Which is a graph of a powerful function. Function

You are familiar with functions y \u003d x, y \u003d x 2 , y \u003d x 3 , y \u003d 1 / xetc. All these functions are special cases of the powerful function, i.e. functions y \u003d X. p. where P is a given valid number. Properties and graphs of the power function depends substantially on the properties of the degree with the actual indicator, and in particular on which values x.and p.it makes sense degree x. p. . Let us proceed to such consideration of various cases depending on the degree p.

Indicator p \u003d 2N.-theful natural number.

In this case, the power function y \u003d X. 2N. where n.- natural number, has the following

properties:

the definition area is all valid numbers, i.e. the set R;

many values \u200b\u200b- non-negative numbers, i.e. y more or equal to 0;

function y \u003d X. 2N. even because x. 2N. \u003d (- x) 2N.

the function is descending on the interval x.<0 and increasing on the interval x\u003e 0.

Schedule function y \u003d X. 2N. has the same kind as such as a function graph y \u003d X. 4 .

2. Indicator p \u003d 2N-1- an odd natural number in this case by the power function y \u003d X. 2N-1 where the natural number has the following properties:

the definition area is the set R;

many values \u200b\u200b- set R;

function y \u003d X. 2N-1 odd since (- x) 2N-1 =x. 2N-1 ;

the function is increasing on the entire valid axis.

Schedule function y \u003d x2n-1it has the same appearance as, for example, a function schedule y \u003d X3..

3.Weider p \u003d -2N.where n -natural number.

In this case, the power function y \u003d X. -2N. \u003d 1 / x 2N. possesses the following properties:

many values \u200b\u200b- positive numbers y\u003e 0;

function Y. \u003d 1 / x 2N. even because 1 / (- X) 2N. =1 / X. 2N. ;

the function is increasing at the interval x<0 и убывающей на промежутке x>0.

Function schedule Y. \u003d 1 / x 2N. It has the same appearance as, for example, the function of the Y function \u003d 1 / x 2 .

4. Address p \u003d - (2N-1)where n.- natural number. In this case, the power function y \u003d X. - (2N-1) Possesses the following properties:

the definition area is the set R, except x \u003d 0;

many values \u200b\u200b- set R, except y \u003d 0;

function y \u003d X. - (2N-1) odd since (- x) - (2N-1) =-x. - (2N-1) ;

the function is descending at intervals x.<0 and x\u003e 0..

Schedule function y \u003d X. - (2N-1) It has the same appearance as, for example, a function schedule y \u003d 1 / X 3 .

1. Inverse trigonometric functions, their properties and graphics.

Inverse trigonometric functions, their properties and graphics.Inverse trigonometric functions (circular functions, arkfunctions) - Mathematical functions that are inverse to trigonometric functions.

1. Arcsin feature

Schedule function .

Arksinus numbers m. called an angle value x., for which

The function is continuous and limited on all its numeric straight. Function is strictly increasing.

1. [Edit] ArcSin function properties

1. [Edit] Getting ARCSIN Functions

Dana feature on all its definition areas she happens to be piecewise monotonous, and, therefore, the opposite The function is not. Therefore, we will look at the segment on which it strictly increases and takes all the values. areas of values -. Since for a function on the interval, each value of the argument corresponds to the only value of the function, then on this segment there is reverse function The graph of which is symmetrical graphics function on the segment relatively straight

On the region of determining the power function y \u003d x p, the following formulas take place:
; ;
;
; ;
; ;
; .

Properties of power functions and their schedules

The power function with an indicator is zero, p \u003d 0

If the indicator of the power function y \u003d x p is zero, p \u003d 0, then the power function is defined for all x ≠ 0 and is a constant equal to one:
y \u003d x p \u003d x 0 \u003d 1, x ≠ 0.

Power function with natural odd indicator, p \u003d n \u003d 1, 3, 5, ...

Consider the power function y \u003d x p \u003d x n with a natural odd indicator of the degree n \u003d 1, 3, 5, .... Such an indicator can also be written in the form: n \u003d 2k + 1, where k \u003d 0, 1, 2, 3, ... is not negative. Below are properties and graphs of such functions.

The graph of the power function y \u003d x n with a natural odd indicator when different values Extent rate n \u003d 1, 3, 5, ....

Domain: -∞ < x < ∞
Many values: -∞ < y < ∞
Parity: odd, y (-X) \u003d - y (x)
Monotone: Monotonously increase
Extremes: not
Convex:
at -∞< x < 0 выпукла вверх
at 0.< x < ∞ выпукла вниз
Points of inflection: x \u003d 0, y \u003d 0
x \u003d 0, y \u003d 0
Limits:
;
Private values:
at x \u003d -1,
y (-1) \u003d (-1) n ≡ (-1) 2k + 1 \u003d -1
at x \u003d 0, y (0) \u003d 0 n \u003d 0
at x \u003d 1, y (1) \u003d 1 n \u003d 1
Reverse function:
for n \u003d 1, the function is the opposite to itself: x \u003d y
at n ≠ 1, inverse function is the root of degree n:

Power function with natural even indicator, p \u003d n \u003d 2, 4, 6, ...

Consider the power function y \u003d x p \u003d x n with a natural even indicator of the degree n \u003d 2, 4, 6, .... Such an indicator can also be written in the form: n \u003d 2k, where k \u003d 1, 2, 3, ... - Natural. Properties and graphs of such functions are given below.

The graph of the power function y \u003d x n with a natural even indicator at different values \u200b\u200bof the degree rate n \u003d 2, 4, 6, ....

Domain: -∞ < x < ∞
Many values: 0 ≤ y.< ∞
Parity: even, y (-x) \u003d y (x)
Monotone:
at x ≤ 0 monotonously decreases
at x ≥ 0 increases monotonically
Extremes: minimum, x \u003d 0, y \u003d 0
Convex: Bashed down
Points of inflection: not
Point of intersection with axes of coordinates: x \u003d 0, y \u003d 0
Limits:
;
Private values:
at x \u003d -1, y (-1) \u003d (-1) n ≡ (-1) 2k \u003d 1
at x \u003d 0, y (0) \u003d 0 n \u003d 0
at x \u003d 1, y (1) \u003d 1 n \u003d 1
Reverse function:
at n \u003d 2, square root:
For n ≠ 2, the root of degree n:

Power function with a whole negative indicator, p \u003d n \u003d -1, -2, -3, ...

Consider the power function y \u003d x p \u003d x n with a whole negative indicator of the degree n \u003d -1, -2, -3, .... If you put n \u003d -k, where k \u003d 1, 2, 3, ... - natural, then it can be represented as:

The graph of the power function y \u003d x n with a whole negative indicator at different values \u200b\u200bof the degree rate n \u003d -1, -2, -3, ....

An odd indicator, n \u003d -1, -3, -5, ...

Below are the properties of the function y \u003d x n with an odd negative indicator N \u003d -1, -3, -5, ....

Domain: x ≠ 0.
Many values: y ≠ 0
Parity: odd, y (-X) \u003d - y (x)
Monotone: Monotonously decrease
Extremes: not
Convex:
With X.< 0 : выпукла вверх
With X\u003e 0: Break down
Points of inflection: not
Point of intersection with axes of coordinates: not
Sign:
With X.< 0, y < 0
With x\u003e 0, y\u003e 0
Limits:
; ; ;
Private values:
at x \u003d 1, y (1) \u003d 1 n \u003d 1
Reverse function:
with n \u003d -1,
with N.< -2 ,

An even indicator, n \u003d -2, -4, -6, ...

Below are the properties of the function y \u003d x n with an even negative indicator N \u003d -2, -4, -6, ....

Domain: x ≠ 0.
Many values: Y\u003e 0.
Parity: even, y (-x) \u003d y (x)
Monotone:
With X.< 0 : монотонно возрастает
With x\u003e 0: Monotonously decreases
Extremes: not
Convex: Bashed down
Points of inflection: not
Point of intersection with axes of coordinates: not
Sign: Y\u003e 0.
Limits:
; ; ;
Private values:
at x \u003d 1, y (1) \u003d 1 n \u003d 1
Reverse function:
at n \u003d -2,
with N.< -2 ,

Power function with rational (fractional) indicator

Consider the power function y \u003d x p with a rational (fractional) indicator, where n is an integer, M\u003e 1 - natural. Moreover, N, M do not have common divisors.

Danger of fractional indicator - odd

Let the denominator of the fractional indicator of the degree: m \u003d 3, 5, 7, .... In this case, the power function x p is defined both for positive and for the negative values \u200b\u200bof the X argument. Consider the properties of such power functions when the parameter P is within certain limits.

The indicator p is negative, p< 0

Let the rational indicator (with an odd denominator M \u003d 3, 5, 7, ...) less than zero :.

Graphs of power functions with a rational negative indicator at different values \u200b\u200bof the indicator of the degree, where m \u003d 3, 5, 7, ... - odd.

An odd numerator, n \u003d -1, -3, -5, ...

We present the properties of the power function y \u003d x p with a rational negative indicator, where n \u003d -1, -3, -5, ... is an odd negative integer, M \u003d 3, 5, 7 ... is an odd natural.

Domain: x ≠ 0.
Many values: y ≠ 0
Parity: odd, y (-X) \u003d - y (x)
Monotone: Monotonously decrease
Extremes: not
Convex:
With X.< 0 : выпукла вверх
With X\u003e 0: Break down
Points of inflection: not
Point of intersection with axes of coordinates: not
Sign:
With X.< 0, y < 0
With x\u003e 0, y\u003e 0
Limits:
; ; ;
Private values:
at x \u003d -1, y (-1) \u003d (-1) n \u003d -1
at x \u003d 1, y (1) \u003d 1 n \u003d 1
Reverse function:

Even numerator, n \u003d -2, -4, -6, ...

The properties of the power function y \u003d x p with a rational negative indicator, where n \u003d -2, -4, -6, ... - even negative integer, m \u003d 3, 5, 7 ... - odd natural.

Domain: x ≠ 0.
Many values: Y\u003e 0.
Parity: even, y (-x) \u003d y (x)
Monotone:
With X.< 0 : монотонно возрастает
With x\u003e 0: Monotonously decreases
Extremes: not
Convex: Bashed down
Points of inflection: not
Point of intersection with axes of coordinates: not
Sign: Y\u003e 0.
Limits:
; ; ;
Private values:
At x \u003d -1, y (-1) \u003d (-1) n \u003d 1
at x \u003d 1, y (1) \u003d 1 n \u003d 1
Reverse function:

P is positive, less than one, 0< p < 1

Graph of a powerful function with a rational indicator (0< p < 1 ) при различных значениях показателя степени , где m = 3, 5, 7, ... - нечетное.

An odd numerator, n \u003d 1, 3, 5, ...

< p < 1 , где n = 1, 3, 5, ... - нечетное натуральное, m = 3, 5, 7 ... - нечетное натуральное.

Domain: -∞ < x < +∞
Many values: -∞ < y < +∞
Parity: odd, y (-X) \u003d - y (x)
Monotone: Monotonously increase
Extremes: not
Convex:
With X.< 0 : выпукла вниз
With X\u003e 0: Built up
Points of inflection: x \u003d 0, y \u003d 0
Point of intersection with axes of coordinates: x \u003d 0, y \u003d 0
Sign:
With X.< 0, y < 0
With x\u003e 0, y\u003e 0
Limits:
;
Private values:
at x \u003d -1, y (-1) \u003d -1
at x \u003d 0, y (0) \u003d 0
at x \u003d 1, y (1) \u003d 1
Reverse function:

Even numerator, n \u003d 2, 4, 6, ...

The properties of the power function y \u003d x p are presented with a rational indicator in the range of 0< p < 1 , где n = 2, 4, 6, ... - четное натуральное, m = 3, 5, 7 ... - нечетное натуральное.

Domain: -∞ < x < +∞
Many values: 0 ≤ y.< +∞
Parity: even, y (-x) \u003d y (x)
Monotone:
With X.< 0 : монотонно убывает
With x\u003e 0: Monotonously increases
Extremes: At least x \u003d 0, y \u003d 0
Convex: Convected up at x ≠ 0
Points of inflection: not
Point of intersection with axes of coordinates: x \u003d 0, y \u003d 0
Sign: At x ≠ 0, Y\u003e 0
Limits:
;
Private values:
at x \u003d -1, y (-1) \u003d 1
at x \u003d 0, y (0) \u003d 0
at x \u003d 1, y (1) \u003d 1
Reverse function:

Eaction p View more units, P\u003e 1

A graph of the powerful function with a rational indicator (p\u003e 1) at different values \u200b\u200bof the indicator of the degree, where m \u003d 3, 5, 7, ... - odd.

An odd numerator, n \u003d 5, 7, 9, ...

The properties of the power function y \u003d x p with a rational indicator, a large unit :. Where n \u003d 5, 7, 9, ... is an odd natural, m \u003d 3, 5, 7 ... - odd natural.

Domain: -∞ < x < ∞
Many values: -∞ < y < ∞
Parity: odd, y (-X) \u003d - y (x)
Monotone: Monotonously increase
Extremes: not
Convex:
at -∞< x < 0 выпукла вверх
at 0.< x < ∞ выпукла вниз
Points of inflection: x \u003d 0, y \u003d 0
Point of intersection with axes of coordinates: x \u003d 0, y \u003d 0
Limits:
;
Private values:
at x \u003d -1, y (-1) \u003d -1
at x \u003d 0, y (0) \u003d 0
at x \u003d 1, y (1) \u003d 1
Reverse function:

Even numerator, n \u003d 4, 6, 8, ...

The properties of the power function y \u003d x p with a rational indicator, a large unit :. Where n \u003d 4, 6, 8, ... - even natural, m \u003d 3, 5, 7 ... - odd natural.

Domain: -∞ < x < ∞
Many values: 0 ≤ y.< ∞
Parity: even, y (-x) \u003d y (x)
Monotone:
With X.< 0 монотонно убывает
With x\u003e 0, monotonously increases
Extremes: At least x \u003d 0, y \u003d 0
Convex: Bashed down
Points of inflection: not
Point of intersection with axes of coordinates: x \u003d 0, y \u003d 0
Limits:
;
Private values:
at x \u003d -1, y (-1) \u003d 1
at x \u003d 0, y (0) \u003d 0
at x \u003d 1, y (1) \u003d 1
Reverse function:

Danger of fractional indicator - even

Let the denominator of the fractional indicator of the degree of degree: m \u003d 2, 4, 6, .... In this case, the power function x p is not defined for negative values \u200b\u200bof the argument. Its properties coincide with the properties of the power function with the irrational indicator (see the next section).

Power function with irrational indicator

Consider the power function y \u003d x p with an irrational indicator of the degree P. The properties of such functions differ from those discussed above in the fact that they are not defined for the negative values \u200b\u200bof the X argument. For positive values \u200b\u200bof the argument, the properties depend only on the value of the degree of the degree P and do not depend on whether p is integer, rational or irrational.

y \u003d x p at different values \u200b\u200bof the parameter p.

Power function with a negative indicator P< 0

Domain: X\u003e 0.
Many values: Y\u003e 0.
Monotone: Monotonously decrease
Convex: Bashed down
Points of inflection: not
Point of intersection with axes of coordinates: not
Limits: ;
Private value: At x \u003d 1, y (1) \u003d 1 p \u003d 1

Power function with positive indicator P\u003e 0

The indicator is less than one 0< p < 1

Domain: x ≥ 0.
Many values: y ≥ 0.
Monotone: Monotonously increase
Convex: Based up
Points of inflection: not
Point of intersection with axes of coordinates: x \u003d 0, y \u003d 0
Limits:
Private values: At x \u003d 0, y (0) \u003d 0 p \u003d 0.
At x \u003d 1, y (1) \u003d 1 p \u003d 1

The indicator is greater than the unit P\u003e 1

Domain: x ≥ 0.
Many values: y ≥ 0.
Monotone: Monotonously increase
Convex: Bashed down
Points of inflection: not
Point of intersection with axes of coordinates: x \u003d 0, y \u003d 0
Limits:
Private values: At x \u003d 0, y (0) \u003d 0 p \u003d 0.
At x \u003d 1, y (1) \u003d 1 p \u003d 1

References:
I.N. Bronstein, K.A. Semendyaev, a reference book on mathematics for engineers and students of the attendants, "Lan", 2009.

See also:

The study of the properties of functions and their graphs occupies a significant place in both school mathematics and subsequent courses. And not only in the courses of mathematical and functional analysis, and not only in other sections higher MathematicsBut in most narrow professional items. For example, in the economy - the functions, costs, demand functions, supply and consumption functions ..., in radio engineering - control functions and response functions, in statistics - distribution functions ... To facilitate further study of special functions, you need to learn how to operate in graphs elementary functions. To do this, after studying the next table, we recommend passing the "Conformation of Function Graphs" link.

In the school course of mathematics are studied the following
elementary functions.

Function name	Formula function	Schedule function	Graphic name	Comment
Linear	y \u003d kx.		Straight	The most simple private case of linear dependence is direct proportionality. y \u003d kx.where k. ≠ 0 - proportionality coefficient. In the picture, an example for k. \u003d 1, i.e. In fact, the given graph illustrates a functional dependence that specifies the equality of the value of the function value of the argument.
Linear	y. = kX. + b.		Straight	General linear dependence: coefficients k. and b. - Any valid numbers. Here k. = 0.5, b. = -1.
Quadratic	y \u003d X. 2		Parabola	The simplest case of quadratic dependence is a symmetric parabola with a vertex at the beginning of the coordinates.
Quadratic	y \u003d AX. 2 + bX. + c.		Parabola	General case of quadratic dependence: coefficient a. - an arbitrary valid number is not zero ( a. belongs r, a. ≠ 0), b., c. - Any valid numbers.
Power	y \u003d X. 3		Cubic Parabola	The easiest case for an odd degree. Cases with coefficients are studied in the "Motion of Function Graphs" section.
Power	y \u003d X. 1/2		Schedule function y. = √x.	The easiest case for fractional degree ( x. 1/2 = √x.). Cases with coefficients are studied in the "Motion of Function Graphs" section.
Power	y \u003d k / x		Hyperbola	The easiest case for a short degree ( 1 / x \u003d x -1) - back-proportional dependence. Here k. = 1.
Indicative	y. = e X.		Exhibitor	An exponential dependence is called an indicative function for the foundation. e. - irrational number approximately equal to 2,7182818284590 ...
Indicative	y \u003d a x		Graph indicative function	a. \u003e 0 I. a. a.. Here is an example for y \u003d 2 x (a. = 2 > 1).
Indicative	y \u003d a x		Graph indicative function	Exponential function Defined for a. \u003e 0 I. a. ≠ 1. Fun graphics significantly depend on the value of the parameter a.. Here is an example for y \u003d 0.5 x (a. = 1/2 < 1).
Logarithmic	y. \u003d LN. x.			Graph Logo Function for Base e. (Natural logarithm) is sometimes called logarithmics.
Logarithmic	y. \u003d Log. A X.		Schedule logarithmic function	Logarithms are defined for a. \u003e 0 I. a. ≠ 1. Fun graphics significantly depend on the value of the parameter a.. Here is an example for y. \u003d log 2. x. (a. = 2 > 1).
Logarithmic	y \u003d Log. A X.		Schedule logarithmic function	Logarithms are defined for a. \u003e 0 I. a. ≠ 1. Fun graphics significantly depend on the value of the parameter a.. Here is an example for y. \u003d log 0.5 x. (a. = 1/2 < 1).
Sinus	y. \u003d SIN x.		Sinusoid	Trigonometric function sinus. Cases with coefficients are studied in the "Motion of Function Graphs" section.
Cosine	y. \u003d COS. x.		Kosinusoid	Trigonometric cosine function. Cases with coefficients are studied in the "Motion of Function Graphs" section.
Tangent	y. \u003d TG. x.		TangentSoid	Trigonometric function Tangent. Cases with coefficients are studied in the "Motion of Function Graphs" section.
Cotangent	y. \u003d CTG. x.		Kothangensoid	Trigonometric Cotangen feature. Cases with coefficients are studied in the "Motion of Function Graphs" section.

Inverse trigonometric functions.

Function name	Formula function	Schedule function	Graphic name

You are familiar with functions y \u003d x, y \u003d x 2, y \u003d x 3, y \u003d 1 / x etc. All these functions are special cases of the powerful function, i.e. functions y \u003d x pwhere P is a given valid number.
Properties and graphs of the power function depends substantially on the properties of the degree with the actual indicator, and in particular on which values x.and P.it makes sense degree x. p. . Let us turn to such consideration of various cases depending on
Indicator p.

1. Indicator p \u003d 2N.-theful natural number.

y \u003d x 2nwhere n. - natural number, has the following

properties:

• the definition area is all valid numbers, i.e. the set R;
• many values \u200b\u200b- non-negative numbers, i.e. y more or equal to 0;
• function y \u003d x 2n even because x 2N.=(- x) 2N.
• the function is descending on the intervalx.<0 and increasing on the intervalx\u003e 0.

Schedule function y \u003d x 2nhas the same kind as such as a function graph y \u003d x 4.

2. Indicator p \u003d 2N-1- odd natural number
In this case, the power function y \u003d x 2n-1where the natural number has the following properties:

• the definition area is the set R;
• many values \u200b\u200b- set R;
• function y \u003d x 2n-1 odd since (- x) 2n-1=x 2n-1;
• the function is increasing on the entire valid axis.

Schedule function y \u003d X. 2N-1 has the same appearance as, for example, a function schedule y \u003d X. 3 .

3.Weider p \u003d -2N.where n -natural number.

In this case, the power function y \u003d x -2n \u003d 1 / x 2npossesses the following properties:

• the definition area is the set R, except x \u003d 0;
• many values \u200b\u200b- positive numbers Y\u003e 0;
• function Y. \u003d 1 / x 2n even because 1 / (- x) 2n=1 / x 2n;
• the function is increasing at the interval x<0 и убывающей на промежутке x>0.

Function schedule Y. \u003d 1 / x 2n It has the same appearance as, for example, the function of the Y function \u003d 1 / x 2.

1. Power function, its properties and graph;

2. Conversion:

Parallel transfer;

Symmetry relative to the axes of coordinates;

Symmetry relative to the start of coordinates;

Symmetry relatively straight y \u003d x;

Stretching and compression along the coordinate axes.

3. Indicative function, its properties and graph, similar transformations;

4. Logarithmic function, its properties and schedule;

5. Trigonometric function, its properties and graph, similar transformations (Y \u003d SIN X; Y \u003d COS X; Y \u003d TG x);

Function: Y \u003d X \\ N - its properties and schedule.

Power function, its properties and schedule

y \u003d x, y \u003d x 2, y \u003d x 3, y \u003d 1 / x etc. All these functions are special cases of the powerful function, i.e. functions y \u003d x pwhere P is a given valid number.
Properties and graphs of the power function depends substantially on the properties of the degree with the actual indicator, and in particular on which values x.and P.it makes sense degree x P.. Let us turn to such consideration of various cases depending on
Indicator p.

1. Indicator p \u003d 2N.- An even natural number.

y \u003d x 2nwhere n. - Natural number, possesses the following properties:

• the definition area is all valid numbers, i.e. the set R;
• many values \u200b\u200b- non-negative numbers, i.e. y more or equal to 0;
• function y \u003d x 2n even because x 2n \u003d (-X) 2N
• the function is descending on the interval x.< 0 and increasing on the interval x\u003e 0.

Schedule function y \u003d x 2nhas the same kind as such as a function graph y \u003d x 4.

2. Indicator p \u003d 2N - 1- odd natural number

In this case, the power function y \u003d x 2n-1where the natural number has the following properties:

• the definition area is the set R;
• many values \u200b\u200b- set R;
• function y \u003d x 2n-1 odd since (- x) 2n-1= x 2n-1;
• the function is increasing on the entire valid axis.

Schedule function y \u003d x 2n-1 y \u003d x 3.

3. Indicator p \u003d -2N.where n -natural number.

In this case, the power function y \u003d x -2n \u003d 1 / x 2npossesses the following properties:

• many values \u200b\u200b- positive numbers y\u003e 0;
• function Y. \u003d 1 / x 2n even because 1 / (- x) 2n= 1 / x 2n;
• the function is increasing in the period x0.

Function schedule Y. \u003d 1 / x 2n It has the same appearance as, for example, the function of the Y function \u003d 1 / x 2.

4. Indicator p \u003d - (2N-1)where n. - natural number.
In this case, the power function y \u003d X - (2N-1) Possesses the following properties:

• the definition area is the set R, except x \u003d 0;
• many values \u200b\u200b- set R, except y \u003d 0;
• function y \u003d X - (2N-1) odd since (- x) - (2N-1) = -x - (2N-1);
• the function is descending at intervals x.< 0 and x\u003e 0..

Schedule function y \u003d X - (2N-1) It has the same appearance as, for example, a function schedule y \u003d 1 / x 3.