Let's see how to examine a function using a graph. It turns out that by looking at the graph, we can find out everything that interests us, namely:

• domain of a function
• function range
• function zeros
• intervals of increasing and decreasing
• maximum and minimum points
• the largest and smallest value of a function on a segment.

Let's clarify the terminology:

Abscissa is the horizontal coordinate of the point.
Ordinate- vertical coordinate.
Abscissa axis- the horizontal axis, most often called the axis.
Y axis- vertical axis, or axis.

Argument- an independent variable on which the function values ​​depend. Most often indicated.
In other words, we choose , substitute functions into the formula and get .

Domain functions - the set of those (and only those) argument values ​​for which the function exists.
Indicated by: or .

In our figure, the domain of definition of the function is the segment. It is on this segment that the graph of the function is drawn. This is the only place where this function exists.

Function Range is the set of values ​​that a variable takes. In our figure, this is a segment - from the lowest to the highest value.

Function zeros- points where the value of the function is zero, that is. In our figure these are points and .

Function values ​​are positive where . In our figure these are the intervals and .
Function values ​​are negative where . For us, this is the interval (or interval) from to .

Key Concepts - increasing and decreasing function on some set. As a set, you can take a segment, an interval, a union of intervals, or the entire number line.

Function increases

In other words, the more , the more, that is, the graph goes to the right and up.

Function decreases on a set if for any and belonging to the set, the inequality implies the inequality .

For a decreasing function, a larger value corresponds to a smaller value. The graph goes to the right and down.

In our figure, the function increases on the interval and decreases on the intervals and .

Let's define what it is maximum and minimum points of the function.

Maximum point- this is an internal point of the domain of definition, such that the value of the function in it is greater than in all points sufficiently close to it.
In other words, a maximum point is a point at which the value of the function more than in neighboring ones. This is a local “hill” on the chart.

In our figure there is a maximum point.

Minimum point- an internal point of the domain of definition, such that the value of the function in it is less than in all points sufficiently close to it.
That is, the minimum point is such that the value of the function in it is less than in its neighbors. This is a local “hole” on the graph.

In our figure there is a minimum point.

The point is the boundary. It is not an internal point of the domain of definition and therefore does not fit the definition of a maximum point. After all, she has no neighbors on the left. In the same way, on our chart there cannot be a minimum point.

The maximum and minimum points together are called extremum points of the function. In our case this is and .

What to do if you need to find, for example, minimum function on the segment? IN in this case answer: . Because minimum function is its value at the minimum point.

Similarly, the maximum of our function is . It is reached at point .

We can say that the extrema of the function are equal to and .

Sometimes problems require finding largest and smallest values ​​of a function on a given segment. They do not necessarily coincide with the extremes.

In our case smallest function value on the segment is equal to and coincides with the minimum of the function. But its greatest value on this segment is equal to . It is reached at the left end of the segment.

In any case, the largest and smallest values continuous function on a segment are achieved either at extremum points or at the ends of the segment.

The methodological material is for reference only and applies to a wide range of topics. The article provides an overview of graphs of basic elementary functions and considers the most important issue - how to build a graph correctly and QUICKLY. In the course of studying higher mathematics without knowledge of basic graphs elementary functions It will be hard, so it is very important to remember what the graphs of a parabola, hyperbola, sine, cosine, etc. look like, and remember some of the function values. We will also talk about some properties of the main functions.

I do not claim completeness and scientific thoroughness of the materials; the emphasis will be placed, first of all, on practice - those things with which one encounters literally at every step, in any topic of higher mathematics. Charts for dummies? One could say so.

Due to numerous requests from readers clickable table of contents:

In addition, there is an ultra-short synopsis on the topic
– master 16 types of charts by studying SIX pages!

Seriously, six, even I was surprised. This summary contains improved graphics and is available for a nominal fee; a demo version can be viewed. It is convenient to print the file so that the graphs are always at hand. Thanks for supporting the project!

And let's start right away:

How to construct coordinate axes correctly?

In practice, tests are almost always completed by students in separate notebooks, lined in a square. Why do you need checkered markings? After all, the work, in principle, can be done on A4 sheets. And the cage is necessary just for high-quality and accurate design of drawings.

Any drawing of a function graph begins with coordinate axes .

Drawings can be two-dimensional or three-dimensional.

Let's first consider the two-dimensional case Cartesian rectangular coordinate system:

1) Draw coordinate axes. The axis is called x-axis , and the axis is y-axis . We always try to draw them neat and not crooked. The arrows should also not resemble Papa Carlo’s beard.

2) We sign the axes with large letters “X” and “Y”. Don't forget to label the axes.

3) Set the scale along the axes: draw a zero and two ones. When making a drawing, the most convenient and frequently used scale is: 1 unit = 2 cells (drawing on the left) - if possible, stick to it. However, from time to time it happens that the drawing does not fit on the notebook sheet - then we reduce the scale: 1 unit = 1 cell (drawing on the right). It’s rare, but it happens that the scale of the drawing has to be reduced (or increased) even more

There is NO NEED to “machine gun” …-5, -4, -3, -1, 0, 1, 2, 3, 4, 5, …. For coordinate plane is not a monument to Descartes, and the student is not a dove. We put zero And two units along the axes. Sometimes instead of units, it is convenient to “mark” other values, for example, “two” on the abscissa axis and “three” on the ordinate axis - and this system (0, 2 and 3) will also uniquely define the coordinate grid.

It is better to estimate the estimated dimensions of the drawing BEFORE constructing the drawing. So, for example, if the task requires drawing a triangle with vertices , , , then it is completely clear that the popular scale of 1 unit = 2 cells will not work. Why? Let's look at the point - here you will have to measure fifteen centimeters down, and, obviously, the drawing will not fit (or barely fit) on a notebook sheet. Therefore, we immediately select a smaller scale: 1 unit = 1 cell.

By the way, about centimeters and notebook cells. Is it true that 30 notebook cells contain 15 centimeters? For fun, measure 15 centimeters in your notebook with a ruler. In the USSR, this may have been true... It is interesting to note that if you measure these same centimeters horizontally and vertically, the results (in the cells) will be different! Strictly speaking, modern notebooks are not checkered, but rectangular. This may seem nonsense, but drawing, for example, a circle with a compass in such situations is very inconvenient. To be honest, at such moments you begin to think about the correctness of Comrade Stalin, who was sent to camps for hack work in production, not to mention the domestic automobile industry, falling planes or exploding power plants.

Speaking of quality, or a brief recommendation on stationery. Today, most of the notebooks on sale are, to say the least, complete crap. For the reason that they get wet, and not only from gel pens, but also from ballpoint pens! They save money on paper. For registration tests I recommend using notebooks from the Arkhangelsk Pulp and Paper Mill (18 sheets, grid) or “Pyaterochka”, although it is more expensive. It is advisable to choose a gel pen; even the cheapest Chinese gel refill is much better than a ballpoint pen, which either smudges or tears the paper. The only “competitive” ballpoint pen I can remember is the Erich Krause. She writes clearly, beautifully and consistently – whether with a full core or with an almost empty one.

Additionally: The vision of a rectangular coordinate system through the eyes of analytical geometry is covered in the article Linear (non) dependence of vectors. Basis of vectors, detailed information about coordinate quarters can be found in the second paragraph of the lesson Linear inequalities.

3D case

It's almost the same here.

1) Draw coordinate axes. Standard: axis applicate – directed upwards, axis – directed to the right, axis – directed downwards to the left strictly at an angle of 45 degrees.

2) Label the axes.

3) Set the scale along the axes. The scale along the axis is two times smaller than the scale along the other axes. Also note that in the right drawing I used a non-standard "notch" along the axis (this possibility has already been mentioned above). From my point of view, this is more accurate, faster and more aesthetically pleasing - there is no need to look for the middle of the cell under a microscope and “sculpt” a unit close to the origin of coordinates.

When making a 3D drawing, again, give priority to scale
1 unit = 2 cells (drawing on the left).

What are all these rules for? Rules are made to be broken. That's what I'll do now. The fact is that subsequent drawings of the article will be made by me in Excel, and the coordinate axes will look incorrect from the point of view correct design. I could draw all the graphs by hand, but it’s actually scary to draw them as Excel is reluctant to draw them much more accurately.

Graphs and basic properties of elementary functions

A linear function is given by the equation. The graph of linear functions is direct. In order to construct a straight line, it is enough to know two points.

Example 1

Construct a graph of the function. Let's find two points. It is advantageous to choose zero as one of the points.

If , then

Let's take another point, for example, 1.

If , then

When completing tasks, the coordinates of the points are usually summarized in a table:

And the values ​​themselves are calculated orally or on a draft, a calculator.

Two points have been found, let’s make the drawing:

When preparing a drawing, we always sign the graphics.

It would be useful to recall special cases of a linear function:

Notice how I placed the signatures, signatures should not allow discrepancies when studying the drawing. In this case, it was extremely undesirable to put a signature next to the point of intersection of the lines, or at the bottom right between the graphs.

1) A linear function of the form () is called direct proportionality. For example, . A direct proportionality graph always passes through the origin. Thus, constructing a straight line is simplified - it is enough to find just one point.

2) An equation of the form specifies a straight line parallel to the axis, in particular, the axis itself is given by the equation. The graph of the function is constructed immediately, without finding any points. That is, the entry should be understood as follows: “the y is always equal to –4, for any value of x.”

3) An equation of the form specifies a straight line parallel to the axis, in particular, the axis itself is given by the equation. The graph of the function is also plotted immediately. The entry should be understood as follows: “x is always, for any value of y, equal to 1.”

Some will ask, why remember 6th grade?! That’s how it is, maybe it’s so, but over the years of practice I’ve met a good dozen students who were baffled by the task of constructing a graph like or.

Constructing a straight line is the most common action when making drawings.

The straight line is discussed in detail in the course of analytical geometry, and those interested can refer to the article Equation of a straight line on a plane.

Graph of a quadratic, cubic function, graph of a polynomial

Parabola. Schedule quadratic function () represents a parabola. Consider the famous case:

Let's recall some properties of the function.

So, the solution to our equation: – it is at this point that the vertex of the parabola is located. Why this is so can be found in the theoretical article on the derivative and the lesson on extrema of the function. In the meantime, let’s calculate the corresponding “Y” value:

Thus, the vertex is at the point

Now we find other points, while brazenly using the symmetry of the parabola. It should be noted that the function – is not even, but, nevertheless, no one canceled the symmetry of the parabola.

In what order to find the remaining points, I think it will be clear from the final table:

This construction algorithm can figuratively be called a “shuttle” or the “back and forth” principle with Anfisa Chekhova.

Let's make the drawing:

From the graphs examined, another useful feature comes to mind:

For a quadratic function () the following is true:

If , then the branches of the parabola are directed upward.

If , then the branches of the parabola are directed downward.

In-depth knowledge about the curve can be obtained in the lesson Hyperbola and parabola.

A cubic parabola is given by the function. Here is a drawing familiar from school:

Let us list the main properties of the function

Graph of a function

It represents one of the branches of a parabola. Let's make the drawing:

Main properties of the function:

In this case, the axis is vertical asymptote for the graph of a hyperbola at .

It would be a GROSS mistake if, when drawing up a drawing, you carelessly allow the graph to intersect with an asymptote.

Also one-sided limits tell us that the hyperbola not limited from above And not limited from below.

Let’s examine the function at infinity: , that is, if we start moving along the axis to the left (or right) to infinity, then the “games” will be in an orderly step infinitely close approach zero, and, accordingly, the branches of the hyperbola infinitely close approach the axis.

So the axis is horizontal asymptote for the graph of a function, if “x” tends to plus or minus infinity.

The function is odd, and, therefore, the hyperbola is symmetrical about the origin. This fact is obvious from the drawing, in addition, it is easily verified analytically: .

The graph of a function of the form () represents two branches of a hyperbola.

If , then the hyperbola is located in the first and third coordinate quarters(see picture above).

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

The indicated pattern of hyperbola residence is easy to analyze from the point of view of geometric transformations of graphs.

Example 3

Construct the right branch of the hyperbola

We use the point-wise construction method, and it is advantageous to select the values ​​so that they are divisible by a whole:

Let's make the drawing:

It will not be difficult to construct the left branch of the hyperbola; the oddness of the function will help here. Roughly speaking, in the table of pointwise construction, we mentally add a minus to each number, put the corresponding points and draw the second branch.

Detailed geometric information about the line considered can be found in the article Hyperbola and parabola.

Graph of an Exponential Function

IN this paragraph I will immediately consider the exponential function, since in problems of higher mathematics in 95% of cases it is the exponential that appears.

Let me remind you that this is an irrational number: , this will be required when constructing a graph, which, in fact, I will build without ceremony. Three points are probably enough:

Let's leave the graph of the function alone for now, more on it later.

Main properties of the function:

Function graphs, etc., look fundamentally the same.

I must say that the second case occurs less frequently in practice, but it does occur, so I considered it necessary to include it in this article.

Graph of a logarithmic function

Consider a function with a natural logarithm.
Let's make a point-by-point drawing:

If you have forgotten what a logarithm is, please refer to your school textbooks.

Main properties of the function:

Domain:

Range of values: .

The function is not bounded from above: although slowly, the branch of the logarithm goes up to infinity.
Let us examine the behavior of the function near zero on the right: . So the axis is vertical asymptote for the graph of a function as “x” tends to zero from the right.

It is imperative to know and remember the typical value of the logarithm: .

In principle, the graph of the logarithm to the base looks the same: , , (decimal logarithm to the base 10), etc. Moreover, the larger the base, the flatter the graph will be.

We won’t consider the case; I don’t remember the last time I built a graph with such a basis. And the logarithm seems to be a very rare guest in problems of higher mathematics.

At the end of this paragraph I will say one more fact: Exponential function and logarithmic function- the two are mutual inverse functions . If you look closely at the graph of the logarithm, you can see that this is the same exponent, it’s just located a little differently.

Graphs of trigonometric functions

Where does trigonometric torment begin at school? Right. From sine

Let's plot the function

This line is called sinusoid.

Let me remind you that “pi” is an irrational number: , and in trigonometry it makes your eyes dazzle.

Main properties of the function:

This function is periodic with period . What does it mean? Let's look at the segment. To the left and right of it, exactly the same piece of the graph is repeated endlessly.

Domain: , that is, for any value of “x” there is a sine value.

Range of values: . The function is limited: , that is, all the “games” sit strictly in the segment .
This does not happen: or, more precisely, it happens, but these equations do not have a solution.

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: Аx 2 – x + А = 0. This equation has a solution when 1 – 4А 2 ≥ 0. From here we find highest value A = 1/2.

Answer: Figure 5, max y(x) = ½.

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National Research University

Department of Applied Geology

Abstract on higher mathematics

On the topic: “Basic elementary functions,

their properties and graphs"

Completed:

Checked:

teacher

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

Let us formulate the main properties exponential function:

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

2. Range - the set (R+) of all positive real numbers.

3. For a > 1, the function increases along the entire number line; at 0<а<1 функция убывает.

4. Is a function of general form.

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

A function of the form y(x)=x 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. Let's consider special cases that are power functions and reflect the basic properties of this type of curve in the following order: power function y=x² (function with an even exponent - a parabola), power function y=x³ (function with an odd exponent - cubic parabola) and function y=√x (x to the power of ½) (function with a fractional exponent), 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=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 of coordinates O(0;0).

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

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

, on the interval xО [-3;3]

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

Power function with negative integer exponent:

If the exponent n is odd, then the graph of such a power function is called a hyperbola. A power function with an integer negative 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 (symmetrical 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 xО [-3;3]

Power function with fractional exponent

A power function with a fractional exponent (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) ОR, if n is an odd number and D(x)= , on the interval xО , on the interval xО [-3;3]

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

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

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

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

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 straight line y = x. Figure 9 shows a graph of the logarithmic function for a > 1, and Figure 10 for 0< a < 1.

; on the interval xО ; on the interval xО

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

The functions y = sin x, y = tan x, y = ctg x are odd, and the function y = cos x is even.

Function y = 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 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 = sin (x) is shown in Figure 11.