Schoolchildren are faced with the task of constructing a graph of a function at the very beginning of studying algebra and continue to build them year after year. Starting from the graph of a linear function, for which you need to know only two points, to a parabola, which already requires 6 points, a hyperbola and a sine wave. Every year the functions become more and more complex and it is no longer possible to construct their graphs using a template; it is necessary to carry out more complex studies using derivatives and limits.

Let's figure out how to find the graph of a function? To do this, let's start with the simplest functions, the graphs of which are plotted point by point, and then consider a plan for constructing more complex functions.

Graphing a Linear Function

To construct the simplest graphs, use a table of function values. The graph of a linear function is a straight line. Let's try to find the points on the graph of the function y=4x+5.

1. To do this, let’s take two arbitrary values ​​of the variable x, substitute them one by one into the function, find the value of the variable y and enter everything into the table.
2. Take the value x=0 and substitute it into the function instead of x - 0. We get: y=4*0+5, that is, y=5, write this value in the table under 0. Similarly, take x=0, we get y=4*1+5 , y=9.
3. Now, to build a graph of the function, you need to plot these points on the coordinate plane. Then you need to draw a straight line.

Graphing a Quadratic Function

A quadratic function is a function of the form y=ax 2 +bx +c, where x is a variable, a,b,c are numbers (a is not equal to 0). For example: y=x 2, y=x 2 +5, y=(x-3) 2, y=2x 2 +3x+5.

To construct the simplest quadratic function y=x 2, 5-7 points are usually taken. Let's take the values ​​for the variable x: -2, -1, 0, 1, 2 and find the values ​​of y in the same way as when constructing the first graph.

The graph of a quadratic function is called a parabola. After constructing graphs of functions, students have new tasks related to the graph.

Example 1: find the abscissa of the graph point of the function y=x 2 if the ordinate is 9. To solve the problem, you need to substitute its value 9 into the function instead of y. We get 9=x 2 and solve this equation. x=3 and x=-3. This can also be seen on the graph of the function.

Researching a function and plotting it

To plot graphs of more complex functions, you need to perform several steps aimed at studying it. To do this you need:

1. Find the domain of definition of the function. The domain of definition is all the values ​​that the variable x can take. Those points at which the denominator becomes 0 or the radical expression becomes negative should be excluded from the definition domain.
2. Set whether the function is even or odd. Recall that an even function is one that meets the condition f(-x)=f(x). Its graph is symmetrical with respect to Oy. A function will be odd if it meets the condition f(-x)=-f(x). In this case, the graph is symmetrical about the origin.
3. Find the points of intersection with the coordinate axes. In order to find the abscissa of the point of intersection with the Ox axis, it is necessary to solve the equation f(x) = 0 (the ordinate is equal to 0). To find the ordinate of the point of intersection with the Oy axis, it is necessary to substitute 0 in the function instead of the variable x (the abscissa is 0).
4. Find the asymptotes of the function. An asyptote is a straight line that the graph approaches indefinitely, but never crosses. Let's figure out how to find the asymptotes of the graph of a function.
   • Vertical asymptote of the straight line x=a
   • Horizontal asymptote - straight line y=a
   • Oblique asymptote - straight line of the form y=kx+b
5. Find the extremum points of the function, the intervals of increase and decrease of the function. Let's find the extremum points of the function. To do this, you need to find the first derivative and equate it to 0. It is at these points that the function can change from increasing to decreasing. Let us determine the sign of the derivative on each interval. If the derivative is positive, then the graph of the function increases; if it is negative, it decreases.
6. Find the inflection points of the function graph, the upward and downward convexity intervals.

Finding inflection points is now easier than ever. You just need to find the second derivative, then equate it to zero. Next we find the sign of the second derivative on each interval. If it is positive, then the graph of the function is convex downward, if it is negative, it is convex upward.

A function graph is a visual representation of the behavior of a function on a coordinate plane. Graphs help you understand various aspects of a function that cannot be determined from the function itself. You can build graphs of many functions, and each of them will be given a specific formula. The graph of any function is built using a specific algorithm (if you have forgotten the exact process of graphing a specific function).

Steps

Graphing a Linear Function

Determine whether the function is linear. The linear function is given by a formula of the form F (x) = k x + b (\displaystyle F(x)=kx+b) or y = k x + b (\displaystyle y=kx+b)(for example, ), and its graph is a straight line. Thus, the formula includes one variable and one constant (constant) without any exponents, root signs, or the like. If a function of a similar type is given, it is quite simple to plot a graph of such a function. Here are other examples of linear functions:

Use a constant to mark a point on the Y axis. The constant (b) is the “y” coordinate of the point where the graph intersects the Y axis. That is, it is a point whose “x” coordinate is equal to 0. Thus, if x = 0 is substituted into the formula, then y = b (constant). In our example y = 2 x + 5 (\displaystyle y=2x+5) the constant is equal to 5, that is, the point of intersection with the Y axis has coordinates (0.5). Plot this point on the coordinate plane.

Find the slope of the line. It is equal to the multiplier of the variable. In our example y = 2 x + 5 (\displaystyle y=2x+5) with the variable “x” there is a factor of 2; thus, the slope coefficient is equal to 2. The slope coefficient determines the angle of inclination of the straight line to the X axis, that is, the greater the slope coefficient, the faster the function increases or decreases.

Write the slope as a fraction. The angular coefficient is equal to the tangent of the angle of inclination, that is, the ratio of the vertical distance (between two points on a straight line) to the horizontal distance (between the same points). In our example, the slope is 2, so we can state that the vertical distance is 2 and the horizontal distance is 1. Write this as a fraction: 2 1 (\displaystyle (\frac (2)(1))).

• If the slope is negative, the function is decreasing.

1. From the point where the straight line intersects the Y axis, plot a second point using vertical and horizontal distances. A linear function can be graphed using two points. In our example, the intersection point with the Y axis has coordinates (0.5); From this point, move 2 spaces up and then 1 space to the right. Mark a point; it will have coordinates (1,7). Now you can draw a straight line.

   Using a ruler, draw a straight line through two points. To avoid mistakes, find the third point, but in most cases the graph can be plotted using two points. Thus, you have plotted a linear function.

   Plotting points on the coordinate plane

   1. Define a function. The function is denoted as f(x). All possible values ​​of the variable "y" are called the domain of the function, and all possible values ​​of the variable "x" are called the domain of the function. For example, consider the function y = x+2, namely f(x) = x+2.

      Draw two intersecting perpendicular lines. The horizontal line is the X axis. The vertical line is the Y axis.

      Label the coordinate axes. Divide each axis into equal segments and number them. The intersection point of the axes is 0. For the X axis: positive numbers are plotted to the right (from 0), and negative numbers to the left. For the Y axis: positive numbers are plotted on top (from 0), and negative numbers on the bottom.

      Find the values ​​of "y" from the values ​​of "x". In our example, f(x) = x+2. Substitute specific x values ​​into this formula to calculate the corresponding y values. If given a complex function, simplify it by isolating the “y” on one side of the equation.

      • -1: -1 + 2 = 1
      • 0: 0 +2 = 2
      • 1: 1 + 2 = 3

   2. Plot the points on the coordinate plane. For each pair of coordinates, do the following: find the corresponding value on the X axis and draw a vertical line (dotted); find the corresponding value on the Y axis and draw a horizontal line (dashed line). Mark the intersection point of the two dotted lines; thus, you have plotted a point on the graph.

      Erase the dotted lines. Do this after plotting all the points on the graph on the coordinate plane. Note: the graph of the function f(x) = x is a straight line passing through the coordinate center [point with coordinates (0,0)]; the graph f(x) = x + 2 is a line parallel to the line f(x) = x, but shifted upward by two units and therefore passing through the point with coordinates (0,2) (because the constant is 2).

   Graphing a Complex Function

   Find the zeros of the function. The zeros of a function are the values ​​of the x variable where y = 0, that is, these are the points where the graph intersects the X-axis. Keep in mind that not all functions have zeros, but they are the first step in the process of graphing any function. To find the zeros of a function, equate it to zero. For example:

   Find and mark the horizontal asymptotes. An asymptote is a line that the graph of a function approaches but never intersects (that is, in this region the function is not defined, for example, when dividing by 0). Mark the asymptote with a dotted line. If the variable "x" is in the denominator of a fraction (for example, y = 1 4 − x 2 (\displaystyle y=(\frac (1)(4-x^(2))))), set the denominator to zero and find “x”. In the obtained values ​​of the variable “x” the function is not defined (in our example, draw dotted lines through x = 2 and x = -2), because you cannot divide by 0. But asymptotes exist not only in cases where the function contains a fractional expression. Therefore, it is recommended to use common sense:

This teaching material is for reference only and relates 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 the graphs of basic elementary functions, it will be difficult, so it is very important to remember what the graphs of a parabola, hyperbola, sine, cosine, etc. look like, and remember some of the meanings of the functions. 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 the 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. To complete tests, I recommend using notebooks from the Arkhangelsk Pulp and Paper Mill (18 sheets, square) 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 of 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. Graph of a 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 section, 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 limited from above: , albeit slowly, but 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– these are two mutually 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.

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. The 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 of the 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 = tan 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.

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

The length of a segment on the coordinate plane is found using the formula:

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

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 a three-dimensional coordinate system - all three formulas) are calculated using the formulas:

Function– this is a correspondence of the form y= f(x) between variable quantities, due to which each considered value of some variable quantity 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 argument value X only one value of the dependent variable can correspond at. However, the same value at can be obtained with different X.

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

Function Range are all possible values ​​of the dependent variable of a given function. Designated E(at).

Function increases on the interval in which a larger value of the argument corresponds to a larger value of the function. The function is decreasing on the interval in which a larger value of the argument corresponds to a smaller value of the function.

Intervals of constant sign of a function- these are the intervals of the independent variable over which the dependent variable retains its positive or negative sign.

Function zeros– these are the values ​​of the argument at which the value of the function is equal to zero. At these points, the function graph intersects the abscissa axis (OX axis). Very often, the need to find the zeros of a function means the need to simply solve the equation. Also, often the need to find intervals of constancy of 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 ​​of the even function are equal. The graph of an even function is always symmetrical with respect to the ordinate 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 holds:

This means that for any opposite values ​​of the argument, the values ​​of the 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 (the points of intersection of the x-axis OX) is always equal to zero, because for every positive root X has a negative root - X.

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

Linear function is 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 occasion k < 0 функция будет убывающей, т.е. прямая будет наклонена в другую сторону - слева направо):

Graph of a 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 only one root, then at this point ( x 0 ; 0) the quadratic function only touches the OX axis, but does not intersect it. The quadratic function always intersects the OY axis at the point with coordinates: (0; c). The graph of a 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 function y = ax 2 + bx + c, then the branches of the parabola are directed upward;
• if a < 0, то ветви параболы направлены вниз.

The coordinates of the vertex of a parabola can be calculated using the following formulas. X tops (p- in the pictures above) parabolas (or the point at which the quadratic trinomial reaches its largest or smallest value):

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

Graphs of other functions

Power function

Here are some examples of graphs of power functions:

Inversely proportional is a function given by the formula:

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

Asymptote is a line that the graph of a function approaches infinitely close to but does not intersect. The asymptotes for the inverse proportionality graphs 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 is a function given by the formula:

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

Logarithmic function is a 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:

Graph of a function y = |x| as follows:

Graphs of periodic (trigonometric) functions

Function at = f(x) is called periodic, if there is such a non-zero number T, What f(x + T) = f(x), for anyone X from the domain of the function 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 period T 1, which is determined by the formula:

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

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

Graph of a function 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 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 right.

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How to successfully prepare for the CT in physics and mathematics?

In order to successfully prepare for the CT in physics and mathematics, among other things, it is necessary to fulfill three most important conditions:

1. Study all topics and complete all tests and assignments given in the educational materials on this site. To do this, you need nothing at all, namely: devote three to four hours every day to preparing for the CT in physics and mathematics, studying theory and solving problems. The fact is that the CT is an exam where it is not enough just to know physics or mathematics, you also need to be able to quickly and without failures solve a large number of problems on different topics and of varying complexity. The latter can only be learned by solving thousands of problems.
2. Learn all the formulas and laws in physics, and formulas and methods in mathematics. In fact, this is also very simple to do; 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 solving most of the CT at the right time. After this, you will only have to think about the most difficult tasks.
3. Attend all three stages of rehearsal testing in physics and mathematics. Each RT can be visited twice to decide on both options. Again, on the CT, in addition to the ability to quickly and efficiently solve problems, and knowledge of formulas and methods, you must also be able to properly plan time, distribute forces, and most importantly, correctly fill out the answer form, without confusing the numbers of answers and problems, or your own last name. Also, during RT, it is important to get used to the style of asking questions in problems, which may seem very unusual to an unprepared person at the DT.

Successful, diligent and responsible implementation of these three points, as well as responsible study of the final training tests, will allow you to show an excellent result at the CT, the maximum of what you are capable of.

Found a mistake?

If you think you have found an error in the training materials, please write about it by email (). In the letter, indicate the subject (physics or mathematics), the name or number of the topic or test, the number of the problem, or the place in the text (page) where, in your opinion, there is an error. Also describe what the suspected error is. Your letter will not go unnoticed, the error will either be corrected, or you will be explained why it is not an error.