**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

- Scope
- Multiple value
- Parity, odd parity
- The frequency
- The intersection with the coordinate axes
- Derivative
- The increase and decrease of
- Extrema
- Asymptotes
- Convexity and points of inflection
- A special case

**a)**

**b)**

**c)** when

when

**a)** when *n* doubles

when *n* is odd

**b)** when *n* doubles

when *n* is odd

**c)** when

when

**a),b)** when *n* doubles — doubles

when *n* is odd — odd

**c)** neither even nor odd

not periodic

**a)**

**b)** no

**c)** when

when — no

**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

**a)** when *n* doubles

when *n* is odd — no

**b)** no

**c)** when

when — no

**a)** no

**b)**

**c)** while — not

when

**a)** when *n* doubles — bulge down

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

**b)** when *n* doubles — — ; —

when *n* is odd — — ; —

**c)** when — ; —

when

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)