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 ​​of the 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 ​​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 (points of intersection of the abscissa axis OX) is always equal to zero, because for every positive root X has a 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 general functions, 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 one ; 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). The graph of a quadratic function (parabola) may look like this (the figure shows examples that far from 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), the value of the 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. 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 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, as it seems to you, found an error in the training materials, then please write about it by mail. You can also write about the error on the 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.

A quadratic function is a function of the form:
y=a*(x^2)+b*x+c,
where a is the coefficient at the highest degree of the unknown x,
b - coefficient at unknown x,
and c is a free member.
The graph of a quadratic function is a curve called a parabola. The general view of the parabola is shown in the figure below.

Fig.1 General view of the parabola.

There are several different ways to graph a quadratic function. We will consider the main and most general of them.

Algorithm for plotting a graph of a quadratic function y=a*(x^2)+b*x+c

1. Build a coordinate system, mark a single segment and label the coordinate axes.

2. Determine the direction of the branches of the parabola (up or down).
To do this, you need to look at the sign of the coefficient a. If plus - then the branches are directed upwards, if minus - then the branches are directed downwards.

3. Determine the x-coordinate of the top of the parabola.
To do this, you need to use the formula Tops = -b / 2 * a.

4. Determine the coordinate at the top of the parabola.
To do this, substitute the value of the Top found in the previous step in the equation of the Top = a * (x ^ 2) + b * x + c instead of x.

5. Put the resulting point on the graph and draw an axis of symmetry through it, parallel to the coordinate axis Oy.

6. Find the points of intersection of the graph with the x-axis.
This requires solving the quadratic equation a*(x^2)+b*x+c = 0 using one of the known methods. If the equation has no real roots, then the graph of the function does not intersect the x-axis.

7. Find the coordinates of the point of intersection of the graph with the Oy axis.
To do this, we substitute the value x = 0 into the equation and calculate the value of y. We mark this and the point symmetrical to it on the graph.

8. Find the coordinates of an arbitrary point A (x, y)
To do this, we choose an arbitrary value of the x coordinate, and substitute it into our equation. We get the value of y at this point. Put a point on the graph. And also mark a point on the graph that is symmetrical to the point A (x, y).

9. Connect the obtained points on the graph with a smooth line and continue the graph beyond the extreme points, to the end of the coordinate axis. Sign the graph either on the callout, or, if space permits, along the graph itself.

An example of plotting a graph

As an example, let's plot a quadratic function given by the equation y=x^2+4*x-1
1. Draw coordinate axes, sign them and mark a single segment.
2. The values ​​of the coefficients a=1, b=4, c= -1. Since a \u003d 1, which is greater than zero, the branches of the parabola are directed upwards.
3. Determine the X coordinate of the top of the parabola Tops = -b/2*a = -4/2*1 = -2.
4. Determine the coordinate At the top of the parabola
Tops = a*(x^2)+b*x+c = 1*((-2)^2) + 4*(-2) - 1 = -5.
5. Mark the vertex and draw an axis of symmetry.
6. We find the points of intersection of the graph of a quadratic function with the Ox axis. We solve the quadratic equation x^2+4*x-1=0.
x1=-2-√3 x2 = -2+√3. We mark the obtained values ​​on the graph.
7. Find the points of intersection of the graph with the Oy axis.
x=0; y=-1
8. Choose an arbitrary point B. Let it have a coordinate x=1.
Then y=(1)^2 + 4*(1)-1= 4.
9. We connect the received points and sign the chart.

Tasks on the properties and graphs of a quadratic function, as practice shows, cause serious difficulties. This is rather strange, because the quadratic function is passed in the 8th grade, and then the entire first quarter of the 9th grade is "tortured" by the properties of the parabola and its graphs are built for various parameters.

This is due to the fact that forcing students to build parabolas, they practically do not devote time to "reading" graphs, that is, they do not practice comprehending the information received from the picture. Apparently, it is assumed that, having built two dozen graphs, a smart student himself will discover and formulate the relationship between the coefficients in the formula and the appearance of the graph. In practice, this does not work. For such a generalization, serious experience in mathematical mini-research is required, which, of course, most ninth-graders do not have. Meanwhile, in the GIA they propose to determine the signs of the coefficients precisely according to the schedule.

We will not demand the impossible from schoolchildren and simply offer one of the algorithms for solving such problems.

So, a function of the form y=ax2+bx+c is called quadratic, its graph is a parabola. As the name suggests, the main component is ax 2. That is a should not be equal to zero, the remaining coefficients ( b and With) can be equal to zero.

Let's see how the signs of its coefficients affect the appearance of the parabola.

The simplest dependence for the coefficient a. Most schoolchildren confidently answer: "if a> 0, then the branches of the parabola are directed upwards, and if a < 0, - то вниз". Совершенно верно. Ниже приведен график квадратичной функции, у которой a > 0.

y = 0.5x2 - 3x + 1

In this case a = 0,5

And now for a < 0:

y = - 0.5x2 - 3x + 1

In this case a = - 0,5

Influence of coefficient With also easy enough to follow. Imagine that we want to find the value of a function at a point X= 0. Substitute zero into the formula:

y = a 0 2 + b 0 + c = c. It turns out that y = c. That is With is the ordinate of the point of intersection of the parabola with the y-axis. As a rule, this point is easy to find on the chart. And determine whether it lies above zero or below. That is With> 0 or With < 0.

With > 0:

y=x2+4x+3

With < 0

y = x 2 + 4x - 3

Accordingly, if With= 0, then the parabola will necessarily pass through the origin:

y=x2+4x

More difficult with the parameter b. The point by which we will find it depends not only on b but also from a. This is the top of the parabola. Its abscissa (axis coordinate X) is found by the formula x in \u003d - b / (2a). In this way, b = - 2ax in. That is, we act as follows: on the graph we find the top of the parabola, determine the sign of its abscissa, that is, we look to the right of zero ( x in> 0) or to the left ( x in < 0) она лежит.

However, this is not all. We must also pay attention to the sign of the coefficient a. That is, to see where the branches of the parabola are directed. And only after that, according to the formula b = - 2ax in determine sign b.

Consider an example:

Branches pointing upwards a> 0, the parabola crosses the axis at below zero means With < 0, вершина параболы лежит правее нуля. Следовательно, x in> 0. So b = - 2ax in = -++ = -. b < 0. Окончательно имеем: a > 0, b < 0, With < 0.

Function of the form , where is called quadratic function.

Graph of quadratic function − parabola.

Consider the cases:

CASE I, CLASSICAL PARABOLA

That is , ,

To build, fill in the table by substituting x values ​​into the formula:

Mark points (0;0); (1;1); (-1;1) etc. on the coordinate plane (the smaller the step we take x values ​​(in this case, step 1), and the more x values ​​we take, the smoother the curve), we get a parabola:

It is easy to see that if we take the case , , , that is, then we get a parabola symmetric about the axis (ox). It is easy to verify this by filling out a similar table:

II CASE, "a" DIFFERENT FROM ONE

What will happen if we take , , ? How will the behavior of the parabola change? With title="(!LANG:Rendered by QuickLaTeX.com" height="20" width="55" style="vertical-align: -5px;"> парабола изменит форму, она “похудеет” по сравнению с параболой (не верите – заполните соответствующую таблицу – и убедитесь сами):!}

The first picture (see above) clearly shows that the points from the table for the parabola (1;1), (-1;1) were transformed into points (1;4), (1;-4), that is, with the same values, the ordinate of each point is multiplied by 4. This will happen to all key points of the original table. We argue similarly in the cases of pictures 2 and 3.

And when the parabola "becomes wider" parabola:

Let's recap:

1)The sign of the coefficient is responsible for the direction of the branches. With title="(!LANG:Rendered by QuickLaTeX.com" height="14" width="47" style="vertical-align: 0px;"> ветви направлены вверх, при - вниз. !}

2) Absolute value coefficient (modulus) is responsible for the “expansion”, “compression” of the parabola. The larger , the narrower the parabola, the smaller |a|, the wider the parabola.

CASE III, "C" APPEARS

Now let's put into play (that is, we consider the case when ), we will consider parabolas of the form . It is easy to guess (you can always refer to the table) that the parabola will move up or down along the axis, depending on the sign:

IV CASE, "b" APPEARS

When will the parabola “tear off” from the axis and will finally “walk” along the entire coordinate plane? When it ceases to be equal.

Here, to construct a parabola, we need formula for calculating the vertex: , .

So at this point (as at the point (0; 0) of the new coordinate system) we will build a parabola, which is already within our power. If we are dealing with the case , then from the top we set aside one unit segment to the right, one up, - the resulting point is ours (similarly, a step to the left, a step up is our point); if we are dealing with, for example, then from the top we set aside one single segment to the right, two - up, etc.

For example, the vertex of a parabola:

Now the main thing to understand is that at this vertex we will build a parabola according to the parabola template, because in our case.

When constructing a parabola after finding the coordinates of the vertex is veryIt is convenient to consider the following points:

1) parabola must pass through the point . Indeed, substituting x=0 into the formula, we get that . That is, the ordinate of the point of intersection of the parabola with the axis (oy), this is. In our example (above), the parabola intersects the y-axis at , since .

2) axis of symmetry parabolas is a straight line, so all points of the parabola will be symmetrical about it. In our example, we immediately take the point (0; -2) and build a parabola symmetrical about the axis of symmetry, we get the point (4; -2), through which the parabola will pass.

3) Equating to , we find out the points of intersection of the parabola with the axis (ox). To do this, we solve the equation. Depending on the discriminant, we will get one (, ), two ( title="(!LANG:Rendered by QuickLaTeX.com" height="14" width="54" style="vertical-align: 0px;">, ) или нИсколько () точек пересечения с осью (ох) !} . In the previous example, we have the root of the discriminant - not an integer, when building it, it doesn’t really make sense for us to find the roots, but we can clearly see that we will have two points of intersection with the (oh) axis (since title = "(!LANG: Rendered by QuickLaTeX.com" height="14" width="54" style="vertical-align: 0px;">), хотя, в общем, это видно и без дискриминанта.!}

So let's work out

Algorithm for constructing a parabola if it is given in the form

1) determine the direction of the branches (a>0 - up, a<0 – вниз)

2) find the coordinates of the vertex of the parabola by the formula , .

3) we find the point of intersection of the parabola with the axis (oy) by the free term, we build a point symmetrical to the given one with respect to the axis of symmetry of the parabola (it should be noted that it happens that it is unprofitable to mark this point, for example, because the value is large ... we skip this point ...)

4) At the found point - the top of the parabola (as at the point (0; 0) of the new coordinate system), we build a parabola. If title="(!LANG:Rendered by QuickLaTeX.com" height="20" width="55" style="vertical-align: -5px;">, то парабола становится у’же по сравнению с , если , то парабола расширяется по сравнению с !}

5) We find the points of intersection of the parabola with the axis (oy) (if they themselves have not yet “surfaced”), solving the equation

Example 1

Example 2

Remark 1. If the parabola is initially given to us in the form , where are some numbers (for example, ), then it will be even easier to build it, because we have already been given the coordinates of the vertex . Why?

Let's take a square trinomial and select a full square in it: Look, here we got that , . We previously called the top of the parabola, that is, now,.

For example, . We mark the top of the parabola on the plane, we understand that the branches are directed downwards, the parabola is expanded (relatively). That is, we perform steps 1; 3; four; 5 from the algorithm for constructing a parabola (see above).

Remark 2. If the parabola is given in a form similar to this (that is, represented as a product of two linear factors), then we immediately see the points of intersection of the parabola with the (x) axis. In this case - (0;0) and (4;0). For the rest, we act according to the algorithm, opening the brackets.