What is the focus of a parabola. Quadratic function

A parabola is the locus of points in the plane equidistant from a given point F and a given straight line d not passing through set point... This geometric definition expresses directory parabola property.

The directory property of a parabola

Point F is called the focus of the parabola, line d is the directrix of the parabola, the middle O of the perpendicular dropped from 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 parabola axis (focal axis of the parabola). The FM segment connecting an arbitrary point M of the parabola with its focus is called the focal radius of the point M. The 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 directory properties and parabolas, we conclude that parabola eccentricity by definition is equal to one (e = 1).

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

Indeed, we introduce a rectangular coordinate system (Figure 3.45, b). The vertex O of the parabola is taken as the origin of the coordinate system; the straight line passing through the focus perpendicular to the directrix is ​​taken as the abscissa axis (the positive direction on it from point O to point F); the straight line perpendicular to the abscissa axis and passing through the vertex of the parabola is taken as the ordinate axis (the direction on the ordinate axis is chosen so that the rectangular coordinate system Oxy is right).

Let's compose the equation of the parabola, using its geometric definition, which expresses the directory property of the parabola. In the selected coordinate system, determine the focus coordinates 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) is the 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} !}... Reducing similar terms, we get canonical parabola equation

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

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

Equation of a parabola in a polar coordinate system

The parabola equation in the polar coordinate system Fr \ varphi (Figure 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 will choose the focus F of the parabola, and as the polar axis - the ray with the origin at point F, perpendicular to the directrix and not crossing it (Figure 3.45, c). Then, for an arbitrary point M (r, \ varphi) belonging to a parabola, according to the geometric definition (directory 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 us explain geometric meaning of the parameter p in the canonical equation of the parabola. Substituting x = \ frac (p) (2) into equation (3.51), we obtain 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 parabola axis.

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 Figure 3.45, c). From the parabola equation in polar coordinates at \ varphi = \ frac (\ pi) (2) we get r = p, i.e. the parameter of the parabola coincides with its focal parameter.

Remarks 3.11.

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

2. The equation y ^ 2 = -2px (for p> 0) defines a parabola, which is located to the left of the ordinate (Fig. 3.47, a). This equation is reduced to the canonical one by changing the direction of the abscissa axis (3.37). In 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), the axis of which 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 a vertex O "(x_0, y_0), the axis of which is parallel to the ordinate axis (Fig. 3.47, c). This equation is reduced to the canonical one using parallel translation (3.36) and renaming the coordinate axes (3.38). 3.47, b, c, the given coordinate systems Oxy and the canonical coordinate systems Ox "y" are shown.

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), the axis of which is parallel to the ordinate axis, the branches of the parabola are directed upward (for a> 0) or down (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 coincide with the sign of the leading coefficient a. This replacement corresponds to composition: parallel transfer (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 the axis of symmetry of the parabola since changing 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), which is symmetric to the point M with respect to 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 the 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 (Figure 3.49). If necessary, we determine the coordinates of some points of the parabola. For example, substituting x = 2 into the parabola equation, we obtain y ^ 2 = 4 ~ \ Leftrightarrow ~ y = \ pm2... Therefore, the points with coordinates (2; 2), \, (2; -2) belong to the parabola.

Comparing for given equation with the 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. Directory property can be used as a single definition of an ellipse, hyperbola, parabola (see Fig. 3.50): the locus of points on 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) a parabola if e = 1.

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

3. Among the general properties of an ellipse, hyperbola and parabola are bisectorial property their tangents. Under tangent to the line at some of its point K is understood the limiting position of the secant KM, when the point M, remaining on the line under consideration, tends to the point K. A straight line perpendicular to the tangent to the line and passing through the tangent point 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 the ellipse or to the hyperbola forms equal angles with the focal radii of the tangent point(Figure 3.51, a, b); the tangent (normal) to the parabola makes equal angles with the focal radius of the tangent point and the perpendicular dropped from it to the directrix(Figure 3.51, c). In other words, the tangent to the ellipse at point K is the bisector of the outer corner of the triangle F_1KF_2 (and the normal is the bisector of the inner corner F_1KF_2 of the triangle); the tangent to the hyperbola is the bisector of the inner angle of the triangle F_1KF_2 (and the normal is the bisector of the outer angle); the tangent to the parabola is the bisector of the inner corner of the triangle FKK_d (and the normal is the bisector of the outer corner). The bisectorial property of the 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 the point at infinity.

4. The bisectorial properties imply optical properties of ellipse, hyperbola and parabola, explaining the physical meaning of the term "focus". 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 ray of light 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. Hence, we obtain the following properties:

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

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

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

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

– the midpoints of the parallel chords of an ellipse (hyperbola) lie on one 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 the diameter of the ellipse (hyperbola, parabola) conjugate to these chords.

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

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

The tangent to the 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 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

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

(fig. 76). (1)

Graph of any square trinomial

is also a parabola; can be done by only shifting the coordinate system (by some vector OO), i.e., transforming

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). We get

We want to choose so that the coefficient at and the free term of the polynomial (with respect to) on the right-hand 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. We get

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

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

(fig. 77).

Let's return to equation (1). It can serve as the definition of a parabola. Let us recall its simplest properties. The curve has an axis of symmetry: if a point satisfies equation (1), then the point is symmetric to point M relative to the ordinate, and also satisfies equation (1) - the curve is symmetric about the ordinate (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 absolute value 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 relative to the abscissa axis to the curve.

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

Finally, we will change the names of the axes, that is, we will switch to a new coordinate system, in which the old abscissa axis will be the ordinate axis, and the old ordinate axis will be the abscissa axis. In this new system, equation (1) will be written as

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

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

We will now establish the geometric meaning of the coefficient. For this we take the point

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

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

Let be an arbitrary point of 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 point M from focus F is

But, therefore

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

Conversely, each point M satisfying condition (8) lies on parabola (4).

Indeed,

Hence,

and, after expanding parentheses and casting similar 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 an arbitrary point F and a straight line d not pass through this point be given on the plane. Let us prove that there is a parabola with focus F and directrix d.

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

Let us 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 origin of which is the midpoint O of the segment

Then the line d also gets an equation.

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

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

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

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

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

A straight 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, relative to which the parabola equation has the form (4).

If a point satisfies equation (4), then this equation is also satisfied by a point symmetric to point M about the abscissa axis.

The point of intersection of a parabola with its axis is called the apex of the parabola; it is the origin of the canonical coordinate system 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 will intersect the parabola at two points (see Fig. 79) and define 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 a locus of points on a plane for which the distance to some fixed point F of this plane is equal to the distance to some fixed straight line. 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:

WITH By definition:

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

45. Curves of the second order and their classification. The main theorem on qp.

There are 8 types of KVP:

1.ellipses

2.hyperboles

3.parabolas

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

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

5.pair of intersecting lines y 2 -k 2 x 2 = 0

6.one straight 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 (main theorem about CVP): Equation of the form

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

can only represent a curve of one of these eight types.

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

Transition to a new coordinate system: 1. Parallel transfer

2. Rotate

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

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

It is assumed that at least one of the coefficients for squares or for 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 KVP, except for the special case when the entire plane is in the section (the 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 specified in suitable coordinate systems. These equations are called canonical (simplest). Construct geometric images corresponding to the canonical equations by the method of parallel sections: Intersect 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.

1). Single-sheet hyperboloid:

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

YOZ:
- hyperbole.

By the XOY plane:
- ellipse.

2). Two-sheet hyperboloid.

The origin is the point of symmetry.

Coordinate planes - 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... Section of a hyperboloid by planes x = 0 and y = 0 are hyperboles.

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

3. Paraboloids.

1). Elliptical paraboloid:

Section by plane z = h there is
, where
... The equation shows that z  0 is an infinite bowl.

Intersection by planes y = h and x= h
is a parabola and in general

2). Hyperbolic paraboloid:

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

4. Cone and cylinders of the second order.

1). The cone is the surface
... The cone is shaped by straight lines passing through the origin of coordinates 0 (0, 0, 0). The cone section is ellipses with semiaxes
.

2). Cylinders of the second order.

This is an elliptical cylinder
.

Whatever line we take that intersects the ellipses and is parallel to the Oz axis, then it satisfies this equation. Moving this line around the ellipse will create a surface.

G hyperbolic cylinder:

On the XOU plane, this is a hyperbola. Move the line intersecting the hyperbola parallel to Oz along the hyperbola.

Parabolic cylinder:

N and the XOU plane is a parabola.

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

10. A pair of intersecting planes

11.A pair of parallel planes

12.
- straight

13. Straight - "cylinder" built on one point

14.One point

15 empty set

The 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 PDSC Oxyz be given and in the plane Oyz the line e determined by the equation F (y, z) = 0 (1). Let's compose the equation of the surface obtained by rotating this line around the Oz axis. Take the point M (y, z) on the line e. When the plane Oyz rotates around Oz, 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. the coordinates of the point N satisfy the equation
... Thus, any point on 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 Eq. (2), then it belongs to the surface under consideration. Now we can say that equation (2) is the required equation for the surface of revolution.

For the rest of the readers, I propose to significantly expand their school knowledge about the parabola and hyperbola. Are hyperbola and parabola easy? ... Can'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 hyperbole and tasks for its construction.

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

I must say, rather unexpectedly ... the equation of the "school" hyperbole does not even come close to resembling the canonical notation. But this riddle will wait for us, but for now, scratch the back of your head and remember what characteristic features the curve in question has? Let's spread our imagination on the screen function graph ….

The hyperbola has two symmetrical branches.

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

Example 4

Construct a hyperbola, 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 "one", so we divide both sides of the original equation by 20:

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

And only after that carry out the reduction:

Select the squares in the denominators:

Why is the transformation better done this way? After all, the fractions of the left side can be immediately reduced and obtained. The fact is that in the example under consideration I was 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, an equation. Here, with divisibility, everything is sadder and without three-story fractions not enough:

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

How to build a hyperbola?

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 it is utopian, so it is much more profitable to re-involve simple calculations to help.

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 a hyperbola is often encountered. This situation considered in the lesson Reducing the equation of the second order line to the canonical form.

Parabola and its canonical equation

It is finished! She is the most. Ready to reveal many secrets. The canonical equation of the 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 the given line, and the function sets the lower branch. Obviously, the parabola is symmetrical about the axis. Actually, why bother:

Example 6

Construct parabola

Solution: the vertex is known, find additional points. The equation defines the upper arc of the parabola, the equation defines the lower arc.

In order to shorten the recording of the calculation, we will carry out "under one comb":

For a compact record, the results could be tabulated.

Before performing an elementary pointwise drawing, we formulate a strict

definition of a parabola:

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

The point is called focus parabolas, straight - headmistress (written with one "es") parabolas. Constant "pe" 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 be "distributed" up and down, infinitely close to the axis. 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 parallel translation of a parabola

The parabola is one of the most common lines in mathematics, and you will have to draw it very often. Therefore, please, pay particular attention to the final paragraph of the lesson, where I will analyze standard variants location of this curve.

! Note : as in the cases with the previous curves, it is more correct to talk about 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.

Let's introduce a rectangular coordinate system, where. Let the axis go through focus F parabola and perpendicular to the directrix, and the axis passes midway between the focus and the directrix. Let us denote by the distance between the focus and the directrix. Then is 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 the point of the hyperbola. Be the distance from the point to the directrix. Then( drawing 27.)

Drawing 27.

By the definition of a parabola. Hence,

Squaring the equation, we get:

(15)

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

Study of the properties of a parabola

1) The vertex of the parabola:

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

2) Symmetry of the parabola:

Let belongs to a parabola, i.e., true equality. The point is symmetrical to the point about the axis, therefore, the parabola is symmetrical about the abscissa axis.

Parabola eccentricity:

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

Since by the definition of a parabola.

4) Tangent parabola:

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

Where ( drawing 28.)

Drawing 28.

Parabola image

Drawing 29.

Using ESP- Mathcad:

drawing 30.)

Drawing 30.

a) Construction without using ICT: To construct a parabola, we set a rectangular coordinate system centered 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 carry out the construction of a circle at a point and a radius equal to the distance from the straight line to the directrix of the parabola. The circle intersects the straight line at points and. We build the parabola so that it passes through the origin and through the points and. ( drawing 31.)

Drawing 31.

b) Using ESP-Mathcad:

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

Drawing 32.

In order to generalize 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 will conclude all data on second-order lines in Table 1.

Table # 1.

Lines of the second order in elementary mathematics

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

The process of informatization, which has embraced all aspects of the life of modern society today, has several priority areas, which, of course, should include the informatization of education. It is the fundamental principle of 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 massiveness and availability of personal computers in Russia, the widespread use of telecommunications, which makes it possible to introduce the developed information technologies of teaching into the educational process, improving and modernizing it, improving the quality of knowledge, increasing the motivation for learning. making the most of the principle of individualization of training. Information technologies of teaching are a necessary tool at this stage of informatization of education.

Information technologies not only facilitate access to information and open up opportunities for variability of educational activity, its individualization and differentiation, but also allow to organize the interaction of all subjects of learning in a new way, to build 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 to create new software and methodological complexes aimed at qualitatively increasing the effectiveness of the lesson. Therefore, for successful and targeted use in educational process information technology tools, educators should know general description principles of functioning and didactic capabilities of software and applied 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 mathematics education appeared. The reasons are as follows:

In the change of priorities in society and in science, that is, the priority of the humanities is currently growing;

Reducing the number of math lessons in school;

In the isolation of the content of mathematical education from life;

Small impact on students' feelings and emotions.

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

A computer is an excellent assistant in the study of such a topic as "Quadratic function", because using special programs you can plot graphs of various functions, examine the function, easily determine the coordinates of intersection points, calculate the areas of closed shapes, etc. For example, at an algebra lesson in the 9th grade, devoted to the transformation of a graph (stretching, compression, translations of the coordinate axes), you can see only the frozen result of the construction, and on the monitor screen you can trace the entire dynamics of the sequential actions of the teacher and the student.

A computer like no other technical means, accurately, visually and fascinatingly reveals ideal mathematical models for the student, i.e. what the child should strive for in his practical actions.

How many difficulties a mathematics teacher has to experience in order to convince students that the tangent to the graph of a quadratic function at the point of tangency practically merges with the graph of the function. It is very simple 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 point of tangency, the graph of the function and the tangent coincide. All these actions take place in front of the students. This example provides an impetus for active reflection in the lesson. Using 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, complete theoretical and practical tasks. The programs are convenient for their versatility. They can be used for both self-control and teacher control.

Reasonable integration of mathematics and computer technology will allow a richer and deeper look at the process of solving a problem, the course of comprehending mathematical laws. In addition, the computer will help form the graphic, mathematical and mental culture of students, and with the help of the computer, you can prepare didactic materials: cards, survey sheets, tests, etc. creativity.

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