Exponential function is a generalization of the product of n numbers equal to a :
y (n) = a n = a a a a,
to the set of real numbers x :
y (x) = x.
Here a is a fixed real number, which is called the base of the exponential function.
An exponential function with base a is also called exponential to base a.

The generalization is carried out as follows.
For natural x = 1, 2, 3,... , the exponential function is the product of x factors:
.
Moreover, it has the properties (1.5-8) (), which follow from the rules for multiplying numbers. At zero and negative values ​​of integers , the exponential function is determined by formulas (1.9-10). For fractional values ​​x = m/n of rational numbers, , it is determined by formula (1.11). For real , the exponential function is defined as the limit of the sequence:
,
where is an arbitrary sequence of rational numbers converging to x : .
With this definition, the exponential function is defined for all , and satisfies the properties (1.5-8), as well as for natural x .

A rigorous mathematical formulation of the definition of an exponential function and a proof of its properties is given on the page "Definition and proof of the properties of an exponential function".

Properties of the exponential function

The exponential function y = a x has the following properties on the set of real numbers () :
(1.1) is defined and continuous, for , for all ;
(1.2) when a ≠ 1 has many meanings;
(1.3) strictly increases at , strictly decreases at ,
is constant at ;
(1.4) at ;
at ;
(1.5) ;
(1.6) ;
(1.7) ;
(1.8) ;
(1.9) ;
(1.10) ;
(1.11) , .

Other useful formulas
.
The formula for converting to an exponential function with a different power base:

For b = e , we get the expression of the exponential function in terms of the exponent:

Private values

, , , , .

The figure shows graphs of the exponential function
y (x) = x
for four values degree bases:a= 2 , a = 8 , a = 1/2 and a = 1/8 . It can be seen that for a > 1 exponential function is monotonically increasing. The larger the base of the degree a, the stronger the growth. At 0 < a < 1 exponential function is monotonically decreasing. The smaller the exponent a, the stronger the decrease.

Ascending, descending

The exponential function at is strictly monotonic, so it has no extrema. Its main properties are presented in the table.

	y = a x , a > 1	y = x, 0 < a < 1
Domain	- ∞ < x < + ∞	- ∞ < x < + ∞
Range of values	0 < y < + ∞	0 < y < + ∞
Monotone	increases monotonically	decreases monotonically
Zeros, y= 0	No	No
Points of intersection with the y-axis, x = 0	y= 1	y= 1
	+ ∞	0
	0	+ ∞

Inverse function

The reciprocal of an exponential function with a base of degree a is the logarithm to base a.

If , then
.
If , then
.

Differentiation of the exponential function

To differentiate an exponential function, its base must be reduced to the number e, apply the table of derivatives and the rule for differentiating a complex function.

To do this, you need to use the property of logarithms
and the formula from the table of derivatives:
.

Let an exponential function be given:
.
We bring it to the base e:

We apply the rule of differentiation of a complex function. To do this, we introduce a variable

Then

From the table of derivatives we have (replace the variable x with z ):
.
Since is a constant, the derivative of z with respect to x is
.
According to the rule of differentiation of a complex function:
.

Derivative of exponential function

.
Derivative of the nth order:
.
Derivation of formulas > > >

An example of differentiating an exponential function

Find the derivative of a function
y= 35 x

Decision

We express the base of the exponential function in terms of the number e.
3 = e log 3
Then
.
We introduce a variable
.
Then

From the table of derivatives we find:
.
Insofar as 5ln 3 is a constant, then the derivative of z with respect to x is:
.
According to the rule of differentiation of a complex function, we have:
.

Answer

Integral

Expressions in terms of complex numbers

Consider the complex number function z:
f (z) = az
where z = x + iy ; i 2 = - 1 .
We express the complex constant a in terms of the modulus r and the argument φ :
a = r e i φ
Then


.
The argument φ is not uniquely defined. In general
φ = φ 0 + 2 pn,
where n is an integer. Therefore, the function f (z) is also ambiguous. Often considered its main importance
.

Expansion in series

.

References:
I.N. Bronstein, K.A. Semendyaev, Handbook of Mathematics for Engineers and Students of Higher Educational Institutions, Lan, 2009.

Hypothesis: If you study the movement of the graph during the formation of the equation of functions, you will notice that all graphs obey common laws, therefore, you can formulate general laws regardless of the functions, which will not only facilitate the construction of graphs of various functions, but also use them in solving problems.

Purpose: To study the movement of graphs of functions:

1) The task of studying literature

2) Learn to build graphs of various functions

3) Learn to convert graphs of linear functions

4) Consider the use of graphs in solving problems

Object of study: Graphs of functions

Subject of research: Movements of graphs of functions

Relevance: The construction of function graphs, as a rule, takes a lot of time and requires attention from the student, but knowing the rules for transforming function graphs and graphs of basic functions, you can quickly and easily build function graphs, which will allow you not only to complete tasks for plotting function graphs, but also solve related problems (to find the maximum (minimum height of time and meeting point))

This project is useful to all students of the school.

Literature review:

The literature discusses ways to construct a graph of various functions, as well as examples of the transformation of graphs of these functions. Graphs of almost all main functions are used in various technical processes, which makes it possible to more clearly present the course of the process and program the result

Permanent function. This function is given by the formula y = b, where b is some number. The graph of a constant function is a straight line parallel to the x-axis and passing through the point (0; b) on the y-axis. The graph of the function y \u003d 0 is the abscissa axis.

Types of function 1Direct proportionality. This function is given by the formula y \u003d kx, where the coefficient of proportionality k ≠ 0. The direct proportionality graph is a straight line passing through the origin.

Linear function. Such a function is given by the formula y = kx + b, where k and b are real numbers. The graph of a linear function is a straight line.

Linear function graphs can intersect or be parallel.

So, the lines of the graphs of linear functions y \u003d k 1 x + b 1 and y \u003d k 2 x + b 2 intersect if k 1 ≠ k 2; if k 1 = k 2 , then the lines are parallel.

2Inverse proportionality is a function that is given by the formula y \u003d k / x, where k ≠ 0. K is called the inverse proportionality coefficient. The inverse proportionality graph is a hyperbola.

The function y \u003d x 2 is represented by a graph called a parabola: on the interval [-~; 0] the function is decreasing, on the interval the function is increasing.

The function y \u003d x 3 increases along the entire number line and is graphically represented by a cubic parabola.

Power function with natural exponent. This function is given by the formula y \u003d x n, where n is a natural number. Graphs of a power function with a natural exponent depend on n. For example, if n = 1, then the graph will be a straight line (y = x), if n = 2, then the graph will be a parabola, etc.

A power function with a negative integer exponent is represented by the formula y \u003d x -n, where n is a natural number. This function is defined for all x ≠ 0. The graph of the function also depends on the exponent n.

Power function with a positive fractional exponent. This function is represented by the formula y \u003d x r, where r is a positive irreducible fraction. This function is also neither even nor odd.

Graph-line that displays the relationship of dependent and independent variables on the coordinate plane. The graph serves to visually display these elements.

An independent variable is a variable that can take on any value in the scope of the functions (where the given function makes sense (cannot be divided by zero))

To plot a function graph,

1) Find ODZ (range of acceptable values)

2) take some arbitrary values ​​for the independent variable

3) Find the value of the dependent variable

4) Build a coordinate plane, mark these points on it

5) Connect their lines, if necessary, investigate the resulting graph. Transformation of graphs of elementary functions.

Graph Conversion

In their pure form, the basic elementary functions are, unfortunately, not so common. Much more often one has to deal with elementary functions obtained from basic elementary functions by adding constants and coefficients. Graphs of such functions can be built by applying geometric transformations to the graphs of the corresponding basic elementary functions (or by switching to a new coordinate system). For example, a quadratic function formula is a quadratic parabola formula, compressed three times relative to the ordinate axis, symmetrically displayed relative to the abscissa axis, shifted against the direction of this axis by 2/3 units and shifted along the direction of the ordinate axis by 2 units.

Let's understand these geometric transformations of the graph of a function step by step using specific examples.

With the help of geometric transformations of the graph of the function f (x), a graph of any function of the form formula can be constructed, where the formula is the compression or expansion coefficients along the oy and ox axes, respectively, the minus signs in front of the coefficients formula and formula indicate a symmetrical display of the graph relative to the coordinate axes , a and b define the shift relative to the abscissa and ordinate axes, respectively.

Thus, there are three types of geometric transformations of the function graph:

The first type is scaling (compression or expansion) along the abscissa and ordinate axes.

The need for scaling is indicated by formula coefficients other than one, if the number is less than 1, then the graph is compressed relative to oy and stretched relative to ox, if the number is greater than 1, then we stretch along the ordinate axis and shrink along the abscissa axis.

The second type is a symmetrical (mirror) display with respect to the coordinate axes.

The need for this transformation is indicated by the minus signs in front of the coefficients of the formula (in this case, we display the graph symmetrically with respect to the ox axis) and the formula (in this case, we display the graph symmetrically with respect to the y axis). If there are no minus signs, then this step is skipped.

Function Graph Transformation

In this article, I will introduce you to linear transformations of function graphs and show how to use these transformations from a function graph to get a function graph.

A linear transformation of a function is a transformation of the function itself and/or its argument to the form , as well as a transformation containing the module of the argument and/or functions.

The following actions cause the greatest difficulties in plotting graphs using linear transformations:

1. The isolation of the base function, in fact, the graph of which we are transforming.
2. Definitions of the order of transformations.

And It is on these points that we will dwell in more detail.

Let's take a closer look at the function

It is based on a function. Let's call her basic function.

When plotting a function we make transformations of the graph of the base function .

If we were to transform the function in the same order in which its value was found for a certain value of the argument, then

Let's consider what types of linear argument and function transformations exist, and how to perform them.

Argument transformations.

1. f(x) f(x+b)

1. We build a graph of a function

2. We shift the graph of the function along the OX axis by |b| units

• left if b>0
• right if b<0

Let's plot the function

1. We plot the function

2. Shift it 2 units to the right:

2. f(x) f(kx)

1. We build a graph of a function

2. Divide the abscissas of the graph points by k, leave the ordinates of the points unchanged.

Let's plot the function.

1. We plot the function

2. Divide all abscissas of the graph points by 2, leave the ordinates unchanged:

3. f(x) f(-x)

1. We build a graph of a function

2. We display it symmetrically about the OY axis.

Let's plot the function.

1. We plot the function

2. We display it symmetrically about the OY axis:

4. f(x) f(|x|)

1. We plot the function

2. We erase the part of the graph located to the left of the OY axis, the part of the graph located to the right of the OY axis We complete it symmetrically about the OY axis:

The graph of the function looks like this:

Let's plot the function

1. We build a function graph (this is a function graph shifted along the OX axis by 2 units to the left):

2. Part of the graph located to the left of the OY (x<0) стираем:

3. The part of the graph located to the right of the OY axis (x>0) is completed symmetrically with respect to the OY axis:

Important! The two main rules for argument conversion.

1. All argument transformations are performed along the OX axis

2. All transformations of the argument are performed "vice versa" and "in reverse order".

For example, in a function, the sequence of argument transformations is as follows:

1. We take the module from x.

2. Add the number 2 to the modulo x.

But we did the plotting in the reverse order:

First, we performed the transformation 2. - shifted the graph by 2 units to the left (that is, the abscissas of the points were reduced by 2, as if "vice versa")

Then we performed the transformation f(x) f(|x|).

Briefly, the sequence of transformations is written as follows:

Now let's talk about function transformation . Transformations are being made

1. Along the OY axis.

2. In the same sequence in which the actions are performed.

These are the transformations:

1. f(x)f(x)+D

2. Shift it along the OY axis by |D| units

• up if D>0
• down if D<0

Let's plot the function

1. We plot the function

2. Move it along the OY axis by 2 units up:

2. f(x)Af(x)

1. We plot the function y=f(x)

2. We multiply the ordinates of all points of the graph by A, we leave the abscissas unchanged.

Let's plot the function

1. Graph the function

2. We multiply the ordinates of all points of the graph by 2:

3.f(x)-f(x)

1. We plot the function y=f(x)

Let's plot the function.

1. We build a function graph.

2. We display it symmetrically about the OX axis.

4. f(x)|f(x)|

1. We plot the function y=f(x)

2. The part of the graph located above the OX axis is left unchanged, the part of the graph located below the OX axis is displayed symmetrically about this axis.

Let's plot the function

1. We build a function graph. It is obtained by shifting the graph of the function along the OY axis by 2 units down:

2. Now we will display the part of the chart located below the OX axis symmetrically about this axis:

And the last transformation, which, strictly speaking, cannot be called a function transformation, since the result of this transformation is no longer a function:

|y|=f(x)

1. We plot the function y=f(x)

2. We erase the part of the graph located below the OX axis, then we complete the part of the graph located above the OX axis symmetrically about this axis.

Let's build a graph of the equation

1. We build a function graph:

2. We erase the part of the graph located below the OX axis:

3. The part of the graph located above the OX axis is completed symmetrically about this axis.

And finally, I suggest you watch the VIDEO LESSON in which I show a step-by-step algorithm for plotting a function graph

The graph of this function looks like this:

Which of these functions have an inverse? For such functions find inverse functions:

4.12. a)	y=x;					b) y = 6 −3x;
d) y =						e) y \u003d 2 x 3 +5;
4.13. a)	y = 4x − 5 ;						y \u003d 9 - 2 x - x 2;
	y = sign x ;						y=1 + lg(x + 2) ;

	y = 2 x 2 +1 ;
			x − 2
						at x< 0
c) y =		−x
						for x ≥ 0

Find out which of these functions are monotonic, which are strictly monotonic, and which are bounded:

4.14. a)

f (x) = c, c R ;

b) f (x) \u003d cos 2 x;

c) f (x) \u003d arctg x;

d) f (x) \u003d e 2 x;

e) f (x) \u003d -x 2 + 2 x;

e) f(x) =

2x+5

y = ctg7 x .

4.15. a)

f(x) = 3−x

b) f(x) =

f(x)=

x + 3

x+6

x< 0,

3x+5

d) f (x) \u003d 3 x 3 - x;

− 10 at

f(x)=

e) f(x) =

x 2 at

x ≥ 0;

x+1

f(x) = tg(sinx).

4.2. elementary functions. Function Graph Transformation

Recall that the graph of the function f (x) in the Cartesian rectangular coordinate system Oxy is the set of all points in the plane with coordinates (x, f (x)).

Often the graph of the function y \u003d f (x) can be built using transformations (shift, stretching) of the graph of some already known function.

In particular, from the graph of the function y \u003d f (x), the graph of the function is obtained:

1) y \u003d f (x) + a - shift along the Oy axis by a units (up if a > 0, and down if a< 0 ;

2) y \u003d f (x − b) - shift along the Ox axis by b units (to the right, if b > 0,

and to the left if b< 0 ;

3) y \u003d kf (x) - by stretching along the Oy axis by k times;

4) y \u003d f (mx) - compression along the Ox axis by m times;

5) y \u003d - f (x) - symmetrical reflection about the axis Ox;

6) y \u003d f (−x) - symmetrical reflection about the axis Oy;

7) y \u003d f (x), as follows: the part of the graph located not

below the Ox axis, remains unchanged, and the “lower” part of the graph is reflected symmetrically about the Ox axis;

8) y = f (x ) , as follows: the right side of the graph (for x ≥ 0 )

remains unchanged, and instead of "left" a symmetrical reflection of the "right" about the axis Oy is built.

The main elementary functions are called:

1) constant function y = c;

2) power function y = x α , α R ;

3) exponential function y \u003d a x, a ≠ 0, a ≠1;

4) logarithmic function y = log a x , a > 0, a ≠ 1 ;

5) trigonometric functions y = sin x , y = cos x , y = tg x ,

y = ctg x , y = sec x (where sec x = cos 1 x ), y = cosec x (where cosec x = sin 1 x );

6) inverse trigonometric functions y \u003d arcsin x, y \u003d arccos x, y \u003d arctg x, y \u003d arcctg x.

elementary functions called functions obtained from the basic elementary functions with the help of a finite number of arithmetic operations (+, − , ÷) and compositions (i.e., the formation of complex functions f g ).

Example 4.6. Plot a function

1) y \u003d x 2 + 6 x + 7; 2) y = −2sin 4 x .

Solution: 1) by highlighting the full square, the function is converted to the form y = (x +3) 2 − 2, so the graph of this function can be obtained from the graph of the function y = x 2 . It is enough to first shift the parabola y \u003d x 2 three units to the left (we get the graph of the function y \u003d (x +3) 2), and then two units down (Fig. 4.1);

	standard	sinusoid						y = sin x	four times along the axis		Ox,
we get the graph of the function y \u003d sin 4 x (Fig. 4.2).
										y=sin4x
										y=sin x

Stretching the resulting graph twice along the Oy axis, we get the graph of the function y \u003d 2sin 4 x (Fig. 4.3). It remains to reflect the last graph relative to the Ox axis. The result will be the desired graph (see Fig. 4.3).

						y=2sin4x

y=–2sin4x

Tasks for independent solution

Construct graphs of the following functions, based on the graphs of the main elementary functions:

4.16. a) y \u003d x 2 -6 x +11;

4.17. a) y = −2sin(x −π ) ;

4.18. a) y = − 4 x −1 ;

4.19. a) y = log 2 (−x ) ;

4.20. a) y = x +5 ;

4.21. a) y \u003d tg x;

4.22. a) y = sign x ;

4.23. a) y = x x + + 4 2 ;

y = 3 - 2 x - x 2 .

y = 2 cos 2 x .

Depending on the conditions of the course of physical processes, some quantities take on constant values ​​and are called constants, others change under certain conditions and are called variables.

A careful study of the environment shows that physical quantities are dependent on each other, that is, a change in some quantities entails a change in others.

Mathematical analysis studies the quantitative relationships of mutually changing quantities, abstracting from the specific physical meaning. One of the basic concepts of mathematical analysis is the concept of a function.

Consider the elements of the set and the elements of the set
(Fig. 3.1).

If some correspondence is established between the elements of the sets
and as a rule , then we note that the function is defined
.

Definition 3.1. Conformity , which is associated with each element not an empty set
some well-defined element not an empty set , is called a function or mapping
in .

Symbolically display
in is written as follows:

.

At the same time, many
is called the domain of the function and is denoted
.

In turn, many is called the range of the function and is denoted
.

In addition, it should be noted that the elements of the set
are called independent variables, the elements of the set are called dependent variables.

Ways to set a function

The function can be defined in the following main ways: tabular, graphical, analytical.

If, on the basis of experimental data, tables are compiled that contain the values ​​of the function and the corresponding values ​​of the argument, then this method of specifying the function is called tabular.

At the same time, if some studies of the result of the experiment are output to the registrar (oscilloscope, recorder, etc.), then it is noted that the function is set graphically.

The most common is the analytical way of defining a function, i.e. a method in which the independent and dependent variables are linked using a formula. In this case, the domain of definition of the function plays an important role:

different, although they are given by the same analytical relations.

If only the function formula is given
, then we consider that the domain of definition of this function coincides with the set of those values ​​of the variable , for which the expression
has the meaning. In this regard, the problem of finding the domain of a function plays a special role.

Task 3.1. Find the scope of a function

Decision

The first term takes real values ​​at
, and the second at. Thus, to find the domain of definition of a given function, it is necessary to solve the system of inequalities:

As a result of the solution of such a system, we obtain . Therefore, the domain of the function is the segment
.

The simplest transformations of graphs of functions

The construction of graphs of functions can be greatly simplified if we use the known graphs of the main elementary functions. The following functions are called basic elementary functions:

1) power function
where
;

2) exponential function
where
and
;

3) logarithmic function
, where - any positive number other than one:
and
;

4) trigonometric functions




;
.

5) inverse trigonometric functions
;
;
;
.

Elementary functions are called functions that are obtained from basic elementary functions using four arithmetic operations and superpositions applied a finite number of times.

Simple geometric transformations also simplify the process of plotting functions. These transformations are based on the following statements:

The graph of the function y=f(x+a) is the graph y=f(x), shifted (for a >0 to the left, for a< 0 вправо) на |a| единиц параллельно осиOx.

Graph of the function y=f(x) +b has graphs y=f(x), shifted (if b>0 up, if b< 0 вниз) на |b| единиц параллельно осиOy.

The graph of the function y = mf(x) (m0) is the graph y = f(x), stretched (for m>1) m times or compressed (for 0

The graph of the function y = f(kx) is the graph y = f(x), compressed (for k > 1) k times or stretched (for 0< k < 1) вдоль оси Ox. При –< k < 0 график функции y = f(kx) есть зеркальное отображение графика y = f(–kx) от оси Oy.