The properties of the Y x 2n function. Power feature, its properties and graphics. Power function, its properties and schedule

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
With N ≠ 1, the 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.

Schedule functiony. = aX. 2 + n. .

Explanation.

y. = 2x. 2 + 4.
y. = 2x. 2, moves to four units up the axis y.. Of course, all values y. naturally increase by 4.

Here is the table of function values y. = 2x. 2:

But the table of values y. = 2x. 2 + 4:

We see on the table that the top of the parabolla of the second function by 4 units above the top of the parabola of the first (its coordinate 0; 4). And meanings y. second function for 4 more values y. first function.

Schedule functiony. = a.(x. – m.) 2 .

Explanation.

For example, you need to build a schedule of function y. = 2 (x. – 6) 2 .
This means that Parabola, which is a graph of the function y. = 2x. 2, moves to six units to the right along the axis x.(on the chart - Red Parabola).

Schedule functiony. = a.(x. – m.) 2 + n.

Two functions lead to us to the third function: y. = a.(x. – m.) 2 + n.

Explanation:

For example, you need to build a schedule of function y. = 2 (x. – 6) 2 + 2.
This means that Parabola, which is a graph of the function y. = 2x. 2, moves to 6 units to the right (value M) and 2 units up (value N). Red Parabola on the chart - the result of these movements.

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.-Bedic 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

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. Exponential function, its properties and schedule, 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 different cases depending on the
Indicator p.

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

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

• definition area - all actual numbers, i.e. 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.

The function y \u003d x2n, where N belongs to a set of entire positive numbers. The power function of this species has a well-positive degree indicator A \u003d 2N. Since it is always x2n \u003d (x) 2n, then the graphs of all such functions are symmetrical with respect to the ordinate axis. All functions of the form y \u003d x2n, n belongs to the set of integers of positive numbers have the following identical properties: x \u003d R x? \u003d (-?;?) Y \u003d)