Definition of a power function, its properties and graphics. Power function, its properties and graph Demonstration material Lesson-lecture Function concept

1. Power function, its properties and graph;

2. Transformations:

Parallel transfer;

Symmetry about the coordinate axes;

Symmetry about the origin;

Symmetry about the line y = x;

Stretching and shrinking along the coordinate axes.

3. An exponential function, its properties and graph, similar transformations;

4. Logarithmic function, its properties and graph;

5. Trigonometric function, its properties and graph, similar transformations (y = sin x; y = cos x; y = tg x);

Function: y = x\n - its properties and graph.

Power function, its properties and graph

y \u003d x, y \u003d x 2, y \u003d x 3, y \u003d 1 / x etc. All these functions are special cases of the power function, i.e., the function y = xp, where p is a given real number.
The properties and graph of a power function essentially depend on the properties of a power with a real exponent, and in particular on the values for which x and p makes sense xp. Let us proceed to a similar consideration of various cases, depending on
exponent p.

Index p = 2n- even natural number.

y=x2n, where n is a natural number and has the following properties:

the domain of definition is 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=x2n even, because x 2n = (-x) 2n
the function is decreasing on the interval x< 0 and increasing on the interval x > 0.

Function Graph y=x2n has the same form as, for example, the graph of a function y=x4.

2. Indicator p = 2n - 1- odd natural number

In this case, the power function y=x2n-1, where is a natural number, has the following properties:

domain of definition - set R;
set of values - set R;
function y=x2n-1 odd because (- x) 2n-1= x 2n-1 ;
the function is increasing on the entire real axis.

Function Graph y=x2n-1 y=x3.

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

In this case, the power function y=x-2n=1/x2n has the following properties:

set of values - positive numbers y>0;
function y = 1/x2n even, because 1/(-x) 2n= 1/x2n;
the function is increasing on the interval x0.

Graph of the function y = 1/x2n has the same form as, for example, the graph of the function y = 1/x2.

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

the domain of definition is the set R, except for x = 0;
set of values - set R, except for y = 0;
function y=x-(2n-1) odd because (- x)-(2n-1) = -x-(2n-1);
the function is decreasing on the intervals x< 0 and x > 0.

Function Graph y=x-(2n-1) has the same form as, for example, the graph of the function y = 1/x3.

). For real values of the base X and indicator a usually consider only the real values of S. f. x a . They exist, at least for everyone. x > 0; if a -rational number with an odd denominator, then they also exist for all x 0; if the denominator of a rational number a even, or if irrational, then x a has no real meaning whatever x 0. When x = 0 power function x a is zero for all a> 0 and is not defined for a 0; 0° has no definite meaning. S. f. (in the range of real values) is unique, except for those cases when a - a rational number represented by an irreducible fraction with an even denominator: in these cases it is two-valued, and its values for the same value of the argument X> 0 are equal in absolute value but opposite in sign. Usually then only the non-negative, or arithmetic, value of the S. f. is considered. For X> 0 S. f. - increasing if a> 0, and decreasing if a x = 0, in case 0 a x a)" = ax a-1 . Further,

View functions y \u003d cx a, where With - constant factor, play an important role in mathematics and its applications; at a= 1, these functions express direct proportionality (their graphs are straight lines passing through the origin, see fig. one), at a =-1 - inverse proportionality (graphs are equilateral hyperbolas centered at the origin, having coordinate axes as their asymptotes, see fig. 2). Many laws of physics are mathematically expressed using functions of the form y = cx a(see fig. 3); for example, y = cx 2 expresses the law of uniformly accelerated or uniformly slow motion ( y - path, X - time, 2 c- acceleration; initial distance and speed are equal to zero).

In the complex region of S. f. z a is defined for all z≠ 0 by the formula:

where k= 0, ± 1, ± 2,.... If a - integer, then S. f. z a is unambiguous:

If a a - rational (and = p/q, where R and q coprime), then S. f. z a accepts q different meanings:

where ε k = - degree roots q from unity: k = 0, 1, ..., q - 1. If a - irrational, then S. f. z a - infinite value: multiplier ε α2κ π ι accepts for different k different meanings. With complex values of a S. f. z a is determined by the same formula (*). For example,

so that, in particular, k = 0, ± 1, ± 2,....

Under the main value ( z a) 0 S. f. its meaning is understood k = 0 if -πz ≤ π (or 0 ≤ arg z z a) = |z a|e ia arg z , (i) 0 \u003d e -π / 2, etc.

Big soviet encyclopedia. - M.: Soviet Encyclopedia. 1969-1978 .

See what the "Power function" is in other dictionaries:

A function of the form y = axn, where a and n are any real numbers ... Big Encyclopedic Dictionary

Power function function, where (exponent) is some real number ... Wikipedia

A function of the form y = axn, where a and p are real. numbers, S. f. covers a large number of regularities in nature. On fig. graphs of S. f are shown. for n \u003d 1, 2, 3, 1/2 and a \u003d 1. To Art. Power function … Big encyclopedic polytechnic dictionary

A function of the form y=axn, where a and n are any real numbers. The figure shows graphs of a power function for n = 1, 2, 3, 1/2 and a = 1. * * * POWER FUNCTION POWER FUNCTION, a function of the form y = axn, where a and n are any real numbers ... encyclopedic Dictionary

power function- laipsninė funkcija statusas T sritis automatika atitikmenys: engl. power function vok. Potenzfunktion, f rus. power function, f pranc. fonction puissance, f … Automatikos terminų žodynas

The function y \u003d x a, where a is a constant number. If a is an integer, then the C. f. special case rational function. With complex values of chi aC. f. is ambiguous if a is not an integer. For fixed real. and a number x a is a power ... Mathematical Encyclopedia

A function of the form y = axn, where a and n are any real numbers. On fig. graphs of S. f are shown. for n= 1, 2, 3, 1/2 and a=1 ... Natural science. encyclopedic Dictionary

demand function- A function that shows how the volume of sales of a particular product changes depending on its price with equal marketing efforts to promote it to the market. demand function A function that reflects ... ... Technical Translator's Handbook

Demand function- a function that reflects the dependence of the volume of demand for individual goods and services (consumer goods) on a set of factors affecting it. Narrower interpretation: F.s. expresses the interdependence between the demand for a product and the price ... ... Economic and Mathematical Dictionary

Y = 1 + x + x2 + x3 + ... is defined for real or complex x values whose moduli are less than one. F. of the form y \u003d p0xn + p1xn 1 + p2xn 2 + ... + pn 1x + pn, where the coefficients, p0, p1, p2, ..., pn, these numbers are called the entire function n oh ... ... Encyclopedia of Brockhaus and Efron

Books

A set of tables. Algebra and the beginnings of analysis. Grade 11. 15 tables + methodology, . The tables are printed on thick polygraphic cardboard measuring 680 x 980 mm. Brochure with guidelines for the teacher. Study album of 15 sheets.…

Recall the properties and graphs of power functions with an integer negative indicator.

For even n, :

Function example:

All graphs of such functions pass through two fixed points: (1;1), (-1;1). A feature of functions of this type is their parity, the graphs are symmetrical with respect to the op-y axis.

Rice. 1. Graph of a function

For odd n, :

Function example:

All graphs of such functions pass through two fixed points: (1;1), (-1;-1). A feature of functions of this type is their oddness, the graphs are symmetrical with respect to the origin.

Rice. 2. Function Graph

Let us recall the main definition.

The degree of a non-negative number a with a rational positive exponent is called a number.

The degree of a positive number a with a rational negative exponent is called a number.

For the following equality holds:

For example: ; - the expression does not exist by definition of a degree with a negative rational exponent; exists, since the exponent is an integer,

Let us turn to the consideration of power functions with a rational negative exponent.

For example:

To plot this function, you can make a table. We will do otherwise: first, we will build and study the graph of the denominator - we know it (Figure 3).

Rice. 3. Graph of a function

The graph of the denominator function goes through fixed point(1;1). When constructing a graph of the original function, this point remains, when the root also tends to zero, the function tends to infinity. And, conversely, as x tends to infinity, the function tends to zero (Figure 4).

Rice. 4. Function Graph

Consider one more function from the family of functions under study.

It is important that by definition

Consider the graph of the function in the denominator: , we know the graph of this function, it increases in its domain of definition and passes through the point (1; 1) (Figure 5).

Rice. 5. Function Graph

When constructing a graph of the original function, the point (1; 1) remains, when the root also tends to zero, the function tends to infinity. And, conversely, as x tends to infinity, the function tends to zero (Figure 6).

Rice. 6. Function Graph

The considered examples help to understand how the graph goes and what are the properties of the function under study - a function with a negative rational exponent.

Graphs of functions of this family pass through the point (1;1), the function decreases over the entire domain of definition.

Function scope:

The function is not bounded from above, but bounded from below. The function has neither a maximum nor a minimum value.

The function is continuous, it takes all positive values from zero to plus infinity.

Convex Down Function (Figure 15.7)

Points A and B are taken on the curve, a segment is drawn through them, the entire curve is below the segment, this condition is satisfied for arbitrary two points on the curve, therefore the function is convex downward. Rice. 7.

Rice. 7. Convexity of a function

It is important to understand that the functions of this family are bounded from below by zero, but they do not have the smallest value.

Example 1 - find the maximum and minimum of the function on the interval )