Municipal budgetary educational institution

"School No. 77"

Sormovsky district of Nizhny Novgorod

Students' Scientific Society

Graphs and their functions

Completed by: Bakanin Timofey,

9th grade student

Scientific supervisor: Grigorenko L.A.

Nizhny Novgorod

2016

Content

Introduction………………………………………………………………………………...3

Functional dependence and function graph. Methods for specifying a function……..4

The simplest elementary functions……………………………………………………………...5

1. 1. Linear function

      Parabola

      Hyperbola

      Power function

3. Geometric transformations of function graphs…………………………………..11

4. Plotting function graphs……………………………………………………….12

5. Application of function graphs to problem solving……………………………………..17

Conclusion………………………………………………………………………………………...22

References………………………………………………………………………………..23

Introduction.

The study of the behavior of functions and the construction of their graphs is an important branch of mathematics. Fluency in charting techniques often helps solve many problems, and sometimes is the only means of solving them. In addition, the ability to construct graphs of functions is of great independent interest. There are various ways to specify functions: analytical, tabular, verbal, parametric, and graphical.

Whenever you need to find out the general nature of the behavior of a function or discover its features, the graph, due to its clarity, is irreplaceable.

Indeed, the graph of a function is a representation of our understanding of how the function behaves. To do this, you need to know elementary functions, their properties and graphs, and master the technique of constructing graphs.

In engineering and physics, the graphical method of specifying a function is often used. A seismologist, analyzing a seismogram, finds out when the earthquake occurred, where it occurred, and determines the strength and nature of the tremors. A doctor examining a patient can judge from a cardiogram about cardiac abnormalities: studying a cardiogram helps to correctly diagnose the disease. A radio electronics engineer, based on the characteristics of a semiconductor element, selects the most suitable mode of its operation. As mathematics develops, the penetration of the graphical method into various areas of human life increases. In particular, the use of functional dependencies and plotting are widely used in economics.

With the development of computer technology, with its excellent graphical tools and high speeds of operations, working with function graphs has become much more interesting, visual, and exciting.

I chose this particular topic for my work because it will help me in passing exams and is interesting in itself.

Functional dependence and function graph. Methods for specifying a function

If each valuexfrom a certain set of numbers a number is assignedy, then we say that the function y(x) is given on this set. Whereinxis called the independent variable or argument, andy– dependent variable or function.

The domain of a function is the set of all values ​​that its argument can take.

If a function is given by a formula, then it is generally accepted that it is defined for all those values ​​of the argument for which this formula makes sense, i.e. all actions indicated in the expression on the right side of the formula are feasible.

The graph of a function is the set of all points of the coordinate plane, the abscissas of which are equal to the values ​​of the independent variable from the domain of definition of this function, and the ordinates are equal to the corresponding values ​​of the function.

Methods for specifying a function:

1) Tabular method

With this method, a series of individual values ​​of the argument,..., and the corresponding series of individual values ​​of the function,..., are specified in the form of a table. Despite its simplicity, this method of specifying a function has a significant drawback, since it does not give a complete picture of the nature of the functional relationship betweenxAndyand is not visual.

2) Verbal method

This method of specifying is usually illustrated with the example of the Dirichlet functiony= D( x): Ifxis a rational number, then the value of the functionD( x) is equal to 1, and if the numberx-irrational, then the meaningD( x) is equal to zero. So to find the valueD() at a given valuex=, it is necessary to establish in some way whether a number is rational or irrational.

3) Graphic method

Functional dependence can be specified using a function graphy= f( x). The advantage of this method of assignment is clarity, which allows you to establish important features of the behavior of the function. The disadvantage of the graphical method is the impossibility of using mathematical tools for a more detailed study of the function.

4) Analytical method

With the analytical method of specifying, a formula is known according to which, for a given value of the argumentxyou can find the corresponding function valuey. In mathematics, the analytical method of specifying functions is most often used. The advantages of this method of setting are compactness, the ability to calculate the valueyat any valuexand the possibility of using mathematical tools for a more detailed study of the behavior of the function. However, the analytical method of specifying a function is characterized by insufficient clarity and possible difficulty in calculating the function values.

The simplest elementary functions.

1) Linear:

Properties:

1. D( y) = (−∞; +∞); E( y) = (−∞; +∞).

2.Ifb= 0, then the function is odd.

Ifb

3. If x = 0, then y =b, if y = 0, then x = −.

4. Ifk> 0, then the function increases for x-any.

Ifk < 0, то функция decreases for x-any.

Construction of a linear function.

In order to construct a straight line, it is enough to know two points. Graph the functiony=2 x+1 .

x

2) Quadratic function:; .

Properties:

1. D( y) = (−∞; +∞).

2. Ifa> 0, thenE( y) = [y V; +∞);

Ifa < 0, то E( y) = (−∞; at V].

3.Ifb= 0, then the function is even.

Ifb≠ 0, then the function is neither even nor odd.

4.If x = 0, then y =c, if y = 0, then x 1,2 =

5. Ifa> 0, then the function increases as x[x V; +∞);

functiondecreases as x(−∞; x V].

Ifa < 0, то функция возрастает at x(−∞; x V];

functiondecreases as x[x V; +∞).

Construction of a parabola.

Determine the direction of the branches of the parabola.

If, then the branches are directed upwards,

If, then the branches are directed downwards.

Find the vertex of the parabola using two formulas in turn: and.

Plot the resulting point on the graph and draw an axis of symmetry through it, parallel to the coordinate axisOy.

Find 4 graph points by substituting valuesxunder the formula.

Create a graph based on the points found.

3) Hyperbole:

Properties:

1. D( y) = (−∞; 0) u(0; +∞)

2. E( y) = (−∞; 0) u(0 ; +∞)

3. The function is odd.

4. x ≠ 0, y ≠ 0.

5. Ifk> 0, then the function decreases

at x(−∞; 0)u(0; +∞).

Ifk < 0, то функция возрастает

at x(−∞; 0)u(0; +∞).

Construction of a hyperbola.

Finding the domain of definition

Functionisodd, which means the hyperbola is symmetrical about the origin.

Graph of a function of the formrepresent two branches of a hyperbola.

If, then the hyperbola is located in the first and third coordinate quarters

If, then the hyperbola is located in the second and fourth coordinate quarters.

We use the point-by-point method of construction, in this case,

valuesxIt is advantageous to select so that it is divided entirely.

4)Function with module:

Constructing a function with a module.

Let's consider the simplest case

For the function coincides with the function, and for x<0 - с функцией.

5)Power function:

Properties:

Ifn = 2 k, WherekЄ Z

1. D( y)=(−∞; +∞).

2. E( y)=.

Ifn = 2 k+1, wherek Є Z

1. D( y)=(−∞; +∞).

2. E( y)=(−∞; +∞).

3. The function is odd.

4. If x = 0, then y = 0.

5. Function increasesat xЄ(−∞; +∞).

Construction of a cubic parabola.

A cubic parabola is given by the function

We find the domain of definition -x- any real number

Function range -y- any real number.

Functionisodd. Ifthe function is odd, then its graphsymmetrical about the origin.

Using the point-by-point construction method, we make a drawing.

Geometric transformations of function graphs.

1)View conversiony = f( x)+ b

y = f(x) onbunits along the ordinate axis.

Ifb> 0, then a shift occurs

Ifb<0, то происходит смещение↓

2)View conversiony = f(x – a)

This is a parallel transfer of the graph of a functiony = f(x) onaunits along the x-axis

If a > 0, then a shift occurs →

If a< 0, то происходит смещение ←

3)View conversiony = kf(x)

This is tension (compression) inkonce function graphicsy = f(x) along the ordinate axis.

If, |k| > 1, then stretching occurs

If, |k| < 1, то происходит compression

4)View conversiony = f(mx)

This is tension (compression) inmtimes graphics functiony = f(x) along the x-axis

If, |m|> 1, then compression occurs

If, |m|< 1, то происходит растяжение

5)View conversiony = | f(x)|

This is the bottom display

function graphicsy = f(x) to the top

half-plane relative to the x-axis

keeping the top of the graph

6)View conversiony = f(| x|)

This is a display of the right side of the graph of the functiony = f(x) to the left half-plane relative to the ordinate axis, preserving the right side of the graph

Plotting function graphs.

1) Graph the functiony=+

y=

x=-1 andx=1 – break points

2) Graph the functiony=

y=

3) Graph the functiony=,

Domain:x≠0

y=

4) Graph the functiony=

Domain:x≠1

1)≥0 2) <0

≥1 <1

≥1 -1< x<1

x≤-1 , x≥1 y==- x-1

becausex≠1, thenx≤-1, x>1

y== x+1

5) Graph the functiony=

Domain:x≠1

=1.5±0.5

=2, =1

1) x-1>0, x>1

y===x-2

y=x-2, x>1

2) x-1<0, x<1

y==-x+2

y=-x+2, x<1

6) Graph the function y=+

y=+=+

x<-2, y=-x+1-x-2=-2x-1

-2≤x≤1, y=-x+1+x+2=3

x≥1, y=x-1+x+2=2x=1

7) Graph the function y=

Definition range: -1≠0,x≠±1

1) x-1>0, x>1, y==

2) x-1<0, x<1, y==

y= (x) =

y=(x) =(x-1) =

y=(x) =(-x)== -

8) Graph the function y=

Domain:x≠0

The function is odd, then the branches of the graph are symmetrical about the origin.

Application of function graphs to problem solving.

1) At what parameter valueskthe equation
= khas two roots?

Solution.

1. k≥0

2. Let's build a graphy=

a) Function definition area:x≠±1.

b)

c) Since the function is even, the hyperbola is symmetrical about the axisOy.

3. Since the equation has 2 roots, the straight liney= kmust intersect the graph at two points. Therefore 1< k<2. Заметим, что при k=2 will be three roots.

2) At what parameter valueskthe equation

= khas 4 roots?

Solution.

The right side of the equation can only be non-negative, that isk≥0.

Let's plot the function

y=

a)

b)

Answer:

and ifk=0, then the equation has 4 roots(-4;-2;2;4)

b) If 1< k<8, то уравнение имеет 4 корня(-5,5;-0,5;0,5;5,5)

3)Solve inequality

x-1 <

Solution.

1. Find the abscissa of the intersection points of the function graphs

y= x-1 andy=.

2. Let's solve the equations:

A)x-1=-5 x+4 B)x-1= -(-5 x+4)

x-1= -+5 x-4

=5, =1 =0

=3, =1.

3. Let's build graphs of functions

y= x-1 andy=

y=

= - = 2,5

=

(2.5;-2.25) is the vertex of the parabola.

y=0: -5 x+4=0

=2.5±1.5

=4; =1

y(0)=4

Answer:x<1,1< x<3, x>5.

4) Solve equation 1-=.

Solution.

Let us depict the graphs of functions in one coordinate systemy=1- andy=

The graphs intersected at the point (-1;2). Therefore, the root of this equationx=-1.

Answer:x=-1

Graph the Function

and determine at what values ​​the straight line will intersect the constructed graph at three points.

Solution.

Let's plot the function

From the graph it is clear that the straight line y = c will have exactly three intersection points with the graph at c belonging to the set: (0;5).

Answer: (0; 5).

Plot the function y = and determine at what valueskthe constructed graph will not have common points with the straight line y=kX.

Solution.

Domain: x and x

Let's transform the function to the form: y = . The graph is a straight line y = x-3 without two points (-3; -6) and (9; 6).

Straight line y=kx will not have common points with the constructed line if it is parallel to it, i.e. whenk=1, and if it passes through the punctured points. A straight line passes through the first of these points ifk=2, and through the second - if

k=

Answer: ; 1;2.

Conclusion

After completing this work, I learned how to construct graphs of functions using geometric transformations. This will help me solve different types of problems (inequalities, equations, parameter problems) in a graphical way. In this way, I am preparing to successfully pass the OGE and USE exams.

Bibliography:

Kramor V.S. Examples with parameters and their solutions. A guide for applicants to universities. – M.: ARKTI, 2001

Dorodnov A.M. and others. Graphs of functions. A textbook for applicants to universities. M., "Higher School", 1972

Gelfand I.M., E.G. Glagoleva, E.E. Shnol Functions and their graphs “Science” Moscow 1971

Vilenkin N.Ya. Functions in nature and technology: M.: Education, 1985

FIPI open task bank

Sh.A. Alimov, Yu.M. Kolyagin Algebra. 9th grade: textbook for general education institutions. M.: Education, 2011

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Coordinate system - these are two mutually perpendicular coordinate lines intersecting at a point, which is the origin of reference for each of them.

Coordinate axes – straight lines forming a coordinate system.

Abscissa axis(x-axis) - horizontal axis.

Y axis(y-axis) is the vertical axis.

Function

Function is a mapping of elements of set X to set Y. In this case, each element x of the set X corresponds to one single value y of the set Y.

Straight

Linear function – a function of the form y = a x + b where a and b are any numbers.

The graph of a linear function is a straight line.

Let's look at what the graph will look like depending on the coefficients a and b:

If a > 0, the straight line will pass through the I and III coordinate quarters.

If a< 0 , прямая будет проходить через II и IV координатные четверти.

b is the point of intersection of the line with the y axis.

If a = 0, the function takes the form y = b.

Let us separately highlight the graph of the equation x = a.

Important: this equation is not a function since the definition of the function is violated (the function associates each element x of the set X with one single value y of the set Y). This equation assigns one element x to an infinite set of elements y. However, it is possible to construct a graph of this equation. Let’s just not call it the proud word “Function”.

Parabola

The graph of the function y = a x 2 + b x + c is parabola .

In order to unambiguously determine how the graph of a parabola is located on a plane, you need to know what the coefficients a, b, c influence:

1. The coefficient a indicates where the branches of the parabola are directed.

• If a > 0, the branches of the parabola are directed upward.
• If a< 0 , ветки параболы направлены вниз.

1. The coefficient c indicates at what point the parabola intersects the y-axis.
2. The coefficient b helps to find x in - the coordinate of the vertex of the parabola.

x in = − b 2 a

1. The discriminant allows you to determine how many points of intersection the parabola has with the axis.

• If D > 0 - two points of intersection.
• If D = 0 - one intersection point.
• If D< 0 — нет точек пересечения.

The graph of the function y = k x is hyperbola .

A characteristic feature of a hyperbola is that it has asymptotes.

Asymptotes of a hyperbola - straight lines to which it strives, going into infinity.

The x-axis is the horizontal asymptote of the hyperbola

The y-axis is the vertical asymptote of the hyperbola.

On the graph, asymptotes are marked with a green dotted line.

If the coefficient k > 0, then the branches of the hyperole pass through the I and III quarters.

If k<     0, ветви гиперболы проходят через II и IV четверти.

The smaller the absolute value of the coefficient k (coefficient k without taking into account the sign), the closer the branches of the hyperbola are to the x and y axes.

Square root

The function y = x has the following graph:

Increasing/descending functions

Function y = f(x) increases over the interval , if a larger argument value (larger x value) corresponds to a larger function value (larger y value).

That is, the more (to the right) the X, the larger (higher) the Y. The graph goes up (look from left to right)

Function y = f(x) decreases on the interval , if a larger argument value (a larger x value) corresponds to a smaller function value (a larger y value).

1. Fractional linear function and its graph

A function of the form y = P(x) / Q(x), where P(x) and Q(x) are polynomials, is called a fractional rational function.

You are probably already familiar with the concept of rational numbers. Likewise rational functions are functions that can be represented as the quotient of two polynomials.

If a fractional rational function is the quotient of two linear functions - polynomials of the first degree, i.e. function of the form

y = (ax + b) / (cx + d), then it is called fractional linear.

Note that in the function y = (ax + b) / (cx + d), c ≠ 0 (otherwise the function becomes linear y = ax/d + b/d) and that a/c ≠ b/d (otherwise the function is constant ). The linear fractional function is defined for all real numbers except x = -d/c. Graphs of fractional linear functions do not differ in shape from the graph y = 1/x you know. A curve that is a graph of the function y = 1/x is called hyperbole. With an unlimited increase in x in absolute value, the function y = 1/x decreases unlimited in absolute value and both branches of the graph approach the abscissa: the right one approaches from above, and the left one from below. The lines to which the branches of a hyperbola approach are called its asymptotes.

Example 1.

y = (2x + 1) / (x – 3).

Solution.

Let's select the whole part: (2x + 1) / (x – 3) = 2 + 7/(x – 3).

Now it is easy to see that the graph of this function is obtained from the graph of the function y = 1/x by the following transformations: shift by 3 unit segments to the right, stretching along the Oy axis 7 times and shifting by 2 unit segments upward.

Any fraction y = (ax + b) / (cx + d) can be written in a similar way, highlighting the “integer part”. Consequently, the graphs of all fractional linear functions are hyperbolas, shifted in various ways along the coordinate axes and stretched along the Oy axis.

To construct a graph of any arbitrary fractional-linear function, it is not at all necessary to transform the fraction defining this function. Since we know that the graph is a hyperbola, it will be enough to find the straight lines to which its branches approach - the asymptotes of the hyperbola x = -d/c and y = a/c.

Example 2.

Find the asymptotes of the graph of the function y = (3x + 5)/(2x + 2).

Solution.

The function is not defined, at x = -1. This means that the straight line x = -1 serves as a vertical asymptote. To find the horizontal asymptote, let’s find out what the values ​​of the function y(x) approach when the argument x increases in absolute value.

To do this, divide the numerator and denominator of the fraction by x:

y = (3 + 5/x) / (2 + 2/x).

As x → ∞ the fraction will tend to 3/2. This means that the horizontal asymptote is the straight line y = 3/2.

Example 3.

Graph the function y = (2x + 1)/(x + 1).

Solution.

Let’s select the “whole part” of the fraction:

(2x + 1) / (x + 1) = (2x + 2 – 1) / (x + 1) = 2(x + 1) / (x + 1) – 1/(x + 1) =

2 – 1/(x + 1).

Now it is easy to see that the graph of this function is obtained from the graph of the function y = 1/x by the following transformations: a shift by 1 unit to the left, a symmetrical display with respect to Ox and a shift by 2 unit segments up along the Oy axis.

Domain D(y) = (-∞; -1)ᴗ(-1; +∞).

Range of values ​​E(y) = (-∞; 2)ᴗ(2; +∞).

Intersection points with axes: c Oy: (0; 1); c Ox: (-1/2; 0). The function increases at each interval of the domain of definition.

Answer: Figure 1.

2. Fractional rational function

Consider a fractional rational function of the form y = P(x) / Q(x), where P(x) and Q(x) are polynomials of degree higher than first.

Examples of such rational functions:

y = (x 3 – 5x + 6) / (x 7 – 6) or y = (x – 2) 2 (x + 1) / (x 2 + 3).

If the function y = P(x) / Q(x) represents the quotient of two polynomials of degree higher than the first, then its graph will, as a rule, be more complex, and it can sometimes be difficult to construct it accurately, with all the details. However, it is often enough to use techniques similar to those we have already introduced above.

Let the fraction be a proper fraction (n< m). Известно, что любую несократимую рациональную дробь можно представить, и притом единственным образом, в виде суммы конечного числа элементарных дробей, вид которых определяется разложением знаменателя дроби Q(x) в произведение действительных сомножителей:

P(x)/Q(x) = A 1 /(x – K 1) m1 + A 2 /(x – K 1) m1-1 + … + A m1 /(x – K 1) + …+

L 1 /(x – K s) ms + L 2 /(x – K s) ms-1 + … + L ms /(x – K s) + …+

+ (B 1 x + C 1) / (x 2 +p 1 x + q 1) m1 + … + (B m1 x + C m1) / (x 2 +p 1 x + q 1) + …+

+ (M 1 x + N 1) / (x 2 +p t x + q t) m1 + … + (M m1 x + N m1) / (x 2 +p t x + q t).

Obviously, the graph of a fractional rational function can be obtained as the sum of graphs of elementary fractions.

Plotting graphs of fractional rational functions

Let's consider several ways to construct graphs of a fractional rational function.

Example 4.

Draw a graph of the function y = 1/x 2 .

Solution.

We use the graph of the function y = x 2 to construct a graph of y = 1/x 2 and use the technique of “dividing” the graphs.

Domain D(y) = (-∞; 0)ᴗ(0; +∞).

Range of values ​​E(y) = (0; +∞).

There are no points of intersection with the axes. The function is even. Increases for all x from the interval (-∞; 0), decreases for x from 0 to +∞.

Answer: Figure 2.

Example 5.

Graph the function y = (x 2 – 4x + 3) / (9 – 3x).

Solution.

Domain D(y) = (-∞; 3)ᴗ(3; +∞).

y = (x 2 – 4x + 3) / (9 – 3x) = (x – 3)(x – 1) / (-3(x – 3)) = -(x – 1)/3 = -x/ 3 + 1/3.

Here we used the technique of factorization, reduction and reduction to a linear function.

Answer: Figure 3.

Example 6.

Graph the function y = (x 2 – 1)/(x 2 + 1).

Solution.

The domain of definition is D(y) = R. Since the function is even, the graph is symmetrical about the ordinate. Before building a graph, let’s transform the expression again, highlighting the whole part:

y = (x 2 – 1)/(x 2 + 1) = 1 – 2/(x 2 + 1).

Note that isolating the integer part in the formula of a fractional rational function is one of the main ones when constructing graphs.

If x → ±∞, then y → 1, i.e. the straight line y = 1 is a horizontal asymptote.

Answer: Figure 4.

Example 7.

Let's consider the function y = x/(x 2 + 1) and try to accurately find its largest value, i.e. the highest point on the right half of the graph. To accurately construct this graph, today's knowledge is not enough. Obviously, our curve cannot “rise” very high, because the denominator quickly begins to “overtake” the numerator. Let's see if the value of the function can be equal to 1. To do this, we need to solve the equation x 2 + 1 = x, x 2 – x + 1 = 0. This equation has no real roots. This means our assumption is incorrect. To find the largest value of the function, you need to find out at what largest A the equation A = x/(x 2 + 1) will have a solution. Let's replace the original equation with a quadratic one: Ax 2 – x + A = 0. This equation has a solution when 1 – 4A 2 ≥ 0. From here we find the largest value A = 1/2.

Answer: Figure 5, max y(x) = ½.

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Basic elementary functions, their inherent properties and corresponding graphs are one of the basics of mathematical knowledge, similar in importance to the multiplication table. Elementary functions are the basis, the support for the study of all theoretical issues.

Yandex.RTB R-A-339285-1

The article below provides key material on the topic of basic elementary functions. We will introduce terms, give them definitions; Let's study each type of elementary functions in detail and analyze their properties.

The following types of basic elementary functions are distinguished:

Definition 1

• constant function (constant);
• nth root;
• power function;
• exponential function;
• logarithmic function;
• trigonometric functions;
• fraternal trigonometric functions.

A constant function is defined by the formula: y = C (C is a certain real number) and also has a name: constant. This function determines the correspondence of any real value of the independent variable x to the same value of the variable y - the value of C.

The graph of a constant is a straight line that is parallel to the abscissa axis and passes through a point having coordinates (0, C). For clarity, we present graphs of constant functions y = 5, y = - 2, y = 3, y = 3 (indicated in black, red and blue colors in the drawing, respectively).

Definition 2

This elementary function is defined by the formula y = x n (n is a natural number greater than one).

Let's consider two variations of the function.

1. nth root, n – even number

For clarity, we indicate a drawing that shows graphs of such functions: y = x, y = x 4 and y = x8. These features are color coded: black, red and blue respectively.

The graphs of a function of even degree have a similar appearance for other values ​​of the exponent.

Definition 3

Properties of the nth root function, n is an even number

• domain of definition – the set of all non-negative real numbers [ 0 , + ∞) ;
• when x = 0, function y = x n has a value equal to zero;
• this function is a function of general form (it is neither even nor odd);
• range: [ 0 , + ∞) ;
• this function y = x n with even root exponents increases throughout the entire domain of definition;
• the function has a convexity with an upward direction throughout the entire domain of definition;
• there are no inflection points;
• there are no asymptotes;
• the graph of the function for even n passes through the points (0; 0) and (1; 1).

1. nth root, n – odd number

Such a function is defined on the entire set of real numbers. For clarity, consider the graphs of the functions y = x 3 , y = x 5 and x 9 . In the drawing they are indicated by colors: black, red and blue are the colors of the curves, respectively.

Other odd values ​​of the root exponent of the function y = x n will give a graph of a similar type.

Definition 4

Properties of the nth root function, n is an odd number

• domain of definition – the set of all real numbers;
• this function is odd;
• range of values ​​– the set of all real numbers;
• the function y = x n for odd root exponents increases over the entire domain of definition;
• the function has concavity on the interval (- ∞ ; 0 ] and convexity on the interval [ 0 , + ∞);
• the inflection point has coordinates (0; 0);
• there are no asymptotes;
• The graph of the function for odd n passes through the points (- 1 ; - 1), (0 ; 0) and (1 ; 1).

Power function

Definition 5

The power function is defined by the formula y = x a.

The appearance of the graphs and the properties of the function depend on the value of the exponent.

• when a power function has an integer exponent a, then the type of graph of the power function and its properties depend on whether the exponent is even or odd, as well as what sign the exponent has. Let's consider all these special cases in more detail below;
• the exponent can be fractional or irrational - depending on this, the type of graphs and properties of the function also vary. We will analyze special cases by setting several conditions: 0< a < 1 ; a > 1 ; - 1 < a < 0 и a < - 1 ;
• a power function can have a zero exponent; we will also analyze this case in more detail below.

Let's analyze the power function y = x a, when a is an odd positive number, for example, a = 1, 3, 5...

For clarity, we indicate the graphs of such power functions: y = x (graphic color black), y = x 3 (blue color of the graph), y = x 5 (red color of the graph), y = x 7 (graphic color green). When a = 1, we get the linear function y = x.

Definition 6

Properties of a power function when the exponent is odd positive

• the function is increasing for x ∈ (- ∞ ; + ∞) ;
• the function has convexity for x ∈ (- ∞ ; 0 ] and concavity for x ∈ [ 0 ; + ∞) (excluding the linear function);
• the inflection point has coordinates (0 ; 0) (excluding linear function);
• there are no asymptotes;
• points of passage of the function: (- 1 ; - 1) , (0 ; 0) , (1 ; 1) .

Let's analyze the power function y = x a, when a is an even positive number, for example, a = 2, 4, 6...

For clarity, we indicate the graphs of such power functions: y = x 2 (graphic color black), y = x 4 (blue color of the graph), y = x 8 (red color of the graph). When a = 2, we obtain a quadratic function, the graph of which is a quadratic parabola.

Definition 7

Properties of a power function when the exponent is even positive:

• domain of definition: x ∈ (- ∞ ; + ∞) ;
• decreasing for x ∈ (- ∞ ; 0 ] ;
• the function has concavity for x ∈ (- ∞ ; + ∞) ;
• there are no inflection points;
• there are no asymptotes;
• points of passage of the function: (- 1 ; 1) , (0 ; 0) , (1 ; 1) .

The figure below shows examples of power function graphs y = x a when a is an odd negative number: y = x - 9 (graphic color black); y = x - 5 (blue color of the graph); y = x - 3 (red color of the graph); y = x - 1 (graphic color green). When a = - 1, we obtain inverse proportionality, the graph of which is a hyperbola.

Definition 8

Properties of a power function when the exponent is odd negative:

When x = 0, we obtain a discontinuity of the second kind, since lim x → 0 - 0 x a = - ∞, lim x → 0 + 0 x a = + ∞ for a = - 1, - 3, - 5, …. Thus, the straight line x = 0 is a vertical asymptote;

• range: y ∈ (- ∞ ; 0) ∪ (0 ; + ∞) ;
• the function is odd because y (- x) = - y (x);
• the function is decreasing for x ∈ - ∞ ; 0 ∪ (0 ; + ∞) ;
• the function has convexity for x ∈ (- ∞ ; 0) and concavity for x ∈ (0 ; + ∞) ;
• there are no inflection points;

k = lim x → ∞ x a x = 0, b = lim x → ∞ (x a - k x) = 0 ⇒ y = k x + b = 0, when a = - 1, - 3, - 5, . . . .

• points of passage of the function: (- 1 ; - 1) , (1 ; 1) .

The figure below shows examples of graphs of the power function y = x a when a is an even negative number: y = x - 8 (graphic color black); y = x - 4 (blue color of the graph); y = x - 2 (red color of the graph).

Definition 9

Properties of a power function when the exponent is even negative:

• domain of definition: x ∈ (- ∞ ; 0) ∪ (0 ; + ∞) ;

When x = 0, we obtain a discontinuity of the second kind, since lim x → 0 - 0 x a = + ∞, lim x → 0 + 0 x a = + ∞ for a = - 2, - 4, - 6, …. Thus, the straight line x = 0 is a vertical asymptote;

• the function is even because y(-x) = y(x);
• the function is increasing for x ∈ (- ∞ ; 0) and decreasing for x ∈ 0; + ∞ ;
• the function has concavity at x ∈ (- ∞ ; 0) ∪ (0 ; + ∞) ;
• there are no inflection points;
• horizontal asymptote – straight line y = 0, because:

k = lim x → ∞ x a x = 0 , b = lim x → ∞ (x a - k x) = 0 ⇒ y = k x + b = 0 when a = - 2 , - 4 , - 6 , . . . .

• points of passage of the function: (- 1 ; 1) , (1 ; 1) .

From the very beginning, pay attention to the following aspect: in the case when a is a positive fraction with an odd denominator, some authors take the interval - ∞ as the domain of definition of this power function; + ∞ , stipulating that the exponent a is an irreducible fraction. At the moment, the authors of many educational publications on algebra and principles of analysis DO NOT DEFINE power functions, where the exponent is a fraction with an odd denominator for negative values ​​of the argument. Further we will adhere to exactly this position: we will take the set [ 0 ; + ∞) . Recommendation for students: find out the teacher’s view on this point in order to avoid disagreements.

So, let's look at the power function y = x a , when the exponent is a rational or irrational number, provided that 0< a < 1 .

Let us illustrate the power functions with graphs y = x a when a = 11 12 (graphic color black); a = 5 7 (red color of the graph); a = 1 3 (blue color of the graph); a = 2 5 (green color of the graph).

Other values ​​of the exponent a (provided 0< a < 1) дадут аналогичный вид графика.

Definition 10

Properties of the power function at 0< a < 1:

• range: y ∈ [ 0 ; + ∞) ;
• the function is increasing for x ∈ [ 0 ; + ∞) ;
• the function is convex for x ∈ (0 ; + ∞);
• there are no inflection points;
• there are no asymptotes;

Let's analyze the power function y = x a, when the exponent is a non-integer rational or irrational number, provided that a > 1.

Let us illustrate with graphs the power function y = x a under given conditions using the following functions as an example: y = x 5 4 , y = x 4 3 , y = x 7 3 , y = x 3 π (black, red, blue, green color of the graphs, respectively).

Other values ​​of the exponent a, provided a > 1, will give a similar graph.

Definition 11

Properties of the power function for a > 1:

• domain of definition: x ∈ [ 0 ; + ∞) ;
• range: y ∈ [ 0 ; + ∞) ;
• this function is a function of general form (it is neither odd nor even);
• the function is increasing for x ∈ [ 0 ; + ∞) ;
• the function has concavity for x ∈ (0 ; + ∞) (when 1< a < 2) и выпуклость при x ∈ [ 0 ; + ∞) (когда a > 2);
• there are no inflection points;
• there are no asymptotes;
• passing points of the function: (0 ; 0) , (1 ; 1) .

Please note! When a is a negative fraction with an odd denominator, in the works of some authors there is an opinion that the domain of definition in this case is the interval - ∞; 0 ∪ (0 ; + ∞) with the caveat that the exponent a is an irreducible fraction. At the moment, the authors of educational materials on algebra and principles of analysis DO NOT DEFINE power functions with an exponent in the form of a fraction with an odd denominator for negative values ​​of the argument. Further, we adhere to exactly this view: we take the set (0 ; + ∞) as the domain of definition of power functions with fractional negative exponents. Recommendation for students: Clarify your teacher's vision at this point to avoid disagreements.

Let's continue the topic and analyze the power function y = x a provided: - 1< a < 0 .

Let us present a drawing of graphs of the following functions: y = x - 5 6, y = x - 2 3, y = x - 1 2 2, y = x - 1 7 (black, red, blue, green color of the lines, respectively).

Definition 12

Properties of the power function at - 1< a < 0:

lim x → 0 + 0 x a = + ∞ when - 1< a < 0 , т.е. х = 0 – вертикальная асимптота;

• range: y ∈ 0 ; + ∞ ;
• this function is a function of general form (it is neither odd nor even);
• there are no inflection points;

The drawing below shows graphs of power functions y = x - 5 4, y = x - 5 3, y = x - 6, y = x - 24 7 (black, red, blue, green colors of the curves, respectively).

Definition 13

Properties of the power function for a< - 1:

• domain of definition: x ∈ 0 ; + ∞ ;

lim x → 0 + 0 x a = + ∞ when a< - 1 , т.е. х = 0 – вертикальная асимптота;

• range: y ∈ (0 ; + ∞) ;
• this function is a function of general form (it is neither odd nor even);
• the function is decreasing for x ∈ 0; + ∞ ;
• the function has a concavity for x ∈ 0; + ∞ ;
• there are no inflection points;
• horizontal asymptote – straight line y = 0;
• point of passage of the function: (1; 1) .

When a = 0 and x ≠ 0, we obtain the function y = x 0 = 1, which defines the line from which the point (0; 1) is excluded (it was agreed that the expression 0 0 will not be given any meaning).

The exponential function has the form y = a x, where a > 0 and a ≠ 1, and the graph of this function looks different based on the value of the base a. Let's consider special cases.

First, let's look at the situation when the base of the exponential function has a value from zero to one (0< a < 1) . A good example is the graphs of functions for a = 1 2 (blue color of the curve) and a = 5 6 (red color of the curve).

The graphs of the exponential function will have a similar appearance for other values ​​of the base under the condition 0< a < 1 .

Definition 14

Properties of the exponential function when the base is less than one:

• range: y ∈ (0 ; + ∞) ;
• this function is a function of general form (it is neither odd nor even);
• an exponential function whose base is less than one is decreasing over the entire domain of definition;
• there are no inflection points;
• horizontal asymptote – straight line y = 0 with variable x tending to + ∞;

Now consider the case when the base of the exponential function is greater than one (a > 1).

Let us illustrate this special case with a graph of exponential functions y = 3 2 x (blue color of the curve) and y = e x (red color of the graph).

Other values ​​of the base, larger units, will give a similar appearance to the graph of the exponential function.

Definition 15

Properties of the exponential function when the base is greater than one:

• domain of definition – the entire set of real numbers;
• range: y ∈ (0 ; + ∞) ;
• this function is a function of general form (it is neither odd nor even);
• an exponential function whose base is greater than one is increasing as x ∈ - ∞; + ∞ ;
• the function has a concavity at x ∈ - ∞; + ∞ ;
• there are no inflection points;
• horizontal asymptote – straight line y = 0 with variable x tending to - ∞;
• point of passage of the function: (0; 1) .

The logarithmic function has the form y = log a (x), where a > 0, a ≠ 1.

Such a function is defined only for positive values ​​of the argument: for x ∈ 0; + ∞ .

The graph of a logarithmic function has a different appearance, based on the value of the base a.

Let us first consider the situation when 0< a < 1 . Продемонстрируем этот частный случай графиком логарифмической функции при a = 1 2 (синий цвет кривой) и а = 5 6 (красный цвет кривой).

Other values ​​of the base, not larger units, will give a similar type of graph.

Definition 16

Properties of a logarithmic function when the base is less than one:

• domain of definition: x ∈ 0 ; + ∞ . As x tends to zero from the right, the function values ​​tend to +∞;
• range of values: y ∈ - ∞ ; + ∞ ;
• this function is a function of general form (it is neither odd nor even);
• logarithmic
• the function has a concavity for x ∈ 0; + ∞ ;
• there are no inflection points;
• there are no asymptotes;

Now let's look at the special case when the base of the logarithmic function is greater than one: a > 1 . The drawing below shows graphs of logarithmic functions y = log 3 2 x and y = ln x (blue and red colors of the graphs, respectively).

Other values ​​of the base greater than one will give a similar type of graph.

Definition 17

Properties of a logarithmic function when the base is greater than one:

• domain of definition: x ∈ 0 ; + ∞ . As x tends to zero from the right, the function values ​​tend to - ∞ ;
• range of values: y ∈ - ∞ ; + ∞ (the entire set of real numbers);
• this function is a function of general form (it is neither odd nor even);
• the logarithmic function is increasing for x ∈ 0; + ∞ ;
• the function is convex for x ∈ 0; + ∞ ;
• there are no inflection points;
• there are no asymptotes;
• point of passage of the function: (1; 0) .

The trigonometric functions are sine, cosine, tangent and cotangent. Let's look at the properties of each of them and the corresponding graphics.

In general, all trigonometric functions are characterized by the property of periodicity, i.e. when the values ​​of the functions are repeated for different values ​​of the argument, differing from each other by the period f (x + T) = f (x) (T is the period). Thus, the item “smallest positive period” is added to the list of properties of trigonometric functions. In addition, we will indicate the values ​​of the argument at which the corresponding function becomes zero.

1. Sine function: y = sin(x)

The graph of this function is called a sine wave.

Definition 18

Properties of the sine function:

• domain of definition: the entire set of real numbers x ∈ - ∞ ; + ∞ ;
• the function vanishes when x = π · k, where k ∈ Z (Z is the set of integers);
• the function is increasing for x ∈ - π 2 + 2 π · k ; π 2 + 2 π · k, k ∈ Z and decreasing for x ∈ π 2 + 2 π · k; 3 π 2 + 2 π · k, k ∈ Z;
• the sine function has local maxima at points π 2 + 2 π · k; 1 and local minima at points - π 2 + 2 π · k; - 1, k ∈ Z;
• the sine function is concave when x ∈ - π + 2 π · k ; 2 π · k, k ∈ Z and convex when x ∈ 2 π · k; π + 2 π k, k ∈ Z;
• there are no asymptotes.

1. Cosine function: y = cos(x)

The graph of this function is called a cosine wave.

Definition 19

Properties of the cosine function:

• domain of definition: x ∈ - ∞ ; + ∞ ;
• smallest positive period: T = 2 π;
• range of values: y ∈ - 1 ; 1 ;
• this function is even, since y (- x) = y (x);
• the function is increasing for x ∈ - π + 2 π · k ; 2 π · k, k ∈ Z and decreasing for x ∈ 2 π · k; π + 2 π k, k ∈ Z;
• the cosine function has local maxima at points 2 π · k ; 1, k ∈ Z and local minima at points π + 2 π · k; - 1, k ∈ z;
• the cosine function is concave when x ∈ π 2 + 2 π · k ; 3 π 2 + 2 π · k , k ∈ Z and convex when x ∈ - π 2 + 2 π · k ; π 2 + 2 π · k, k ∈ Z;
• inflection points have coordinates π 2 + π · k; 0 , k ∈ Z
• there are no asymptotes.

1. Tangent function: y = t g (x)

The graph of this function is called tangent.

Definition 20

Properties of the tangent function:

• domain of definition: x ∈ - π 2 + π · k ; π 2 + π · k, where k ∈ Z (Z is the set of integers);
• Behavior of the tangent function on the boundary of the domain of definition lim x → π 2 + π · k + 0 t g (x) = - ∞ , lim x → π 2 + π · k - 0 t g (x) = + ∞ . Thus, the straight lines x = π 2 + π · k k ∈ Z are vertical asymptotes;
• the function vanishes when x = π · k for k ∈ Z (Z is the set of integers);
• range of values: y ∈ - ∞ ; + ∞ ;
• this function is odd, since y (- x) = - y (x) ;
• the function is increasing as - π 2 + π · k ; π 2 + π · k, k ∈ Z;
• the tangent function is concave for x ∈ [π · k; π 2 + π · k) , k ∈ Z and convex for x ∈ (- π 2 + π · k ; π · k ] , k ∈ Z ;
• inflection points have coordinates π · k ; 0 , k ∈ Z ;

1. Cotangent function: y = c t g (x)

The graph of this function is called a cotangentoid. .

Definition 21

Properties of the cotangent function:

• domain of definition: x ∈ (π · k ; π + π · k) , where k ∈ Z (Z is the set of integers);

Behavior of the cotangent function on the boundary of the domain of definition lim x → π · k + 0 t g (x) = + ∞ , lim x → π · k - 0 t g (x) = - ∞ . Thus, the straight lines x = π · k k ∈ Z are vertical asymptotes;

• smallest positive period: T = π;
• the function vanishes when x = π 2 + π · k for k ∈ Z (Z is the set of integers);
• range of values: y ∈ - ∞ ; + ∞ ;
• this function is odd, since y (- x) = - y (x) ;
• the function is decreasing for x ∈ π · k ; π + π k, k ∈ Z;
• the cotangent function is concave for x ∈ (π · k; π 2 + π · k ], k ∈ Z and convex for x ∈ [ - π 2 + π · k ; π · k), k ∈ Z ;
• inflection points have coordinates π 2 + π · k; 0 , k ∈ Z ;
• There are no oblique or horizontal asymptotes.

The inverse trigonometric functions are arcsine, arccosine, arctangent and arccotangent. Often, due to the presence of the prefix “arc” in the name, inverse trigonometric functions are called arc functions .

1. Arc sine function: y = a r c sin (x)

Definition 22

Properties of the arcsine function:

• this function is odd, since y (- x) = - y (x) ;
• the arcsine function has a concavity for x ∈ 0; 1 and convexity for x ∈ - 1 ; 0 ;
• inflection points have coordinates (0; 0), which is also the zero of the function;
• there are no asymptotes.

1. Arc cosine function: y = a r c cos (x)

Definition 23

Properties of the arc cosine function:

• domain of definition: x ∈ - 1 ; 1 ;
• range: y ∈ 0 ; π;
• this function is of a general form (neither even nor odd);
• the function is decreasing over the entire domain of definition;
• the arc cosine function has a concavity at x ∈ - 1; 0 and convexity for x ∈ 0; 1 ;
• inflection points have coordinates 0; π 2;
• there are no asymptotes.

1. Arc tangent function: y = a r c t g (x)

Definition 24

Properties of the arctangent function:

• domain of definition: x ∈ - ∞ ; + ∞ ;
• range of values: y ∈ - π 2 ; π 2;
• this function is odd, since y (- x) = - y (x) ;
• the function is increasing over the entire domain of definition;
• the arctangent function has concavity for x ∈ (- ∞ ; 0 ] and convexity for x ∈ [ 0 ; + ∞);
• the inflection point has coordinates (0; 0), which is also the zero of the function;
• horizontal asymptotes are straight lines y = - π 2 as x → - ∞ and y = π 2 as x → + ∞ (in the figure, the asymptotes are green lines).

1. Arc tangent function: y = a r c c t g (x)

Definition 25

Properties of the arccotangent function:

• domain of definition: x ∈ - ∞ ; + ∞ ;
• range: y ∈ (0; π) ;
• this function is of a general form;
• the function is decreasing over the entire domain of definition;
• the arc cotangent function has a concavity for x ∈ [ 0 ; + ∞) and convexity for x ∈ (- ∞ ; 0 ] ;
• the inflection point has coordinates 0; π 2;
• horizontal asymptotes are straight lines y = π at x → - ∞ (green line in the drawing) and y = 0 at x → + ∞.

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