Power function properties. Exponential function - properties, graphs, formulas. The denominator of the fractional exponent is odd

The functions y = ax, y = ax 2, y = a / x - are particular forms of the power function for n = 1, n = 2, n = -1 .

If n fractional number p/ q with even denominator q and odd numerator R, then the value may have two signs, and the graph has one more part at the bottom of the abscissa axis X, and it is symmetrical to the top.

We see a graph of a two-valued function y = ± 2x 1/2, i.e. represented by a parabola with a horizontal axis.

Function graphs y = xn at n = -0,1; -1/3; -1/2; -1; -2; -3; -10 ... These graphs pass through point (1; 1).

When n = -1 we get hyperbole... At n < - 1 the graph of the power function is located first above the hyperbola, i.e. between x = 0 and x = 1, and then below (for x> 1). If n> -1 the graph is reversed. Negative values X and fractional values n are similar for positive n.

All graphs approach unlimitedly as to the abscissa axis X, and to the ordinate axis at without touching them. Due to the similarity to the hyperbola, these graphs are called hyperbolas. n th order.

1. Power function, its properties and graph;

2. Transformations:

Parallel transfer;

Symmetry about the coordinate axes;

Symmetry about the origin;

Symmetry about the straight line y = x;

Stretch and shrink along the coordinate axes.

3. 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 = tan x);

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

Power function, its properties and graph

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

1. Indicator p = 2n- an even natural number.

y = x 2n, where n- a natural number, has the following properties:

• domain of definition - all real numbers, that is, the set R;
• the set of values ​​is non-negative numbers, i.e. y is greater than or equal to 0;
• function y = x 2n even since x 2n = (-x) 2n
• the function is decreasing in the interval x< 0 and increasing in the interval x> 0.

Function graph y = x 2n has the same form as, for example, a 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, since (- x) 2n-1= x 2n-1;
• the function is increasing along the whole real axis.

Function graph y = x 2n-1 y = x 3.

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 since 1 / (- x) 2n= 1 / x 2n;
• the function is increasing on the interval x0.

Function y plot = 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 for x = 0;
• set of values ​​- set R, except for y = 0;
• function y = x - (2n-1) odd, since (- x) - (2n-1) = -x - (2n-1);
• the function is decreasing in 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 / x 3.

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 a power function, that is, the functions y = x p, where p is a given real number.
The properties and graph of the power function essentially depends on the properties of the power with a real exponent, and in particular on what values x and p makes sense degree x p... Let us proceed to a similar consideration of various cases, depending on
exponent p.

1. Indicator p = 2n is an even natural number.

y = x 2n, where n- a natural number, has the following

properties:

• domain of definition - all real numbers, that is, the set R;
• the set of values ​​is non-negative numbers, i.e. y is greater than or equal to 0;
• function y = x 2n even since x 2n=(- x) 2n
• the function is decreasing in the interval x<0 and increasing in the interval x> 0.

Function graph y = x 2n has the same form as, for example, a 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, since (- x) 2n-1=x 2n-1;
• the function is increasing along the whole real axis.

Function graph y = x 2n-1 has the same form as, for example, the graph of the function y = x 3 .

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

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

• domain of definition - set R, except for x = 0;
• set of values ​​- positive numbers y> 0;
• function y = 1 / x 2n even since 1 / (- x) 2n=1 / x 2n;
• the function is increasing on the interval x<0 и убывающей на промежутке x>0.

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

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 a power function, that is, the functions y = x p, where p is a given real number.
The properties and graph of the power function essentially depends on the properties of the power with a real exponent, and in particular on what values x and p makes sense degree x p... Let us proceed to a similar consideration of various cases, depending on
exponent p.

1. Indicator p = 2n is an even natural number.

y = x 2n, where n- a natural number, has the following

properties:

• domain of definition - all real numbers, that is, the set R;
• the set of values ​​is non-negative numbers, i.e. y is greater than or equal to 0;
• function y = x 2n even since x 2n=(- x) 2n
• the function is decreasing in the interval x<0 and increasing in the interval x> 0.

Function graph y = x 2n has the same form as, for example, a 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, since (- x) 2n-1=x 2n-1;
• the function is increasing along the whole real axis.

Function graph y = x 2n-1 has the same form as, for example, the graph of the function y = x 3 .

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

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

• domain of definition - set R, except for x = 0;
• set of values ​​- positive numbers y> 0;
• function y = 1 / x 2n even since 1 / (- x) 2n=1 / x 2n;
• the function is increasing on the interval x<0 и убывающей на промежутке x>0.

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

Grade 10

POWER FUNCTION

Exponential calledfunction given by formulawhere, p – some real number.

I ... Indicatoris an even natural number. Then the power function wheren

D ( y )= (−; +).

2) The range of values ​​of a function is a set of non-negative numbers if:

set of non-positive numbers if:

3) ) . Hence, the functionOy .

4) If, then the function decreases asX (-; 0] and increases atX and decreases atX }