What is the focus of a parabola. quadratic function

A parabola is the locus of points in a plane equidistant from a given point F and a given line dnot passing through given point. This geometric definition expresses parabola directory property.

The directory property of a parabola

The point F is called the focus of the parabola, the line d is called the directrix of the parabola, the midpoint O of the perpendicular dropped from the focus to the directrix is ​​the vertex of the parabola, the distance p from the focus to the directrix is ​​the parameter of the parabola, and the distance \frac(p)(2) from the vertex of the parabola to its focus - focal length (Fig. 3.45, a). The straight line perpendicular to the directrix and passing through the focus is called the axis of the parabola (the focal axis of the parabola). The segment FM connecting an arbitrary point M of the parabola with its focus is called the focal radius of the point M . The line segment connecting two points of the parabola is called the chord of the parabola.

For an arbitrary point of the parabola, the ratio of the distance to the focus to the distance to the directrix is ​​equal to one. Comparing the directory properties of , and parabolas, we conclude that parabola eccentricity is by definition equal to one (e=1) .

Geometric definition of a parabola, expressing its directory property, is equivalent to its analytical definition - the line given by the canonical equation of the parabola:

Indeed, let's introduce a rectangular coordinate system (Fig. 3.45, b). Let us take the vertex O of the parabola as the origin of the coordinate system; the straight line passing through the focus perpendicular to the directrix, we will take as the abscissa axis (positive direction on it from the point O to the point F); a straight line perpendicular to the abscissa axis and passing through the vertex of the parabola, we will take as the ordinate axis (the direction on the ordinate axis is chosen so that the rectangular coordinate system Oxy is right).

Let us compose the equation of a parabola using its geometric definition, which expresses the directorial property of the parabola. In the selected coordinate system, we determine the coordinates of the focus F\!\left(\frac(p)(2);\,0\right) and the directrix equation x=-\frac(p)(2) . For an arbitrary point M(x,y) belonging to a parabola, we have:

FM=MM_d,

where M_d\!\left(\frac(p)(2);\,y\right)- orthogonal projection of the point M(x,y) onto the directrix. We write this equation in coordinate form:

\sqrt((\left(x-\frac(p)(2)\right)\^2+y^2}=x+\frac{p}{2}. !}

We square both sides of the equation: (\left(x-\frac(p)(2)\right)\^2+y^2=x^2+px+\frac{p^2}{4} !}. Bringing like terms, we get canonical parabola equation

y^2=2\cdot p\cdot x, those. the chosen coordinate system is canonical.

By reasoning in reverse order, it can be shown that all the points whose coordinates satisfy the equation (3.51), and only they, belong to the locus of points, called the parabola. Thus, the analytic definition of a parabola is equivalent to its geometric definition, which expresses the directory property of a parabola.

Parabola equation in polar coordinates

The parabola equation in the polar coordinate system Fr \ varphi (Fig. 3.45, c) has the form

r=\frac(p)(1-e\cdot\cos\varphi), where p is the parameter of the parabola and e=1 is its eccentricity.

In fact, as the pole of the polar coordinate system, we choose the focus F of the parabola, and as the polar axis - a ray with the origin at the point F, perpendicular to the directrix and not crossing it (Fig. 3.45, c). Then for an arbitrary point M(r,\varphi) belonging to a parabola, according to the geometric definition (directorial property) of a parabola, we have MM_d=r . Insofar as MM_d=p+r\cos\varphi, we obtain the parabola equation in coordinate form:

p+r\cdot\cos\varphi \quad \Leftrightarrow \quad r=\frac(p)(1-\cos\varphi),

Q.E.D. Note that in polar coordinates, the equations of the ellipse, hyperbola, and parabola coincide, but describe different lines, since they differ in eccentricities (0\leqslant e<1 для , e=1 для параболы, e>1 for ).

The geometric meaning of the parameter in the parabola equation

Let's explain geometric meaning of the parameter p in the canonical parabola equation. Substituting x=\frac(p)(2) into equation (3.51), we get y^2=p^2 , i.e. y=\pm p . Therefore, the parameter p is half the length of the parabola chord passing through its focus perpendicular to the axis of the parabola.

The focal parameter of the parabola, as well as for an ellipse and for a hyperbola, is called half the length of the chord passing through its focus perpendicular to the focal axis (see Fig. 3.45, c). From the parabola equation in polar coordinates at \varphi=\frac(\pi)(2) we get r=p , i.e. parabola parameter coincides with its focal parameter.

Remarks 3.11.

1. The parameter p of a parabola characterizes its shape. The more p, the wider the branches of the parabola, the closer p to zero, the narrower the branches of the parabola (Fig. 3.46).

2. The equation y^2=-2px (for p>0) defines a parabola, which is located to the left of the y-axis (Fig. 3.47, a). This equation is reduced to the canonical one by changing the direction of the x-axis (3.37). On fig. 3.47,a shows the given coordinate system Oxy and the canonical Ox"y" .

3. Equation (y-y_0)^2=2p(x-x_0),\,p>0 defines a parabola with vertex O "(x_0, y_0) whose axis is parallel to the abscissa axis (Fig. 3.47.6). This equation is reduced to the canonical one using parallel translation (3.36).

The equation (x-x_0)^2=2p(y-y_0),\,p>0, also defines a parabola with vertex O "(x_0, y_0) , whose axis is parallel to the ordinate axis (Fig. 3.47, c). This equation is reduced to the canonical one by means of parallel translation (3.36) and renaming of the coordinate axes (3.38). In fig. 3.47, b, c shows the given coordinate systems Oxy and the canonical coordinate systems Ox "y" .

4. y=ax^2+bx+c,~a\ne0 is a parabola with apex at the point O"\!\left(-\frac(b)(2a);\,-\frac(b^2-4ac)(4a)\right), whose axis is parallel to the y-axis, the branches of the parabola are directed upwards (for a>0) or downwards (for a<0 ). Действительно, выделяя полный квадрат, получаем уравнение

y=a\left(x+\frac(b)(2a)\right)^2-\frac(b^2)(4a)+c \quad \Leftrightarrow \quad \!\left(x+\frac(b) (2a)\right)^2=\frac(1)(a)\left(y+\frac(b^2-4ac)(4a)\right)\!,

which is reduced to the canonical form (y")^2=2px" , where p=\left|\frac(1)(2a)\right|, by replacing y"=x+\frac(b)(2a) And x"=\pm\!\left(y+\frac(b^2-4ac)(4a)\right).

The sign is chosen to match the sign of the leading coefficient a . This replacement corresponds to the composition: parallel translation (3.36) with x_0=-\frac(b)(2a) And y_0=-\frac(b^2-4ac)(4a), renaming the coordinate axes (3.38), and in the case of a<0 еще и изменения направления координатной оси (3.37). На рис.3.48,а,б изображены заданные системы координат Oxy и канонические системы координат O"x"y" для случаев a>0 and a<0 соответственно.

5. The abscissa axis of the canonical coordinate system is axis of symmetry of the parabola, since changing the variable y to -y does not change equation (3.51). In other words, the coordinates of the point M (x, y) belonging to the parabola, and the coordinates of the point M "(x, -y), symmetrical to the point M about the abscissa axis, satisfy equation (3.S1). The axes of the canonical coordinate system are called the main axes of the parabola.

Example 3.22. Draw a parabola y^2=2x in the canonical coordinate system Oxy . Find the focal parameter, focus coordinates and directrix equation.

Solution. We build a parabola, taking into account its symmetry about the abscissa axis (Fig. 3.49). If necessary, we determine the coordinates of some points of the parabola. For example, substituting x=2 into the parabola equation, we get y^2=4~\Leftrightarrow~y=\pm2. Therefore, the points with coordinates (2;2),\,(2;-2) belong to the parabola.

Comparing for given equation with canonical (3.S1), we define the focal parameter: p=1 . Focus coordinates x_F=\frac(p)(2)=\frac(1)(2),~y_F=0, i.e. F\!\left(\frac(1)(2),\,0\right). We compose the directrix equation x=-\frac(p)(2) , i.e. x=-\frac(1)(2) .

General properties of an ellipse, hyperbola, parabola

1. The directory property can be used as a single definition of an ellipse, hyperbola, parabola (see Fig. 3.50): the locus of points in the plane, for each of which the ratio of the distance to a given point F (focus) to the distance to a given straight line d (directrix) that does not pass through a given point is constant and equal to the eccentricity e, is called:

a) if 0\leqslant e<1 ;

b) if e>1 ;

c) parabola if e=1.

2. Ellipse, hyperbola, parabola are obtained in sections of a circular cone by planes and therefore are called conic sections. This property can also serve as a geometric definition of an ellipse, hyperbola, parabola.

3. Common properties of an ellipse, hyperbola and parabola include bisector property their tangents. Under tangent to the line at some of its point K is understood as the limiting position of the secant KM, when the point M, remaining on the line under consideration, tends to the point K. A line perpendicular to the tangent line and passing through the point of contact is called normal to this line.

The bisectorial property of tangents (and normals) to an ellipse, hyperbola, and parabola is formulated as follows: the tangent (normal) to an ellipse or hyperbola forms equal angles with the focal radii of the tangent point(Fig. 3.51, a, b); the tangent (normal) to the parabola forms equal angles with the focal radius of the tangent point and the perpendicular dropped from it to the directrix(Fig. 3.51, c). In other words, the tangent to the ellipse at point K is the bisector of the triangle's external angle F_1KF_2 (and the normal is the bisector of the triangle's internal angle F_1KF_2); the tangent to the hyperbola is the bisector of the internal angle of the triangle F_1KF_2 (and the normal is the bisector of the external angle); the tangent to the parabola is the bisector of the interior angle of the triangle FKK_d (and the normal is the bisector of the exterior angle). The bisectorial property of a tangent to a parabola can be formulated in the same way as for an ellipse and a hyperbola, if we assume that the parabola has a second focus at infinity.

4. Bisectorial properties imply optical properties of the ellipse, hyperbola and parabola, explaining the physical meaning of the term "focus". Let us imagine surfaces formed by the rotation of an ellipse, hyperbola or parabola around the focal axis. If a reflective coating is applied to these surfaces, then elliptical, hyperbolic and parabolic mirrors are obtained. According to the law of optics, the angle of incidence of a light beam on a mirror is equal to the angle of reflection, i.e. the incident and reflected rays form equal angles with the normal to the surface, and both rays and the axis of rotation are in the same plane. From this we get the following properties:

- if the light source is in one of the foci of the elliptical mirror, then the rays of light, reflected from the mirror, are collected in another focus (Fig. 3.52, a);

- if the light source is in one of the foci of the hyperbolic mirror, then the rays of light, reflected from the mirror, diverge as if they came from another focus (Fig. 3.52, b);

- if the light source is at the focus of a parabolic mirror, then the rays of light, reflected from the mirror, go parallel to the focal axis (Fig. 3.52, c).

5. Diametral property ellipse, hyperbola and parabola can be formulated as follows:

– the midpoints of the parallel chords of the ellipse (hyperbola) lie on the same straight line passing through the center of the ellipse (hyperbola);

– the midpoints of the parallel chords of the parabola lie on a straight line, collinear to the axis of symmetry of the parabola.

The locus of the midpoints of all parallel chords of an ellipse (hyperbola, parabola) is called ellipse diameter (hyperbolas, parabolas) conjugate to these chords.

This is the definition of diameter in the narrow sense (see example 2.8). Previously, the definition of diameter was given in a broad sense, where the diameter of an ellipse, hyperbola, parabola, and other second-order lines is a straight line containing the midpoints of all parallel chords. In a narrow sense, the diameter of an ellipse is any chord passing through its center (Fig. 3.53, a); the diameter of a hyperbola is any straight line passing through the center of the hyperbola (with the exception of asymptotes), or part of such a straight line (Fig. 3.53.6); the diameter of a parabola is any ray emanating from some point of the parabola and collinear with the axis of symmetry (Fig. 3.53, c).

Two diameters, each of which bisects all the chords parallel to the other diameter, are called conjugate. In Fig. 3.53, bold lines show the conjugate diameters of an ellipse, hyperbola, and parabola.

The tangent to an ellipse (hyperbola, parabola) at point K can be defined as the limiting position of parallel secants M_1M_2 when points M_1 and M_2, remaining on the line under consideration, tend to point K. It follows from this definition that the tangent parallel to the chords passes through the end of the diameter conjugate to these chords.

6. Ellipse, hyperbola and parabola have, in addition to the above, numerous geometric properties and physical applications. For example, Fig. 3.50 can serve as an illustration of the trajectories of motion of space objects located in the vicinity of the center F of attraction.

Throughout this chapter, it is assumed that a certain scale has been chosen in the plane (in which all the figures considered below lie); only rectangular coordinate systems with this scale are considered.

§ 1. Parabola

A parabola is known to the reader from a school mathematics course as a curve that is a graph of a function

(Fig. 76). (one)

Graph of any square trinomial

is also a parabola; is possible by means of only one shift of the coordinate system (by some vector OO), i.e., transformations

achieve that the graph of the function (in the second coordinate system) coincides with the graph (2) (in the first coordinate system).

Indeed, let us substitute (3) into equality (2). Get

We want to choose so that the coefficient at and the free term of the polynomial (with respect to ) on the right side of this equality are equal to zero. To do this, we determine from the equation

which gives

Now we determine from the condition

into which we substitute the already found value . Get

So, by means of shift (3), in which

we moved on to new system coordinates, in which the parabola equation (2) takes the form

(Fig. 77).

Let's return to equation (1). It can serve as a definition of a parabola. We recall its simplest properties. The curve has an axis of symmetry: if the point satisfies equation (1), then the point symmetrical to the point M about the y-axis also satisfies equation (1) - the curve is symmetrical about the y-axis (Fig. 76).

If , then the parabola (1) lies in the upper half-plane , having a single common point O with the abscissa axis.

With an unlimited increase in the module of the abscissa, the ordinate also increases indefinitely. Give a general view of the curve in fig. 76 a.

If (Fig. 76, b), then the curve is located in the lower half-plane symmetrically with respect to the abscissa axis to the curve.

If we switch to a new coordinate system obtained from the old one by replacing the positive direction of the ordinate axis with the opposite one, then the parabola, which has the equation in the old system, will receive the equation y in the new coordinate system. Therefore, when studying parabolas, we can restrict ourselves to equations (1), in which .

Finally, let's change the names of the axes, i.e., let's move on to the new coordinate system, in which the y-axis will be the old abscissa axis, and the abscissa axis will be the old y-axis. In this new system, equation (1) will be written in the form

Or, if the number is denoted by , in the form

Equation (4) is called in analytic geometry the canonical equation of a parabola; the rectangular coordinate system in which the given parabola has equation (4) is called the canonical coordinate system (for this parabola).

Now we will establish the geometric meaning of the coefficient . For this we take a point

called the focus of the parabola (4), and the straight line d defined by the equation

This line is called the directrix of the parabola (4) (see Fig. 78).

Let be an arbitrary point of the parabola (4). From equation (4) it follows that Therefore, the distance of the point M from the directrix d is the number

The distance of the point M from the focus F is

But, therefore

So, all points M of the parabola are equidistant from its focus and directrix:

Conversely, each point M that satisfies condition (8) lies on the parabola (4).

Indeed,

Consequently,

and, after opening the brackets and bringing like terms,

We have proved that each parabola (4) is the locus of points equidistant from the focus F and from the directrix d of this parabola.

At the same time, we also established the geometric meaning of the coefficient in equation (4): the number is equal to the distance between the focus and the directrix of the parabola.

Now let a point F and a line d not passing through this point be given arbitrarily on the plane. Let us prove that there exists a parabola with focus F and directrix d.

To do this, we draw a line g through the point F (Fig. 79), perpendicular to the line d; the point of intersection of both lines will be denoted by D; the distance (i.e., the distance between the point F and the line d) is denoted by .

We turn the straight line g into an axis, taking the direction DF on it as positive. We will make this axis the abscissa axis of a rectangular coordinate system, the beginning of which is the midpoint O of the segment

Then the line d gets the equation .

Now we can write the canonical parabola equation in the chosen coordinate system:

moreover, the point F will be the focus, and the straight line d will be the directrix of the parabola (4).

We established above that a parabola is the locus of points M equidistant from the point F and the line d. So, we can give such a geometric (i.e., independent of any coordinate system) definition of a parabola.

Definition. A parabola is the locus of points equidistant from some fixed point (the "focus" of the parabola) and some fixed line (the "directrix" of the parabola).

Denoting the distance between the focus and the directrix of the parabola as , we can always find a rectangular coordinate system that is canonical for a given parabola, i.e., one in which the equation of the parabola has a canonical form:

Conversely, any curve having such an equation in some rectangular coordinate system is a parabola (in the geometric sense just established).

The distance between the focus and the directrix of the parabola is called the focal parameter, or simply the parabola parameter.

The line passing through the focus perpendicular to the directrix of the parabola is called its focal axis (or simply the axis); it is the axis of symmetry of the parabola - this follows from the fact that the axis of the parabola is the abscissa axis in the coordinate system, with respect to which the parabola equation has the form (4).

If the point satisfies equation (4), then this equation also satisfies the point , symmetrical to the point M about the x-axis.

The point of intersection of a parabola with its axis is called the top of the parabola; it is the origin of the coordinate system that is canonical for the given parabola.

Let us give one more geometric interpretation of the parabola parameter.

Let us draw a straight line through the focus of the parabola perpendicular to the axis of the parabola; it intersects the parabola at two points (see Fig. 79) and determines the so-called focal chord of the parabola (that is, the chord passing through the focus parallel to the directrix of the parabola). Half the length of the focal chord is the parameter of the parabola.

Indeed, half the length of the focal chord is the absolute value of the ordinate of any of the points, the abscissa of each of which is equal to the abscissa of the focus, i.e. . Therefore, for the ordinate of each of the points we have

Q.E.D.

Definition: A parabola is the locus of points in a plane for which the distance to some fixed point F of this plane is equal to the distance to some fixed straight line. The point F is called the focus of the parabola, and the fixed line is called the directrix of the parabola.

To derive the equation, we construct:

FROM by definition:

Since 2 >=0, then the parabola lies in the right half-plane. As x increases from 0 to infinity
. The parabola is symmetrical with respect to Ox. The point of intersection of a parabola with its axis of symmetry is called the vertex of the parabola.

45. Curves of the second order and their classification. The main theorem about kvp.

There are 8 types of KVP:

1.ellipses

2.hyperbolas

3.parabolas

Curves 1,2,3 are canonical sections. If we intersect the cone with a plane parallel to the axis of the cone, we get a hyperbola. If the plane is parallel to the generatrix, then a parabola. All planes do not pass through the vertex of the cone. If any other plane then an ellipse.

4. pair of parallel lines y 2 + a 2 =0, a0

5. a pair of intersecting lines y 2 -k 2 x 2 \u003d 0

6.one line y 2 =0

7.one point x 2 + y 2 =0

8. empty set - empty curve (cr. without points) x 2 + y 2 +1=0 or x 2 + 1=0

Theorem (the main theorem about KVP): Type equation

a 11 x 2 + 2a 12 x y + a 22 y 2 + 2a 1 x + 2a 2 y+a 0 = 0

can only represent a curve of one of the specified eight types.

Idea of ​​proof is to move to such a coordinate system in which the KVP equation will take on the simplest form, when the type of curve it represents becomes obvious. The theorem is proved by rotating the coordinate system through such an angle at which the term with the product of coordinates vanishes. And with the help of a parallel translation of the coordinate system, in which either the member with the variable x or the member with the variable y disappears.

Transition to a new coordinate system: 1. Parallel transfer

2. Turn

45. Surfaces of the second order and their classification. The main theorem about pvp. Surfaces of revolution.

P VP - a set of points whose rectangular coordinates satisfy the equation of the 2nd degree: (1)

It is assumed that at least one of the coefficients at squares or at products is different from 0. The equation is invariant with respect to the choice of the coordinate system.

Theorem Any plane intersects the PVP along the PVP, except for the special case when the section contains the entire plane. (PVP can be a plane or a pair of planes).

There are 15 types of PVP. We list them by indicating the equations by which they are given in suitable coordinate systems. These equations are called canonical (simple). Build geometric images corresponding to the canonical equations by the method of parallel sections: Cross the surface with coordinate planes and planes parallel to them. The result is sections and curves that give an idea of ​​the shape of the surface.

1. Ellipsoid.

If a=b=c then we get a sphere.

2. Hyperboloids.

one). One-sheeted hyperboloid:

Section of a one-sheeted hyperboloid by coordinate planes: XOZ:
- hyperbole.

YOZ:
- hyperbole.

Plane XOY:
- ellipse.

2). Two-sheeted hyperboloid.

The origin of coordinates is a point of symmetry.

Coordinate planes are planes of symmetry.

Plane z = h intersects the hyperboloid in an ellipse
, i.e. plane z = h begins to intersect the hyperboloid at | h |  c. Cross section of a hyperboloid by planes x = 0 And y = 0 are hyperbole.

Numbers a,b,c in equations (2),(3),(4) are called semiaxes of ellipsoids and hyperboloids.

3. Paraboloids.

one). Elliptical paraboloid:

Plane section z = h eat
, where
. It can be seen from the equation that z  0 is an infinite bowl.

Plane intersection y = h And x= h
is a parabola and

2). Hyperbolic paraboloid:

Obviously, the XOZ and YOZ planes are planes of symmetry, and the z axis is the axis of the paraboloid. Intersection of a paraboloid with a plane z = h- hyperbole:
,
. Plane z=0 intersects a hyperbolic paraboloid along two axes
which are asymptotes.

4. Cone and cylinders of the second order.

one). The cone is the surface
. The cone is framed by straight lines passing through the origin 0 (0, 0, 0). The section of a cone is an ellipse with semi-axes
.

2). Cylinders of the second order.

It's an elliptical cylinder
.

Whatever line we take intersecting the ellipses and parallel to the Oz axis, then it satisfies this equation. By moving this line around the ellipse we get a surface.

G hyperbolic cylinder:

On the HOW plane, this is a hyperbola. We move the line intersecting the hyperbola parallel to Oz along the hyperbola.

Parabolic cylinder:

H and the HOW plane is a parabola.

Cylindrical surfaces are formed by a straight line (generator) moving parallel to itself along a certain straight line (guide).

10. A pair of intersecting planes

11. Pair of parallel planes

12.
- straight

13. Straight line - "cylinder" built on one point

14.One point

15. Empty set

Main theorem about PVP: Each PVP belongs to one of the 15 types discussed above. There are no other PVPs.

Surfaces of revolution. Let the PDCS Oxyz be given and in the plane Oyz the line e defined by the equation F(y,z)=0 (1). Let us compose the equation of the surface obtained by the rotation of this line around the Oz axis. Take a point M(y, z) on the line e. When the plane Oyz rotates around Oz, the point M will describe a circle. Let N(X,Y,Z) be an arbitrary point of this circle. It is clear that z=Z.

.

Substituting the found values ​​of z and y into equation (1), we obtain the correct equality:
those. point N coordinates satisfy the equation
. Thus, any point of the surface of revolution satisfies equation (2). It is not difficult to prove that if the point N(x 1 ,y 1 ,z 1) satisfies equation (2) then it belongs to the considered surface. Now we can say that equation (2) is the desired equation for the surface of revolution.

For the rest of the readers, I propose to significantly replenish their school knowledge about parabola and hyperbola. Hyperbola and parabola - is it simple? … Don't wait =)

Hyperbola and its canonical equation

The general structure of the presentation of the material will resemble the previous paragraph. Let's start with general concept hyperbolas and tasks for its construction.

The canonical equation of a hyperbola has the form , where are positive real numbers. Note that, unlike ellipse, the condition is not imposed here, that is, the value of "a" may be less than the value of "be".

I must say, quite unexpectedly ... the equation of the "school" hyperbole does not even closely resemble the canonical record. But this riddle will still have to wait for us, but for now let's scratch the back of our head and remember what characteristic features the curve under consideration has? Let's spread it on the screen of our imagination function graph ….

A hyperbola has two symmetrical branches.

Good progress! Any hyperbole has these properties, and now we will look with genuine admiration at the neckline of this line:

Example 4

Build a hyperbole given by the equation

Solution: at the first step, we bring this equation to the canonical form . Please remember the typical procedure. On the right, you need to get a “one”, so we divide both parts of the original equation by 20:

Here you can reduce both fractions, but it is more optimal to make each of them three-story:

And only after that to carry out the reduction:

We select the squares in the denominators:

Why is it better to carry out transformations in this way? After all, the fractions of the left side can be immediately reduced and get. The fact is that in the example under consideration, we were a little lucky: the number 20 is divisible by both 4 and 5. In the general case, such a number does not work. Consider, for example, the equation . Here, with divisibility, everything is sadder and without three-story fractions no longer needed:

So, let's use the fruit of our labors - the canonical equation:

How to build a hyperbole?

There are two approaches to constructing a hyperbola - geometric and algebraic.
From a practical point of view, drawing with a compass ... I would even say utopian, so it is much more profitable to again bring simple calculations to the rescue.

It is advisable to adhere to the following algorithm, first finished drawing, then comments:

In practice, a combination of rotation through an arbitrary angle and parallel translation of the hyperbola is often encountered. This situation considered in class Reduction of the 2nd order line equation to the canonical form.

Parabola and its canonical equation

It's done! She is the most. Ready to reveal many secrets. The canonical equation of a parabola has the form , where is a real number. It is easy to see that in its standard position the parabola "lies on its side" and its vertex is at the origin. In this case, the function sets the upper branch of this line, and the function sets the lower branch. Obviously, the parabola is symmetrical about the axis. Actually, what to bathe:

Example 6

Build a parabola

Solution: the vertex is known, let's find additional points. The equation determines the upper arc of the parabola, the equation determines the lower arc.

In order to shorten the record, we will carry out calculations “under the same brush”:

For compact notation, the results could be summarized in a table.

Before performing an elementary point-by-point drawing, we formulate a strict

definition of a parabola:

A parabola is the set of all points in a plane that are equidistant from a given point and a given line that does not pass through the point.

The point is called focus parabolas, straight line headmistress (written with one "es") parabolas. "pe" constant canonical equation called focal parameter, which is equal to the distance from the focus to the directrix. In this case . In this case, the focus has coordinates , and the directrix is ​​given by the equation .
In our example:

The definition of a parabola is even easier to understand than the definitions of an ellipse and a hyperbola. For any point of the parabola, the length of the segment (the distance from the focus to the point) is equal to the length of the perpendicular (the distance from the point to the directrix):

Congratulations! Many of you have made a real discovery today. It turns out that the hyperbola and parabola are not at all graphs of "ordinary" functions, but have a pronounced geometric origin.

Obviously, with an increase in the focal parameter, the branches of the graph will “spread out” up and down, approaching the axis infinitely close. With a decrease in the value of "pe", they will begin to shrink and stretch along the axis

The eccentricity of any parabola is equal to one:

Rotation and translation of a parabola

The parabola is one of the most common lines in mathematics, and you will have to build it really often. Therefore, please pay special attention to the final paragraph of the lesson, where I will analyze typical options location of this curve.

! Note : as in the cases with the previous curves, it is more correct to speak of rotation and parallel translation of the coordinate axes, but the author will limit himself to simplified version presentation, so that the reader has an elementary understanding of these transformations.

We introduce a rectangular coordinate system, where . Let the axis go through the focus F parabola and is perpendicular to the directrix, and the axis passes midway between the focus and the directrix. Denote by the distance between the focus and the directrix. Then the directrix equation.

The number is called the focal parameter of the parabola. Let be the current point of the parabola. Let be the focal radius of a hyperbola point. be the distance from the point to the directrix. Then( drawing 27.)

Drawing 27.

By definition of a parabola. Consequently,

Let's square the equation, we get:

(15)

where (15) is the canonical equation of a parabola symmetric about the axis and passing through the origin.

Investigation of the properties of a parabola

1) Top of the parabola:

Equation (15) is satisfied by numbers and, therefore, the parabola passes through the origin.

2) Parabola symmetry:

Let it belong to a parabola, i.e. a true equality. The point is symmetric to the point about the axis, therefore, the parabola is symmetric about the x-axis.

Parabola eccentricity:

Definition 4.2. The eccentricity of a parabola is a number equal to one.

Since by definition a parabola .

4) Tangent of a parabola:

The tangent to the parabola at the point of tangency is given by the equation

Where ( drawing 28.)

Drawing 28.

Picture of a parabola

Drawing 29.

Using ESO-Mathcad:

drawing 30.)

Drawing 30.

a) Construction without the use of ICT: To construct a parabola, we set a rectangular coordinate system with a center at point O and a unit segment. We mark the focus on the OX axis, since we draw such that, and the directrix of the parabola. We build a circle at a point and with a radius equal to the distance from the straight line to the directrix of the parabola. The circle intersects the line at points. We build a parabola so that it passes through the origin and through the points. ( drawing 31.)

Drawing 31.

b) Using ESO-Mathcad:

The resulting equation has the form: . To construct a second-order line in Mathcad, we bring the equation to the form: .( drawing 32.)

Drawing 32.

To summarize the work on the theory of second-order lines in elementary mathematics and for the convenience of using information about lines in solving problems, we conclude all data on second-order lines in Table No. 1.

Table number 1.

Second order lines in elementary mathematics

Possibilities of using ICT in the study of second-order lines

The process of informatization, which today has covered all aspects of the life of modern society, has several priority areas, which, of course, include the informatization of education. It is the fundamental basis for the global rationalization of human intellectual activity through the use of information and communication technologies (ICT).

The middle of the 90s of the last century and up to the present day, is characterized by the mass character and availability of personal computers in Russia, the widespread use of telecommunications, which makes it possible to introduce the developed information technologies of education into the educational process, improving and modernizing it, improving the quality of knowledge, increasing motivation for learning, making maximum use of the principle of individualization of education. Information technologies of education are a necessary tool at this stage of informatization of education.

Information technologies not only facilitate access to information and open up opportunities for the variability of educational activities, its individualization and differentiation, but also allow organizing the interaction of all subjects of education in a new way, building educational system in which the student would be an active and equal participant in educational activities.

Formation of new information technologies within the framework of subject lessons stimulate the need for the creation of new software and methodological complexes aimed at improving the quality of the lesson. Therefore, for successful and purposeful use in educational process information technology tools, teachers should know general description principles of functioning and didactic capabilities of software and application tools, and then, based on their experience and recommendations, "embed" them into the educational process.

The study of mathematics is currently associated with a number of features and developmental difficulties. school education in our country.

The so-called crisis of mathematical education appeared. Its reasons are as follows:

In the change of priorities in society and in science, that is, at present there is an increase in the priority of the humanities;

In reducing the number of mathematics lessons at school;

In isolation of the content of mathematical education from life;

In a small impact on the feelings and emotions of students.

Today, the question remains open: "How to most effectively use the potential of modern information and communication technologies in teaching schoolchildren, including teaching mathematics?"

A computer is an excellent assistant in studying a topic such as “Quadratic function”, because using special programs you can plot various functions, explore a function, easily determine the coordinates of intersection points, calculate the areas of closed figures, etc. For example, in an algebra lesson in the 9th grade, dedicated to the transformation of the graph (stretching, compression, shifting of the coordinate axes), you can see only the frozen result of the construction, and the entire dynamics of the successive actions of the teacher and student can be traced on the monitor screen.

Computer like no other technical means, accurately, visually and excitingly opens up ideal mathematical models for the student, i.e. what the child should strive for in his practical actions.

How many difficulties a teacher of mathematics has to experience in order to convince students that the tangent to the graph of a quadratic function at the point of contact practically merges with the graph of the function. It is very easy to demonstrate this fact on a computer - it is enough to narrow the interval along the Ox axis and find that in a very small neighborhood of the tangent point, the graph of the function and the tangent coincide. All these activities take place in front of the students. This example gives an impetus to active reflection in the lesson. The use of a computer is possible both in the course of explaining new material in the lesson, and at the control stage. With the help of these programs, for example, "My Test", the student can independently check his level of knowledge in theory, perform theoretical and practical tasks. Programs are convenient for their versatility. They can be used both for self-control and for teacher control.

A reasonable integration of mathematics and computer technology will allow a richer and deeper look at the process of solving a problem, the course of understanding mathematical patterns. In addition, the computer will help to form the graphic, mathematical and mental culture of students, and using the computer you can prepare didactic materials: cards, survey sheets, tests, etc. At the same time, give the children the opportunity to independently develop tests on the topic, during which interest and creativity.

Thus, there is a need to use the computer, if possible, in mathematics lessons more widely than it is. The use of information technology will improve the quality of knowledge, expand the horizons of studying the quadratic function, and therefore help to find new perspectives to maintain students' interest in the subject and the topic, and therefore to a better, more attentive attitude to it. Today, modern information technologies are becoming the most important tool for modernizing the school as a whole - from management to education and ensuring the availability of education.