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Power function, its properties and graph presentation for an algebra lesson (grade 10) on the topic. Power function, its properties and graphs Power function definition and graphs

Are you familiar with the functions y=x, y=x 2 , y=x 3 , y=1/x etc. All these functions are special cases of the power function, i.e. the function y=x p, where p is a given real number. The properties and graph of a power function significantly depend on the properties of a power with a real exponent, and in particular on the values for which x And p degree makes sense x p. Let us proceed to a similar consideration of various cases depending on the exponent p.

Index p=2n-even natural number.

In this case, the power function y=x 2n, Where n- a natural number, has the following

properties:

domain of definition - all real numbers, i.e. the set R;

set of values - non-negative numbers, i.e. y is greater than or equal to 0;

function y=x 2n even, because x 2n =(-x) 2n

the function is decreasing on the interval x<0 and increasing on the interval x>0.

Graph of a function y=x 2n has the same form as, for example, the graph of a function y=x 4 .

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

domain of definition - set R;

set of values - set R;

function y=x 2n-1 odd because (- x) 2n-1 =x 2n-1 ;

the function is increasing on the entire real axis.

Graph of a function y=x2n-1 has the same form as, for example, the graph of a function y=x3.

3.Indicator p=-2n, Where n- natural number.

In this case, the power function y=x -2n =1/x 2n has the following properties:

set of values - positive numbers y>0;

function y =1/x 2n even, because 1/(-x) 2n =1/x 2n ;

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

Graph of function y =1/x 2n has the same form as, for example, the graph of the function y =1/x 2 .

4.Indicator p=-(2n-1), Where n- natural number. In this case, the power function y=x -(2n-1) has the following properties:

domain of definition - set R, except x=0;

set of values - set R, except y=0;

function y=x -(2n-1) odd because (- x) -(2n-1) =-x -(2n-1) ;

the function is decreasing on intervals x<0 And x>0.

Graph of a function y=x -(2n-1) has the same form as, for example, the graph of a function y=1/x 3 .

Inverse trigonometric functions, their properties and graphs.

Reverse trigonometric functions, their properties and graphs.Inverse trigonometric functions (circular functions, arc functions) - mathematical functions that are the inverse of trigonometric functions.

arcsin function

Graph of a function .

Arcsine numbers m this angle value is called x, for which

The function is continuous and bounded along its entire number line. Function is strictly increasing.

[Edit]Properties of the arcsin function

[Edit]Getting the arcsin function

Given the function Throughout its entire domain of definition she happens to be piecewise monotonic, and, therefore, the inverse correspondence is not a function. Therefore, we will consider the segment on which it strictly increases and takes on all values range of values- . Since for a function on an interval each value of the argument corresponds to a single value of the function, then on this interval there is inverse function whose graph is symmetrical to the graph of a function on a segment relative to a straight line

On the domain of definition of the power function y = x p the following formulas hold:
; ;
;
; ;
; ;
; .

Properties of power functions and their graphs

Power function with exponent equal to zero, p = 0

If the exponent of the power function y = x p equal to zero, p = 0, then the power function is defined for all x ≠ 0 and is a constant equal to unity:
y = x p = x 0 = 1, x ≠ 0.

Power function with natural odd exponent, p = n = 1, 3, 5, ...

Consider a power function y = x p = x n with a natural odd exponent n = 1, 3, 5, ... . This indicator can also be written in the form: n = 2k + 1, where k = 0, 1, 2, 3, ... is a non-negative integer. Below are the properties and graphs of such functions.

Graph of a power function y = x n with a natural odd exponent for various values of the exponent n = 1, 3, 5, ....

Domain: -∞ < x < ∞
Multiple meanings: -∞ < y < ∞
Parity: odd, y(-x) = - y(x)
Monotone: monotonically increases
Extremes: No
Convex:
at -∞< x < 0 выпукла вверх
at 0< x < ∞ выпукла вниз
Inflection points: x = 0, y = 0
x = 0, y = 0
Limits:
;
Private values:
at x = -1,
y(-1) = (-1) n ≡ (-1) 2k+1 = -1
at x = 0, y(0) = 0 n = 0
for x = 1, y(1) = 1 n = 1
Reverse function:
for n = 1, the function is its inverse: x = y
for n ≠ 1, inverse function is the root of degree n:

Power function with natural even exponent, p = n = 2, 4, 6, ...

Consider a power function y = x p = x n with a natural even exponent n = 2, 4, 6, ... . This indicator can also be written in the form: n = 2k, where k = 1, 2, 3, ... - natural. The properties and graphs of such functions are given below.

Graph of a power function y = x n with a natural even exponent for various values of the exponent n = 2, 4, 6, ....

Domain: -∞ < x < ∞
Multiple meanings: 0 ≤ y< ∞
Parity: even, y(-x) = y(x)
Monotone:
for x ≤ 0 monotonically decreases
for x ≥ 0 monotonically increases
Extremes: minimum, x = 0, y = 0
Convex: convex down
Inflection points: No
Intersection points with coordinate axes: x = 0, y = 0
Limits:
;
Private values:
at x = -1, y(-1) = (-1) n ≡ (-1) 2k = 1
at x = 0, y(0) = 0 n = 0
for x = 1, y(1) = 1 n = 1
Reverse function:
for n = 2, Square root:
for n ≠ 2, root of degree n:

Power function with negative integer exponent, p = n = -1, -2, -3, ...

Consider a power function y = x p = x n with an integer negative exponent n = -1, -2, -3, ... . If we put n = -k, where k = 1, 2, 3, ... is a natural number, then it can be represented as:

Graph of a power function y = x n with a negative integer exponent for various values of the exponent n = -1, -2, -3, ... .

Odd exponent, n = -1, -3, -5, ...

Below are the properties of the function y = x n with an odd negative exponent n = -1, -3, -5, ....

Domain: x ≠ 0
Multiple meanings: y ≠ 0
Parity: odd, y(-x) = - y(x)
Monotone: monotonically decreases
Extremes: No
Convex:
at x< 0 : выпукла вверх
for x > 0: convex downward
Inflection points: No
Intersection points with coordinate axes: No
Sign:
at x< 0, y < 0
for x > 0, y > 0
Limits:
; ; ;
Private values:
for x = 1, y(1) = 1 n = 1
Reverse function:
when n = -1,
at n< -2 ,

Even exponent, n = -2, -4, -6, ...

Below are the properties of the function y = x n with an even negative exponent n = -2, -4, -6, ....

Domain: x ≠ 0
Multiple meanings: y > 0
Parity: even, y(-x) = y(x)
Monotone:
at x< 0 : монотонно возрастает
for x > 0: monotonically decreases
Extremes: No
Convex: convex down
Inflection points: No
Intersection points with coordinate axes: No
Sign: y > 0
Limits:
; ; ;
Private values:
for x = 1, y(1) = 1 n = 1
Reverse function:
at n = -2,
at n< -2 ,

Power function with rational (fractional) exponent

Consider a power function y = x p with a rational (fractional) exponent, where n is an integer, m > 1 is a natural number. Moreover, n, m do not have common divisors.

The denominator of the fractional indicator is odd

Let the denominator of the fractional exponent be odd: m = 3, 5, 7, ... . In this case, the power function x p is defined for both positive and negative values of the argument x. Let us consider the properties of such power functions when the exponent p is within certain limits.

The p-value is negative, p< 0

Let the rational exponent (with odd denominator m = 3, 5, 7, ...) be less than zero: .

Graphs of power functions with a rational negative exponent for various values of the exponent, where m = 3, 5, 7, ... is odd.

Odd numerator, n = -1, -3, -5, ...

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

Domain: x ≠ 0
Multiple meanings: y ≠ 0
Parity: odd, y(-x) = - y(x)
Monotone: monotonically decreases
Extremes: No
Convex:
at x< 0 : выпукла вверх
for x > 0: convex downward
Inflection points: No
Intersection points with coordinate axes: No
Sign:
at x< 0, y < 0
for x > 0, y > 0
Limits:
; ; ;
Private values:
at x = -1, y(-1) = (-1) n = -1
for x = 1, y(1) = 1 n = 1
Reverse function:

Even numerator, n = -2, -4, -6, ...

Properties of the power function y = x p with a rational negative exponent, where n = -2, -4, -6, ... is an even negative integer, m = 3, 5, 7 ... is an odd natural integer.

Domain: x ≠ 0
Multiple meanings: y > 0
Parity: even, y(-x) = y(x)
Monotone:
at x< 0 : монотонно возрастает
for x > 0: monotonically decreases
Extremes: No
Convex: convex down
Inflection points: No
Intersection points with coordinate axes: No
Sign: y > 0
Limits:
; ; ;
Private values:
at x = -1, y(-1) = (-1) n = 1
for x = 1, y(1) = 1 n = 1
Reverse function:

The p-value is positive, less than one, 0< p < 1

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

Odd numerator, n = 1, 3, 5, ...

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

Domain: -∞ < x < +∞
Multiple meanings: -∞ < y < +∞
Parity: odd, y(-x) = - y(x)
Monotone: monotonically increases
Extremes: No
Convex:
at x< 0 : выпукла вниз
for x > 0: convex upward
Inflection points: x = 0, y = 0
Intersection points with coordinate axes: x = 0, y = 0
Sign:
at x< 0, y < 0
for x > 0, y > 0
Limits:
;
Private values:
at x = -1, y(-1) = -1
at x = 0, y(0) = 0
for x = 1, y(1) = 1
Reverse function:

Even numerator, n = 2, 4, 6, ...

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

Domain: -∞ < x < +∞
Multiple meanings: 0 ≤ y< +∞
Parity: even, y(-x) = y(x)
Monotone:
at x< 0 : монотонно убывает
for x > 0: increases monotonically
Extremes: minimum at x = 0, y = 0
Convex: convex upward for x ≠ 0
Inflection points: No
Intersection points with coordinate axes: x = 0, y = 0
Sign: for x ≠ 0, y > 0
Limits:
;
Private values:
at x = -1, y(-1) = 1
at x = 0, y(0) = 0
for x = 1, y(1) = 1
Reverse function:

The p index is greater than one, p > 1

Graph of a power function with a rational exponent (p > 1) for various values of the exponent, where m = 3, 5, 7, ... - odd.

Odd numerator, n = 5, 7, 9, ...

Properties of the power function y = x p with a rational exponent greater than one: . Where n = 5, 7, 9, ... - odd natural, m = 3, 5, 7 ... - odd natural.

Domain: -∞ < x < ∞
Multiple meanings: -∞ < y < ∞
Parity: odd, y(-x) = - y(x)
Monotone: monotonically increases
Extremes: No
Convex:
at -∞< x < 0 выпукла вверх
at 0< x < ∞ выпукла вниз
Inflection points: x = 0, y = 0
Intersection points with coordinate axes: x = 0, y = 0
Limits:
;
Private values:
at x = -1, y(-1) = -1
at x = 0, y(0) = 0
for x = 1, y(1) = 1
Reverse function:

Even numerator, n = 4, 6, 8, ...

Properties of the power function y = x p with a rational exponent greater than one: . Where n = 4, 6, 8, ... - even natural, m = 3, 5, 7 ... - odd natural.

Domain: -∞ < x < ∞
Multiple meanings: 0 ≤ y< ∞
Parity: even, y(-x) = y(x)
Monotone:
at x< 0 монотонно убывает
for x > 0 monotonically increases
Extremes: minimum at x = 0, y = 0
Convex: convex down
Inflection points: No
Intersection points with coordinate axes: x = 0, y = 0
Limits:
;
Private values:
at x = -1, y(-1) = 1
at x = 0, y(0) = 0
for x = 1, y(1) = 1
Reverse function:

The denominator of the fractional indicator is even

Let the denominator of the fractional exponent be even: m = 2, 4, 6, ... . In this case, the power function x p is not defined for negative values of the argument. Its properties coincide with the properties of a power function with an irrational exponent (see the next section).

Power function with irrational exponent

Consider a power function y = x p with an irrational exponent p. The properties of such functions differ from those discussed above in that they are not defined for negative values of the argument x. For positive values of the argument, the properties depend only on the value of the exponent p and do not depend on whether p is integer, rational, or irrational.

y = x p for different values of the exponent p.

Power function with negative exponent p< 0

Domain: x > 0
Multiple meanings: y > 0
Monotone: monotonically decreases
Convex: convex down
Inflection points: No
Intersection points with coordinate axes: No
Limits: ;
Private meaning: For x = 1, y(1) = 1 p = 1

Power function with positive exponent p > 0

Indicator less than one 0< p < 1

Domain: x ≥ 0
Multiple meanings: y ≥ 0
Monotone: monotonically increases
Convex: convex upward
Inflection points: No
Intersection points with coordinate axes: x = 0, y = 0
Limits:
Private values: For x = 0, y(0) = 0 p = 0 .
For x = 1, y(1) = 1 p = 1

The indicator is greater than one p > 1

Domain: x ≥ 0
Multiple meanings: y ≥ 0
Monotone: monotonically increases
Convex: convex down
Inflection points: No
Intersection points with coordinate axes: x = 0, y = 0
Limits:
Private values: For x = 0, y(0) = 0 p = 0 .
For x = 1, y(1) = 1 p = 1

References:
I.N. Bronstein, K.A. Semendyaev, Handbook of mathematics for engineers and college students, “Lan”, 2009.

Linear function y=x 1 (y=x)

The graph is a straight line passing through the point (0;0) at an angle of 45 degrees to the positive direction of the Ox axis.

The graph is presented below.

Basic properties of a linear function:

The function is increasing and defined on the entire number line.
It has no maximum or minimum values.

Quadratic function y=x 2

The graph of a quadratic function is a parabola.

Basic properties of a quadratic function:

1. At x =0, y=0, and y>0 at x0
2. The quadratic function reaches its minimum value at its vertex. Ymin at x=0; It should also be noted that the function does not have a maximum value.
3. The function decreases on the interval (-∞;0] and increases on the interval)