44. Definition of a parabola. Conclusion canonical equation parabolas.

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:

With 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 under 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 there is
, 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.

Consider a line in the plane and a point that does not lie on this line. And ellipse, and hyperbola can be defined in a unified way as the locus of points for which the ratio of the distance to a given point to the distance to a given straight line is a constant

rank ε. At 0 1 - hyperbole. The parameter ε is the eccentricity of both the ellipse and the hyperbola. Of the possible positive values ​​of the parameter ε, one, namely ε = 1, turns out to be unused. This value corresponds to the locus of points equidistant from the given point and from the given line.

Definition 8.1. The locus of points in a plane equidistant from a fixed point and from a fixed line is called parabola.

The fixed point is called focus of the parabola, and the straight line directrix of the parabola. At the same time, it is assumed that parabola eccentricity is equal to one.

From geometric considerations it follows that the parabola is symmetrical with respect to a straight line perpendicular to the directrix and passing through the focus of the parabola. This line is called the axis of symmetry of the parabola or simply parabola axis. The parabola intersects with its axis of symmetry at a single point. This point is called top of the parabola. It is located in the middle of the segment connecting the focus of the parabola with the point of intersection of its axis with the directrix (Fig. 8.3).

Parabola equation. To derive the parabola equation, we choose on the plane origin at the top of the parabola, as abscissa- the axis of the parabola, the positive direction on which is given by the position of the focus (see Fig. 8.3). This coordinate system is called canonical for the parabola under consideration, and the corresponding variables are canonical.

Let us denote the distance from the focus to the directrix as p. He's called parabola focal parameter.

Then the focus has coordinates F(p/2; 0), and the directrix d is described by the equation x = - p/2. The locus of points M(x; y), equidistant from the point F and from the line d, is given by the equation

We square equation (8.2) and give similar ones. We get the equation

which is called the canonical equation of the parabola.

Note that squaring in this case is an equivalent transformation of equation (8.2), since both parts of the equation are non-negative, as is the expression under the radical.

Type of parabola. If the parabola y 2 \u003d x, the form of which we consider known, is compressed with a coefficient 1 / (2p) along the abscissa axis, then we get a parabola of a general form, which is described by equation (8.3).

Example 8.2. Let us find the coordinates of the focus and the equation of the directrix of the parabola if it passes through a point whose canonical coordinates are (25; 10).

In canonical coordinates, the parabola equation has the form y 2 = 2px. Since the point (25; 10) is on the parabola, then 100 = 50p and therefore p = 2. Therefore, y 2 = 4x is the canonical equation of the parabola, x = - 1 is the equation of its directrix, and the focus is at the point (1; 0 ).

Optical property of a parabola. The parabola has the following optical property. If a light source is placed at the focus of the parabola, then all light rays after reflection from the parabola will be parallel to the axis of the parabola (Fig. 8.4). The optical property means that at any point M of the parabola normal vector the tangent makes the same angles with the focal radius MF and the abscissa axis.

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 switched to a new coordinate system, in which the parabola equation (2) took 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. General form give a 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 old abscissa axis will be the y-axis, and the old y-axis will be the abscissa 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,

Hence,

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).

Everyone knows what a parabola is. But how to use it correctly, competently in solving various practical problems, we will understand below.

First, let us denote the basic concepts that algebra and geometry give to this term. Consider all possible types of this graph.

We learn all the main characteristics of this function. Let's understand the basics of constructing a curve (geometry). Let's learn how to find the top, other basic values ​​of the graph of this type.

We will find out: how the required curve is correctly constructed according to the equation, what you need to pay attention to. Let's see the main practical application of this unique value in human life.

What is a parabola and what does it look like

Algebra: This term refers to the graph of a quadratic function.

Geometry: This is a second-order curve that has a number of specific features:

Canonical parabola equation

The figure shows a rectangular coordinate system (XOY), an extremum, the direction of the function drawing branches along the abscissa axis.

The canonical equation is:

y 2 \u003d 2 * p * x,

where the coefficient p is the focal parameter of the parabola (AF).

In algebra, it is written differently:

y = a x 2 + b x + c (recognizable pattern: y = x 2).

Properties and Graph of a Quadratic Function

The function has an axis of symmetry and a center (extremum). The domain of definition is all values ​​of the x-axis.

The range of values ​​of the function - (-∞, M) or (M, +∞) depends on the direction of the curve branches. The parameter M here means the value of the function at the top of the line.

How to determine where the branches of a parabola are directed

To find the direction of this type of curve from an expression, you need to specify the sign in front of the first parameter of the algebraic expression. If a ˃ 0, then they are directed upwards. Otherwise, down.

How to find the vertex of a parabola using the formula

Finding the extremum is the main step in solving many practical problems. Of course, you can open special online calculators, but it's better to be able to do it yourself.

How to define it? There is a special formula. When b is not equal to 0, we must look for the coordinates of this point.

Formulas for finding the top:

• x 0 \u003d -b / (2 * a);
• y 0 = y (x 0).

Example.

There is a function y \u003d 4 * x 2 + 16 * x - 25. Let's find the vertices of this function.

For such a line:

• x \u003d -16 / (2 * 4) \u003d -2;
• y = 4 * 4 - 16 * 2 - 25 = 16 - 32 - 25 = -41.

We get the coordinates of the vertex (-2, -41).

Parabola offset

The classic case is when in a quadratic function y = a x 2 + b x + c, the second and third parameters are 0, and = 1 - the vertex is at the point (0; 0).

Movement along the abscissa or ordinate axes is due to a change in the parameters b and c, respectively. The shift of the line on the plane will be carried out exactly by the number of units, which is equal to the value of the parameter.

Example.

We have: b = 2, c = 3.

This means that the classical view of the curve will shift by 2 unit segments along the abscissa axis and by 3 along the ordinate axis.

How to build a parabola using a quadratic equation

It is important for schoolchildren to learn how to correctly draw a parabola according to the given parameters.

By analyzing expressions and equations, you can see the following:

1. The point of intersection of the desired line with the ordinate vector will have a value equal to c.
2. All points of the graph (along the x-axis) will be symmetrical with respect to the main extremum of the function.

In addition, the intersections with OX can be found by knowing the discriminant (D) of such a function:

D \u003d (b 2 - 4 * a * c).

To do this, you need to equate the expression to zero.

The presence of parabola roots depends on the result:

• D ˃ 0, then x 1, 2 = (-b ± D 0.5) / (2 * a);
• D \u003d 0, then x 1, 2 \u003d -b / (2 * a);
• D ˂ 0, then there are no points of intersection with the vector OX.

We get the algorithm for constructing a parabola:

• determine the direction of the branches;
• find the coordinates of the vertex;
• find the intersection with the y-axis;
• find the intersection with the x-axis.

Example 1

Given a function y \u003d x 2 - 5 * x + 4. It is necessary to build a parabola. We act according to the algorithm:

1. a \u003d 1, therefore, the branches are directed upwards;
2. extremum coordinates: x = - (-5) / 2 = 5/2; y = (5/2) 2 - 5 * (5/2) + 4 = -15/4;
3. intersects with the y-axis at the value y = 4;
4. find the discriminant: D = 25 - 16 = 9;
5. looking for roots

• X 1 \u003d (5 + 3) / 2 \u003d 4; (4, 0);
• X 2 \u003d (5 - 3) / 2 \u003d 1; (ten).

Example 2

For the function y \u003d 3 * x 2 - 2 * x - 1, you need to build a parabola. We act according to the above algorithm:

1. a \u003d 3, therefore, the branches are directed upwards;
2. extremum coordinates: x = - (-2) / 2 * 3 = 1/3; y = 3 * (1/3) 2 - 2 * (1/3) - 1 = -4/3;
3. with the y-axis will intersect at the value y \u003d -1;
4. find the discriminant: D \u003d 4 + 12 \u003d 16. So the roots:

• X 1 \u003d (2 + 4) / 6 \u003d 1; (1;0);
• X 2 \u003d (2 - 4) / 6 \u003d -1/3; (-1/3; 0).

From the obtained points, you can build a parabola.

Directrix, eccentricity, focus of a parabola

Based on the canonical equation, the focus F has coordinates (p/2, 0).

Straight line AB is a directrix (a kind of parabola chord of a certain length). Her equation is x = -p/2.

Eccentricity (constant) = 1.

Conclusion

We looked at the topic that students study in high school. Now you know, looking at the quadratic function of a parabola, how to find its vertex, in which direction the branches will be directed, whether there is an offset along the axes, and, having a construction algorithm, you can draw its graph.

Function of the form , where is called quadratic function.

Graph of quadratic function − parabola.

Consider the cases:

CASE I, CLASSICAL PARABOLA

I.e , ,

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 that is symmetric about the (x) axis. 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 single 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 a root from 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; 4; 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.