odd

c) when

when

3. Parity, odd parity

a),b) when n doubles — doubles

when n is odd — odd

c) neither even nor odd

4. The frequency

not periodic

5. The intersection with the coordinate axes

a)

b) no

c) when

when — no

6. Derivative



7. The increase and decrease of

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

8. Extrema

a) when n doubles

when n is odd — no

b) no

c) when

when — no

9. Asymptotes

a) no

b)

c) while — not

when

10. Convexity and points of inflection

a) when n doubles — bulge down

when n is odd, — ; — ; 0 — inflection point

b) when n doubles — — ; —

when n is odd — — ; —

c) when — ; —

when

11. A special case

If , (when )

Graphs of power functions

(green)

(green)