A parabola is an infinite curve that consists of points equidistant from a given line, called the directrix of the parabola, and a given point, the focus of the parabola. A parabola is a conic section, that is, it represents the intersection of a plane and a circular cone.
IN general view the mathematical equation of a parabola has the form: y=ax^2+bx+c, where a is not equal to zero, b reflects the horizontal displacement of the function graph relative to the origin, and c is the vertical displacement of the function graph relative to the origin. Moreover, if a>0, then when plotting the graph they will be directed upward, and if aProperties of the parabola
A parabola is a second-order curve that has an axis of symmetry passing through the focus of the parabola and perpendicular to the directrix of the parabola.
A parabola has a special optical property, which consists in focusing light rays parallel to its axis of symmetry and directed into the parabola at the vertex of the parabola and defocusing a beam of light directed at the vertex of the parabola into parallel light rays relative to the same axis.
If you reflect a parabola relative to any tangent, then the image of the parabola will appear on its directrix. All parabolas are similar to each other, that is, for every two points A and B of one parabola, there are points A1 and B1 for which the statement |A1,B1| = |A,B|*k, where k is the similarity coefficient, which in numerical value is always greater than zero.
Manifestation of a parabola in life
Some cosmic bodies, such as comets or asteroids, passing near large space objects on high speed have a trajectory in the shape of a parabola. This property of small cosmic bodies is used in gravitational maneuvers of spacecraft.
To train future cosmonauts, special aircraft flights are carried out on the ground along a parabolic trajectory, thereby achieving the effect of weightlessness in the gravitational field of the earth.
In everyday life, parabolas can be found in various lighting fixtures. This is due to the optical property of a parabola. One of the latest ways to use a parabola, based on its properties of focusing and defocusing light rays, is solar panels, which are increasingly included in the energy supply sector in the southern regions of Russia.
A function of the form where is called quadratic function.
Graph of a quadratic function – parabola.
Let's consider the cases:
I CASE, CLASSICAL PARABOLA
That is , ,
To construct, fill out the table by substituting the x values into the formula:
Mark the points (0;0); (1;1); (-1;1), etc. on the coordinate plane (the smaller the step we take the x values (in in this case step 1), and the more x values we take, the smoother the curve will be), we get a parabola:
It is easy to see that if we take the case , , , that is, then we get a parabola that is symmetrical about the axis (oh). It’s easy to verify this by filling out a similar table:
II CASE, “a” IS DIFFERENT FROM UNIT
What will happen if we take , , ? How will the behavior of the parabola change? With title="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):!}
In the first picture (see above) it is clearly visible 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 reason similarly in the cases of pictures 2 and 3.
And when the parabola “becomes wider” than the parabola:
Let's summarize:
1)The sign of the coefficient determines the direction of the branches. With title="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” and “compression” of the parabola. The larger , the narrower the parabola; the smaller |a|, the wider the parabola.
III CASE, “C” APPEARS
Now let's introduce into the game (that is, consider the case when), we will consider parabolas of the form . It is not difficult to guess (you can always refer to the table) that the parabola will shift up or down along the axis depending on the sign:
IV CASE, “b” APPEARS
When will the parabola “break away” from the axis and finally “walk” along the entire coordinate plane? When will it stop being equal?
Here to construct a parabola we need formula for calculating the vertex: , .
So at this point (as at point (0;0) new system coordinates) we will build a parabola, which we can already do. If we are dealing with the case, then from the vertex we put one unit segment to the right, one up, - the resulting point is ours (similarly, a step to the left, a step up is our point); if we are dealing with, for example, then from the vertex we put one unit segment to the right, two - upward, 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 pattern, because in our case.
When constructing a parabola after finding the coordinates of the vertex veryIt is convenient to consider the following points:
1) parabola will definitely pass through the point . Indeed, substituting x=0 into the formula, we obtain that . That is, the ordinate of the point of intersection of the parabola with the axis (oy) is . In our example (above), the parabola intersects the ordinate at point , 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 it symmetrical relative to the symmetry axis of the parabola, 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 (oh). To do this, we solve the equation. Depending on the discriminant, we will get one (, ), two ( title="Rendered by QuickLaTeX.com" height="14" width="54" style="vertical-align: 0px;">, ) или нИсколько () точек пересечения с осью (ох) !} . In the previous example, our root of the discriminant is not an integer; when constructing, it doesn’t make much sense for us to find the roots, but we clearly see that we will have two points of intersection with the axis (oh) (since title="Rendered by QuickLaTeX.com" height="14" width="54" style="vertical-align: 0px;">), хотя, в общем, это видно и без дискриминанта.!}
So let's work it 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) we find the coordinates of the vertex of the parabola using the formula , .
3) we find the point of intersection of the parabola with the axis (oy) using the free term, construct a point symmetrical to this point with respect to the symmetry axis 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 vertex of the parabola (as at the point (0;0) of the new coordinate system) we construct a parabola. If title="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 have not yet “surfaced”) by solving the equation
Example 1
Example 2
Note 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 construct it, because we have already been given the coordinates of the vertex . Why?
Let's take a quadratic trinomial and isolate the complete square in it: Look, we got that , . You and I previously called the vertex of a parabola, that is, now,.
For example, . We mark the vertex of the parabola on the plane, we understand that the branches are directed downward, the parabola is expanded (relative to ). That is, we carry out points 1; 3; 4; 5 from the algorithm for constructing a parabola (see above).
Note 2. If the parabola is given in a form similar to this (that is, presented as a product of two linear factors), then we immediately see the points of intersection of the parabola with the axis (ox). In this case – (0;0) and (4;0). For the rest, we act according to the algorithm, opening the brackets.
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 given point. This geometric definition expresses directorial property of a parabola.
Directorial property of a parabola
Point F is called the focus of the parabola, line d is the directrix of the parabola, the midpoint O of the perpendicular lowered 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 is the focal length (Fig. 3.45a). The straight line perpendicular to the directrix and passing through the focus is called the axis of the parabola (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 segment connecting two points of a parabola is called a chord of the parabola.
For an arbitrary point of a parabola, the ratio of the distance to the focus to the distance to the directrix is equal to one. Comparing the directorial properties of , and parabolas, we conclude that parabola eccentricity by definition equal to one (e=1).
Geometric definition of a parabola, expressing its directorial property, is equivalent to its analytical definition - the line given by canonical equation parabolas:
Indeed, let us introduce a rectangular coordinate system (Fig. 3.45, b). We take the vertex O of the parabola as the origin of the coordinate system; we take the straight line passing through the focus perpendicular to the directrix as the abscissa axis (the positive direction on it is from point O to point F); Let us take the straight line perpendicular to the abscissa axis and passing through the vertex of the parabola as the ordinate axis (the direction on the ordinate axis is chosen so that the rectangular coordinate system Oxy is right).
Let's create an equation for a parabola using its geometric definition, which expresses the directorial property of a 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) - orthographic projection points M(x,y) to 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 similar 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 points whose coordinates satisfy equation (3.51), and only they, belong to the locus of points called a parabola. Thus, the analytical definition of a parabola is equivalent to its geometric definition, which expresses the directorial property of a parabola.
Parabola equation in polar coordinate system
The equation of a parabola 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 a beginning at point F, perpendicular to the directrix and not intersecting it (Fig. 3.45, c). Then for an arbitrary point M(r,\varphi) belonging to a parabola, according to the geometric definition (directional property) of a parabola, we have MM_d=r. Because the 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 ).
Geometric meaning of the parameter in the parabola equation
Let's explain geometric meaning parameter p in the canonical parabola equation. 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 chord of the parabola passing through its focus perpendicular to the axis of the parabola.
The focal parameter of the parabola, as well as for an ellipse and 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. the parameter of the parabola coincides with its focal parameter.
Notes 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 (Fig. 3.46).
2. The equation y^2=-2px (for p>0) defines a parabola, which is located to the left of the ordinate axis (Fig. 3.47,a). This equation is reduced to the canonical one by changing the direction of the x-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 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). In Fig. 3.47,b,c depict the given coordinate systems Oxy and the canonical coordinate systems Ox"y".
4. y=ax^2+bx+c,~a\ne0 is a parabola with vertex 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 downward (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|, using replacement 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 the 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 x-axis of the canonical coordinate system is axis of symmetry of the parabola, since replacing the variable y with -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 relative to the x-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, focal coordinates and directrix equation.
Solution. We construct a parabola, taking into account its symmetry relative to the abscissa axis (Fig. 3.49). If necessary, 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. Consequently, points with coordinates (2;2),\,(2;-2) belong to the parabola.
Comparing the given equation with the canonical one (3.S1), we determine 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 equation of the directrix x=-\frac(p)(2) , i.e. x=-\frac(1)(2) .
General properties of ellipse, hyperbola, parabola
1. The directorial 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) not passing through a given point is constant and equal to eccentricity e, is called:
a) if 0\leqslant e<1 ;
b) if e>1;
c) parabola if e=1.
2. An ellipse, hyperbola, and parabola are obtained as planes in sections of a circular cone and are therefore called conic sections. This property can also serve as a geometric definition of an ellipse, hyperbola, and parabola.
3. Common properties of the ellipse, hyperbola and parabola include bisectoral property their tangents. Under tangent to a line at some point K is understood to be 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 a tangent to a line and passing through the point of tangency is called normal to this line.
The bisectoral property of tangents (and normals) to an ellipse, hyperbola and parabola is formulated as follows: the tangent (normal) to an ellipse or to a 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 point of tangency 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 external angle of the triangle F_1KF_2 (and the normal is the bisector of the internal angle F_1KF_2 of the triangle); 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 internal angle of the triangle FKK_d (and the normal is the bisector of the external angle). The bisectoral 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 a point at infinity.
4. From the bisectoral properties it follows optical properties of ellipse, hyperbola and parabola, explaining the physical meaning of the term "focus". Let us imagine surfaces formed by rotating an ellipse, hyperbola or parabola around the focal axis. If a reflective coating is applied to these surfaces, elliptical, hyperbolic and parabolic mirrors are obtained. According to the law of optics, the angle of incidence of a light ray 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 here we get the following properties:
– if the light source is located at one of the focuses of an elliptical mirror, then the rays of light, reflected from the mirror, are collected at another focus (Fig. 3.52, a);
– if the light source is located in one of the focuses of a 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 light rays, reflected from the mirror, go parallel to the focal axis (Fig. 3.52, c).
5. Diametric property ellipse, hyperbola and parabola can be formulated as follows:
– the midpoints of parallel chords of an ellipse (hyperbola) lie on one straight line passing through the center of the ellipse (hyperbola);
– the midpoints of parallel chords of a parabola lie on the straight, collinear axis of symmetry of the parabola.
The geometric locus of the midpoints of all parallel chords of an ellipse (hyperbola, parabola) is called diameter of the ellipse (hyperbola, parabola), conjugate to these chords.
This is the definition of diameter in the narrow sense (see example 2.8). Previously, a 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 a certain point of the parabola and collinear to the axis of symmetry (Fig. 3.53, c).
Two diameters, each of which bisects all 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 the ellipse (hyperbola, parabola) at point K can be defined as the limit position of parallel secants M_1M_2, when points M_1 and M_2, remaining on the line under consideration, tend to point K. From this definition it follows that a 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 those given 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 of gravity F.
Consider a line on the plane and a point not lying on this line. AND ellipse, And hyperbola can be defined in a unified way as the geometric 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 value
rank ε. At 0 1 - hyperbola. The parameter ε is eccentricity of both ellipse and hyperbola. Of the possible positive values of the parameter ε, one, namely ε = 1, turns out to be unused. This value corresponds to the geometric locus of points equidistant from a given point and from a 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 a parabola. At the same time, it is believed that parabola eccentricity equal to one.
From geometric considerations it follows that the parabola is symmetrical with respect to the straight line perpendicular to the directrix and passing through the focus of the parabola. This straight line is called the axis of symmetry of the parabola or simply the axis of the parabola. A parabola intersects its axis of symmetry at a single point. This point is called the vertex 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 equation of a parabola, we choose on the plane origin at the vertex of the parabola, as x-axis- the axis of the parabola, the positive direction on which is specified by the position of the focus (see Fig. 8.3). This coordinate system is called canonical for the parabola in question, and the corresponding variables are canonical.
Let us denote the distance from the focus to the directrix by p. He is called focal parameter of the parabola.
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
Let us square equation (8.2) and present similar ones. We get the equation
which is called canonical parabola equation.
Note that squaring in this case is an equivalent transformation of equation (8.2), since both sides of the equation are non-negative, as is the expression under the radical.
Type of parabola. If the parabola y 2 = x, the form of which we consider known, is compressed with a coefficient 1/(2р) along the abscissa axis, then a parabola of general form is obtained, 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 a parabola if it passes through a point whose canonical coordinates are (25; 10).
In canonical coordinates, the equation of the parabola 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 equal angles with the focal radius MF and the abscissa axis.
Definition 1. Parabola is the set of all points of the plane, each of which is equally distant from a given point, called focus, and from a given line not passing through a given point and called headmistress.
Let's create an equation for a parabola with focus at a given point F and whose directrix is the line d, not passing through F. Let us choose a rectangular coordinate system as follows: axis Oh let's go through the focus F perpendicular to the director d in the direction from d To F, and the origin ABOUT Let's place it in the middle between the focus and the directrix (Fig. 1).
Definition 2. Focus distance F to the headmistress d called parabola parameter and is denoted by p(p> 0).
From Fig. 1 it is clear that p = FK, therefore the focus has coordinates F (p/2; 0), and the directrix equation has the form X= – r/2, or
Let M(x;y) is an arbitrary point of the parabola. Let's connect the dots M With F and we'll spend MN d. Directly from Fig. 1 it is clear that
and according to the formula for the distance between two points 
According to the definition of a parabola, MF = MN, (1)
hence, 
(2)
Equation (2) is the required parabola equation. To simplify equation (2), we transform it as follows:
those.,
Coordinates X And at points M parabolas satisfy condition (1), and therefore equation (3).
Definition 3. Equation (3) is called the canonical equation of a parabola.
2. Study of the shape of a parabola using its equation. Let us determine the shape of the parabola using its canonical equation (3).
1) Point coordinates O (0; 0) satisfy equation (3), therefore, the parabola defined by this equation passes through the origin.
2) Since in equation (3) the variable at included only in even degree, then the parabola y 2 = 2px symmetrical about the abscissa axis.
3) Since p > 0, then from (3) it follows x ≥ 0. Consequently, the parabola y 2 = 2px located to the right of the axis OU.
4) As the abscissa increases X from 0 to +∞ ordinate at varies from 0 before ± ∞, i.e. the points of the parabola move unlimitedly away from the axis Oh, and from the axis OU.
Parabola y 2 = 2px has the shape shown in Fig. 2.
Definition 4. Axis Oh called axis of symmetry of the parabola. Dot O (0; 0) the intersection of a parabola with the axis of symmetry is called the vertex of the parabola. Line segment FM called focal radius points M.
Comment. To create a parabola equation of the form y 2 = 2px we specially chose a rectangular coordinate system (see point 1). If the coordinate system is chosen in a different way, then the equation of the parabola will have a different form.
A
So, for example, if you direct the axis Oh from focus to director (Fig. 3, a
y 2 = –2px. (4)
F(–р/2; 0), and the headmistress d given by the equation x = p/2.
If the axis OU let's go through the focus F d in the direction from d To F, and the origin ABOUT place it in the middle between the focus and the directrix (Fig. 3, b), then the equation of a parabola is an example of the form
x 2 = 2ru . (5)
The focus of such a parabola has coordinates F (0; p/2), and the headmistress d given by the equation y=–p/2.
If the axis OU let's go through the focus F perpendicular to the director d in the direction from F To d(Fig. 3, V), then the equation of the parabola takes the form
x 2 = –2ru (6)
Its focus coordinates will be F (0; –р/2), and the directrix equation d will y = p/2.
Equations (4), (5), (6) are said to have the simplest form.
3. Parallel transfer of a parabola. Let a parabola be given with its vertex at the point O" (a; b), the axis of symmetry of which is parallel to the axis OU, and the branches are directed upward (Fig. 4). You need to create an equation for a parabola.
(9)
Definition 5. Equation (9) is called equation of a parabola with a displaced vertex.
Let's transform this equation as follows:
Putting 
will have 
(10)
It is not difficult to show that for any A, B, C schedule quadratic trinomial(10) is a parabola in the sense of Definition 1. A parabola equation of the form (10) was studied in a school algebra course.
EXERCISES FOR INDEPENDENT SOLUTION
No. 1. Write the equation of a circle:
a. with center at the origin and radius 7;
b. with center at point (-1;4) and radius 2.
Construct the circle data in a rectangular Cartesian coordinate system.
No. 2. Compose the canonical equation of an ellipse with vertices
and tricks 
No. 3. Construct an ellipse given by the canonical equation:
1) 2) 
No. 4. Compose the canonical equation of an ellipse with vertices
and tricks 
No. 5. Compose the canonical equation of a hyperbola with vertices
and tricks
No. 6. Compose the canonical equation of a hyperbola if:
1. distance between foci and between vertices
2. real semi-axis, and eccentricity;
3. focuses on the axis, the real axis is 12, and the imaginary axis is 8.
No. 7. Construct a hyperbola given by the canonical equation:
1) 2) 
.
No. 8. Write the canonical equation of a parabola if:
1) the parabola is located in the right half-plane symmetrically relative to the axis and its parameter;
2) the parabola is located in the left half-plane symmetrically relative to the axis and its parameter is .
Construct these parabolas, their foci and directrixes.
No. 9. Determine the type of line if its equation is:
SELF-TEST QUESTIONS
1. Vectors in space.
1.1. What is a vector?
1.2. What is the absolute magnitude of a vector?
1.3. What types of vectors in space do you know?
1.4. What actions can you perform with them?
1.5. What are vector coordinates? How to find them?
2. Actions on vectors specified by their coordinates.
2.1. What actions can be performed with vectors given in coordinate form (rules, equalities, examples); how to find the absolute value of such a vector.
2.2. Properties:
2.2.1 collinear;
2.2.2 perpendicular;
2.2.3 coplanar;
2.2.4 equal vectors.
(formulations, equalities).
3. Equation of a straight line. Applied problems.
3.1. What types of equations of a straight line do you know (be able to write and interpret from the recording);
3.2. How to examine for parallelism - perpendicularity two straight lines specified by equations with an angular coefficient or general equations?
3.3. How to find the distance from a point to a line between two points?
3.4. How to find the angle between lines given by general line equations or slope equations?
3.5. How to find the coordinates of the midpoint of a segment and the length of this segment?
4. Equation of a plane. Applied problems.
4.1. What types of plane equations do you know (be able to write and interpret from the recording)?
4.2. How to examine parallelism and perpendicularity of straight lines in space?
4.3. How to find the distance from a point to a plane and the angle between the planes?
4.4. How to explore mutual arrangement straight line and plane in space?
4.5. Types of equation of a line in space: general, canonical, parametric, passing through two given points.
4.6. How to find the angle between straight lines and the distance between points in space?
5. Lines of the second order.
5.1. Ellipse: definition, foci, vertices, major and minor axes, focal radii, eccentricity, directrix equations, simplest (or canonical) equations of the ellipse; drawing.
5.2. Hyperbola: definition, foci, vertices, real and imaginary axes, focal radii, eccentricity, directrix equations, simplest (or canonical) hyperbola equations; drawing.
5.3. Parabola: definition, focus, directrix, vertex, parameter, axis of symmetry, simplest (or canonical) equations of a parabola; drawing.
Note to 4.1, 4.2, 4.3: For each 2nd order line, be able to describe the construction.
SELF-TEST TASKS
1.Given points: 
, where N is the student number on the list.
3) find the distance from point M to plane P.
4. Construct a second-order line given by its canonical equation:
.
LITERATURE
1. Higher mathematics for economists - Textbook for universities, ed. N.Sh. Kremer et al., Moscow, UNITY, 2003.
2. Barkovsky V.V., Barkovska N.V. - Vischa mathematics for economists – Kiev, TsUL, 2002.
3. Suvorov I.F. - Course of higher mathematics. - M., Higher School, 1967.
4. Tarasov N.P. - Course of higher mathematics for technical schools. - M.; Science, 1969.
5. Zaitsev I.L. - Elements of higher mathematics for technical schools. - M.; Science, 1965.
6. Valutse N.N., Diligul G.D. - Mathematics for technical schools. - M.; Science, 1990.
7. Shipachev V.S. - Higher mathematics. Textbook for universities - M.: Higher School, 2003.
