The length of the segment on the coordinate axis is found by the formula:

The length of the segment on the coordinate plane is sought by the formula:

To find the length of a segment in a three-dimensional coordinate system, the following formula is used:

The coordinates of the middle of the segment (for the coordinate axis only the first formula is used, for the coordinate plane - the first two formulas, for the three-dimensional coordinate system - all three formulas) are calculated by the formulas:

Function is a correspondence of the form y= f(x) between variables, due to which each considered value of some variable x(argument or independent variable) corresponds to a certain value of another variable, y(dependent variable, sometimes this value is simply called the value of the function). Note that the function assumes that one value of the argument X there can only be one value of the dependent variable at. However, the same value at can be obtained with various X.

Function scope are all values ​​of the independent variable (function argument, usually X) for which the function is defined, i.e. its meaning exists. The domain of definition is indicated D(y). By and large, you are already familiar with this concept. The scope of a function is otherwise called the domain of valid values, or ODZ, which you have been able to find for a long time.

Function range are all possible values ​​of the dependent variable of this function. Denoted E(at).

Function rises on the interval on which the larger value of the argument corresponds to the larger value of the function. Function Decreasing on the interval on which the larger value of the argument corresponds to the smaller value of the function.

Function intervals are the intervals of the independent variable at which the dependent variable retains its positive or negative sign.

Function zeros are those values ​​of the argument for which the value of the function is equal to zero. At these points, the graph of the function intersects the abscissa axis (OX axis). Very often, the need to find the zeros of a function means simply solving the equation. Also, often the need to find intervals of constant sign means the need to simply solve the inequality.

Function y = f(x) are called even X

This means that for any opposite values ​​of the argument, the values even function are equal. The graph of an even function is always symmetrical about the y-axis of the op-amp.

Function y = f(x) are called odd, if it is defined on a symmetric set and for any X from the domain of definition the equality is fulfilled:

This means that for any opposite values ​​of the argument, the values odd function are also opposite. The graph of an odd function is always symmetrical about the origin.

The sum of the roots of even and odd functions (points of intersection of the abscissa axis OX) is always equal to zero, because for every positive root X account for negative root –X.

It is important to note that some function does not have to be even or odd. There are many functions that are neither even nor odd. Such functions are called functions general view , and none of the above equalities or properties hold for them.

Linear function is called a function that can be given by the formula:

The graph of a linear function is a straight line and in the general case looks like this (an example is given for the case when k> 0, in this case the function is increasing; for the case k < 0 функция будет убывающей, т.е. прямая будет наклонена в другую сторону - слева направо):

Graph of Quadratic Function (Parabola)

The graph of a parabola is given by a quadratic function:

A quadratic function, like any other function, intersects the OX axis at the points that are its roots: ( x 1 ; 0) and ( x 2; 0). If there are no roots, then the quadratic function does not intersect the OX axis, if there is one root, then at this point ( x 0; 0) the quadratic function only touches the OX axis, but does not intersect it. A quadratic function always intersects the OY axis at a point with coordinates: (0; c). Schedule quadratic function(parabola) may look like this (the figure shows examples that do not exhaust all possible types of parabolas):

Wherein:

• if the coefficient a> 0, in the function y = ax 2 + bx + c, then the branches of the parabola are directed upwards;
• if a < 0, то ветви параболы направлены вниз.

Parabola vertex coordinates can be calculated using the following formulas. X tops (p- in the figures above) of a parabola (or the point at which the square trinomial reaches its maximum or minimum value):

Y tops (q- in the figures above) of a parabola or the maximum if the branches of the parabola are directed downwards ( a < 0), либо минимальное, если ветви параболы направлены вверх (a> 0), value square trinomial:

Graphs of other functions

power function

Here are some examples of graphs of power functions:

Inversely proportional dependence call the function given by the formula:

Depending on the sign of the number k An inversely proportional graph can have two fundamental options:

Asymptote is the line to which the line of the graph of the function approaches infinitely close, but does not intersect. Asymptotes for Graphs inverse proportionality shown in the figure above are the coordinate axes to which the graph of the function approaches infinitely close, but does not intersect them.

exponential function with base A call the function given by the formula:

a the graph of an exponential function can have two fundamental options (we will also give examples, see below):

logarithmic function call the function given by the formula:

Depending on whether the number is greater or less than one a The graph of a logarithmic function can have two fundamental options:

Function Graph y = |x| as follows:

Graphs of periodic (trigonometric) functions

Function at = f(x) is called periodical, if there exists such a non-zero number T, What f(x + T) = f(x), for anyone X out of function scope f(x). If the function f(x) is periodic with period T, then the function:

Where: A, k, b are constant numbers, and k not equal to zero, also periodic with a period T 1 , which is determined by the formula:

Most examples of periodic functions are trigonometric functions. Here are the graphs of the main trigonometric functions. The following figure shows part of the graph of the function y= sin x(the whole graph continues indefinitely to the left and right), the graph of the function y= sin x called sinusoid:

Function Graph y= cos x called cosine wave. This graph is shown in the following figure. Since the graph of the sine, it continues indefinitely along the OX axis to the left and to the right:

Function Graph y=tg x called tangentoid. This graph is shown in the following figure. Like the graphs of other periodic functions, this graph repeats indefinitely along the OX axis to the left and to the right.

And finally, the graph of the function y=ctg x called cotangentoid. This graph is shown in the following figure. Like the graphs of other periodic and trigonometric functions, this graph repeats indefinitely along the OX axis to the left and to the right.

Learn all formulas and laws in physics, and formulas and methods in mathematics. In fact, it is also very simple to do this, there are only about 200 necessary formulas in physics, and even a little less in mathematics. In each of these subjects there are about a dozen standard methods for solving problems of a basic level of complexity, which can also be learned, and thus, completely automatically and without difficulty, solve most of the digital transformation at the right time. After that, you will only have to think about the most difficult tasks.

Attend all three stages of rehearsal testing in physics and mathematics. Each RT can be visited twice to solve both options. Again, on the CT, in addition to the ability to quickly and efficiently solve problems, and the knowledge of formulas and methods, it is also necessary to be able to properly plan time, distribute forces, and most importantly fill out the answer form correctly, without confusing either the numbers of answers and tasks, or your own name. Also, during the RT, it is important to get used to the style of posing questions in tasks, which may seem very unusual to an unprepared person on the DT.

Successful, diligent and responsible implementation of these three points will allow you to show an excellent result on the CT, the maximum of what you are capable of.

Found an error?

If you think you have found an error in training materials, then write, please, about it by mail. You can also report a bug in social network(). In the letter, indicate the subject (physics or mathematics), the name or number of the topic or test, the number of the task, or the place in the text (page) where, in your opinion, there is an error. Also describe what the alleged error is. Your letter will not go unnoticed, the error will either be corrected, or you will be explained why it is not a mistake.

1. Linear fractional 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. Similarly rational functions are functions that can be represented as a quotient of two polynomials.

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

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 a constant ). The linear-fractional function is defined for all real numbers, except for x = -d/c. Graphs of linear-fractional functions do not differ in form from the graph you know y = 1/x. The curve that is the 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 indefinitely in absolute value and both branches of the graph approach the abscissa axis: the right one approaches from above, and the left one approaches from below. The lines approached by the branches of a hyperbola are called its asymptotes.

Example 1

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

Solution.

Let's select the integer 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, stretch along the Oy axis by 7 times and shift by 2 unit segments up.

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

To plot a graph of some arbitrary linear-fractional function, it is not at all necessary to transform the fraction that defines this function. Since we know that the graph is a hyperbola, it will be enough to find the lines to which its branches approach - the hyperbola asymptotes 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, when x = -1. Hence, the 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, we divide the numerator and denominator of the fraction by x:

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

As x → ∞ the fraction tends to 3/2. Hence, the horizontal asymptote is the straight line y = 3/2.

Example 3

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

Solution.

We 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 of 1 unit to the left, a symmetric display with respect to Ox, and a shift of 2 unit intervals up along the Oy axis.

Domain of definition 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 on each of the intervals 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 the first.

Examples of such rational functions:

y \u003d (x 3 - 5x + 6) / (x 7 - 6) or y \u003d (x - 2) 2 (x + 1) / (x 2 + 3).

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

Let the fraction be proper (n< m). Известно, что любую несократимую rational fraction can be represented, and moreover in a unique way, as a sum of a finite number of elementary fractions, the form of which is determined by expanding the denominator of the fraction Q(x) into a product of real factors:

P(x) / Q(x) \u003d 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 fractional rational functions

Consider several ways to plot a fractional-rational function.

Example 4

Plot the function y = 1/x 2 .

Solution.

We use the graph of the function y \u003d x 2 to plot the graph y \u003d 1 / x 2 and use the method 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

Plot the function y = (x 2 - 4x + 3) / (9 - 3x).

Solution.

Domain D(y) = (-∞; 3)ᴗ(3; +∞).

y \u003d (x 2 - 4x + 3) / (9 - 3x) \u003d (x - 3) (x - 1) / (-3 (x - 3)) \u003d - (x - 1) / 3 \u003d -x / 3 + 1/3.

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

Answer: figure 3.

Example 6

Plot the function y \u003d (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 y-axis. Before plotting, we again transform the expression by highlighting the integer part:

y \u003d (x 2 - 1) / (x 2 + 1) \u003d 1 - 2 / (x 2 + 1).

Note that the selection of the integer part in the formula of a fractional-rational function is one of the main ones when plotting graphs.

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

Answer: figure 4.

Example 7

Consider the function y = x/(x 2 + 1) and try to find exactly its largest value, i.e. most high point right half of the graph. To accurately build this graph, today's knowledge is not enough. It is obvious that our curve cannot "climb" very high, since 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, you need to solve the equation x 2 + 1 \u003d x, x 2 - x + 1 \u003d 0. This equation has no real roots. So our assumption is wrong. To find the most great importance function, you need to find out for which largest A the equation A \u003d 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 highest value A = 1/2.

Answer: Figure 5, max y(x) = ½.

Do you have any questions? Don't know how to build function graphs?
To get the help of a tutor - register.
The first lesson is free!

site, with full or partial copying of the material, a link to the source is required.

Definition: A numerical function is a correspondence that maps to each number x from some given set singular y.

Designation:

where x is an independent variable (argument), y is a dependent variable (function). The set of values ​​x is called the domain of the function (denoted D(f)). The set of values ​​y is called the range of the function (denoted by E(f)). The graph of a function is the set of points in the plane with coordinates (x, f(x))

Ways to set a function.

1. analytical method (using a mathematical formula);
2. tabular method (using a table);
3. descriptive method (using a verbal description);
4. graphical method (using a graph).

Basic properties of the function.

1. Even and odd

A function is called even if
– the domain of definition of the function is symmetric with respect to zero
f(-x) = f(x)

The graph of an even function is symmetrical about the axis 0y

A function is called odd if
– the domain of definition of the function is symmetric with respect to zero
– for any x from the domain of definition f(-x) = -f(x)

The graph of an odd function is symmetrical about the origin.

2. Periodicity

The function f(x) is called periodic with a period if for any x from the domain of definition f(x) = f(x+T) = f(x-T) .

The graph of a periodic function consists of infinitely repeating identical fragments.

3. Monotony (increase, decrease)

The function f(x) increases on the set P if for any x 1 and x 2 from this set, such that x 1

The function f(x) is decreasing on the set P if for any x 1 and x 2 from this set, such that x 1 f(x 2) .

4. Extremes

The point X max is called the maximum point of the function f (x) if for all x from some neighborhood X max , the inequality f (x) f (X max) is satisfied.

The value Y max =f(X max) is called the maximum of this function.

X max - maximum point
Max has a maximum

The point X min is called the minimum point of the function f (x) if for all x from some neighborhood X min, the inequality f (x) f (X min) is satisfied.

The value of Y min =f(X min) is called the minimum of this function.

X min - minimum point
Y min - minimum

X min , X max - extremum points
Y min , Y max - extrema.

5. Function zeros

The zero of the function y = f(x) is the value of the argument x at which the function vanishes: f(x) = 0.

X 1, X 2, X 3 are zeros of the function y = f(x).

Tasks and tests on the topic "Basic properties of a function"

• Function properties - Numerical functions Grade 9

  Lessons: 2 Assignments: 11 Tests: 1

• Properties of logarithms - Exponential and logarithmic functions Grade 11

  Lessons: 2 Assignments: 14 Tests: 1

• Square root function, its properties and graph - Function square root. Square root properties Grade 8

  Lessons: 1 Assignments: 9 Tests: 1

• Functions - Important topics for repeating the exam in mathematics

  Tasks: 24

• Power functions, their properties and graphs - Degrees and roots. Power functions Grade 11

  Lessons: 4 Assignments: 14 Tests: 1

After studying this topic, you should be able to find the domain various functions, determine with the help of graphs the intervals of monotonicity of a function, examine functions for evenness and oddness. Consider the solution of such problems on the following examples.

Examples.

1. Find the domain of the function.

Solution: the scope of the function is found from the condition

hence the function f(x) is even.

Answer: even.

D(f) = [-1; 1] is symmetric with respect to zero.

2)

hence the function is neither even nor odd.

Answer: neither even nor even.

National Research University

Department of Applied Geology

Essay on higher mathematics

On the topic: "Basic elementary functions,

their properties and graphs"

Completed:

Checked:

teacher

Definition. The function given by the formula y=a x (where a>0, a≠1) is called exponential function with base a.

Let us formulate the main properties of the exponential function:

1. The domain of definition is the set (R) of all real numbers.

2. The range of values ​​is the set (R+) of all positive real numbers.

3. When a > 1, the function increases on the entire real line; at 0<а<1 функция убывает.

4. Is a general function.

, on the interval xн [-3;3]
, on the interval xн [-3;3]

A function of the form y(х)=х n , where n is the number ОR, is called a power function. The number n can take on different values: both integer and fractional, both even and odd. Depending on this, the power function will have a different form. Consider special cases that are power functions and reflect the main properties of this type of curves in the following order: power function y \u003d x² (a function with an even exponent - a parabola), a power function y \u003d x³ (a function with an odd exponent - a cubic parabola) and function y \u003d √ x (x to the power of ½) (function with a fractional exponent), a function with a negative integer exponent (hyperbola).

Power function y=x²

1. D(x)=R – the function is defined on the entire numerical axis;

2. E(y)= and increases on the interval

Power function y=x³

1. The graph of the function y \u003d x³ is called a cubic parabola. The power function y=x³ has the following properties:

2. D(x)=R – the function is defined on the entire numerical axis;

3. E(y)=(-∞;∞) – the function takes all values ​​in its domain of definition;

4. When x=0 y=0 – the function passes through the origin O(0;0).

5. The function increases over the entire domain of definition.

6. The function is odd (symmetric about the origin).

, on the interval xн [-3;3]

Depending on the numerical factor in front of x³, the function can be steep / flat and increase / decrease.

Power function with integer negative exponent:

If the exponent n is odd, then the graph of such a power function is called a hyperbola. A power function with a negative integer exponent has the following properties:

1. D(x)=(-∞;0)U(0;∞) for any n;

2. E(y)=(-∞;0)U(0;∞) if n is an odd number; E(y)=(0;∞) if n is an even number;

3. The function decreases over the entire domain of definition if n is an odd number; the function increases on the interval (-∞;0) and decreases on the interval (0;∞) if n is an even number.

4. The function is odd (symmetric about the origin) if n is an odd number; a function is even if n is an even number.

5. The function passes through the points (1;1) and (-1;-1) if n is an odd number and through the points (1;1) and (-1;1) if n is an even number.

, on the interval xн [-3;3]

Power function with fractional exponent

A power function with a fractional exponent of the form (picture) has a graph of the function shown in the figure. A power function with a fractional exponent has the following properties: (picture)

1. D(x) нR if n is an odd number and D(x)=
, on the interval xн
, on the interval xн [-3;3]

The logarithmic function y \u003d log a x has the following properties:

1. Domain of definition D(x)н (0; + ∞).

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

3. The function is neither even nor odd (general).

4. The function increases on the interval (0; + ∞) for a > 1, decreases on (0; + ∞) for 0< а < 1.

The graph of the function y = log a x can be obtained from the graph of the function y = a x using a symmetry transformation about the line y = x. In Figure 9, a plot of the logarithmic function for a > 1 is plotted, and in Figure 10 - for 0< a < 1.

; on the interval xО
; on the interval xО

The functions y \u003d sin x, y \u003d cos x, y \u003d tg x, y \u003d ctg x are called trigonometric functions.

The functions y \u003d sin x, y \u003d tg x, y \u003d ctg x are odd, and the function y \u003d cos x is even.

Function y \u003d sin (x).

1. Domain of definition D(x) ОR.

2. Range of values ​​E(y) О [ - 1; 1].

3. The function is periodic; the main period is 2π.

4. The function is odd.

5. The function increases on the intervals [ -π/2 + 2πn; π/2 + 2πn] and decreases on the intervals [ π/2 + 2πn; 3π/2 + 2πn], n О Z.

The graph of the function y \u003d sin (x) is shown in Figure 11.