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 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 the point (0;0) of the new coordinate system) 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.
Class 10 . Second order curves.
10.1. Ellipse. Canonical equation. Semi-axes, eccentricity, graph.
10.2. Hyperbola. Canonical equation. Semi-axes, eccentricity, asymptotes, graph.
10.3. Parabola. Canonical equation. Parabola parameter, graph.
Second-order curves on a plane are lines whose implicit definition has the form:
Where 
- given real numbers, 
- coordinates of the curve points. The most important lines among second-order curves are the ellipse, hyperbola, and parabola.
10.1. Ellipse. Canonical equation. Semi-axes, eccentricity, graph.
Definition of an ellipse.An ellipse is a plane curve whose sum of distances from two fixed points is 
plane to any point 
(those.). Points 
are called the foci of the ellipse.
Canonical ellipse equation:
.
(2)
(or axis 
) goes through tricks 
, and the origin is the point 
- is located in the center of the segment 
(Fig. 1). Ellipse (2) is symmetrical about the coordinate axes and the origin (the center of the ellipse). Permanent 
,
are called semi-axes of the ellipse.
If the ellipse is given by equation (2), then the foci of the ellipse are found like this.
1) First, we determine where the foci lie: the foci lie on the coordinate axis on which the major semi-axes are located.
2) Then the focal length is calculated 
(distance from foci to origin).
At 
foci lie on the axis 
;
;
.
At 
foci lie on the axis 
;
;
.
Eccentricity ellipse is called the quantity: 
(at 
);
(at 
).
The ellipse always 
. Eccentricity serves as a characteristic of the compression of the ellipse.
If the ellipse (2) is moved so that the center of the ellipse hits the point 
,
, then the equation of the resulting ellipse has the form
.
10.2. Hyperbola. Canonical equation. Semi-axes, eccentricity, asymptotes, graph.
Definition of hyperbole.A hyperbola is a plane curve in which the absolute value of the difference in distances from two fixed points is 
plane to any point 
this curve has a constant value independent of the point 
(those.). Points 
are called the foci of a hyperbola.
Canonical hyperbola equation:
or 
.
(3)
This equation is obtained if the coordinate axis 
(or axis 
) goes through tricks 
, and the origin is the point 
- is located in the center of the segment 
. Hyperbolas (3) are symmetrical about the coordinate axes and the origin. Permanent 
,
are called semi-axes of the hyperbola.
The foci of a hyperbole are found like this.
At the hyperbole 
foci lie on the axis 
:
(Fig. 2.a).
At the hyperbole 
foci lie on the axis 
:
(Fig. 2.b)
Here 
- focal length (distance from the foci to the origin). It is calculated by the formula: 
.
Eccentricity hyperbola is the quantity:
(For 
);
(For 
).
Hyperbole always has 
.
Asymptotes of hyperbolas(3) are two straight lines: 
. Both branches of the hyperbola approach the asymptotes without limit with increasing 
.
The construction of a hyperbola graph should be carried out as follows: first along the semi-axes 
we build an auxiliary rectangle with sides parallel to the coordinate axes; then draw straight lines through the opposite vertices of this rectangle, these are the asymptotes of the hyperbola; finally we depict the branches of the hyperbola, they touch the midpoints of the corresponding sides of the auxiliary rectangle and get closer with growth 
to asymptotes (Fig. 2).
If hyperbolas (3) are moved so that their center hits the point 
, and the semi-axes will remain parallel to the axes 
,
, then the equation of the resulting hyperbolas will be written in the form
,
.
10.3. Parabola. Canonical equation. Parabola parameter, graph.
Definition of a parabola.A parabola is a plane curve for which, for any point 
this curve is the distance from 
to a fixed point 
plane (called the focus of the parabola) is equal to the distance from 
to a fixed straight line on the plane(called the directrix of the parabola) .
Canonical parabola equation:
,
(4)
Where 
- a constant called parameter parabolas.
Dot 
parabola (4) is called the vertex of the parabola. Axis 
is the axis of symmetry. The focus of the parabola (4) is at the point 
, directrix equation 
. Parabola graphs (4) with meanings 
And 
are shown in Fig. 3.a and 3.b respectively.
The equation 
also defines a parabola on the plane 
, whose axes, compared to parabola (4), 
,
switched places.
If parabola (4) is moved so that its vertex hits the point 
, and the axis of symmetry will remain parallel to the axis 
, then the equation of the resulting parabola has the form
.
Let's move on to examples.
Example 1. The second order curve is given by the equation 
. Give a name to this curve. Find its foci and eccentricity. Draw a curve and its foci on a plane 
.
Solution. This curve is an ellipse centered at the point 
and axle shafts 
. This can be easily verified by replacing 
. This transformation means a transition from a given Cartesian coordinate system 
to a new Cartesian coordinate system 
, whose axis 
parallel to the axes 
,
. This coordinate transformation is called a system shift 
exactly 
. In the new coordinate system 
the equation of the curve is transformed into the canonical equation of the ellipse 
, its graph is shown in Fig. 4.
Let's find tricks. 
, so the tricks 
ellipse located on the axis 
.. In the coordinate system 
:
. Because 
, in the old coordinate system 
foci have coordinates.
Example 2. Give the name of the second-order curve and provide its graph.
Solution. Let us select perfect squares based on terms containing variables 
And 
.
Now, the equation of the curve can be rewritten as follows:
Therefore, the given curve is an ellipse centered at the point 
and axle shafts 
. The information obtained allows us to draw its graph.
Example 3. Give a name and graph of the line 
.
Solution. . This is the canonical equation of an ellipse centered at the point 
and axle shafts 
.
Because the, 
, we conclude: the given equation determines on the plane 
the lower half of the ellipse (Fig. 5).
Example 4. Give the name of the second order curve 
. Find its focuses, eccentricity. Give a graph of this curve.
- canonical equation of a hyperbola with semi-axes 
.
Focal length.
The minus sign precedes the term with 
, so the tricks 
hyperbolas lie on the axis 
:. The branches of the hyperbola are located above and below the axis 
.
- eccentricity of the hyperbola.
Asymptotes of a hyperbola: .
The construction of a graph of this hyperbola is carried out in accordance with the procedure outlined above: we build an auxiliary rectangle, draw asymptotes of the hyperbola, draw branches of the hyperbola (see Fig. 2.b).
Example 5. Find out the type of curve given by the equation 
and plot it.
- hyperbola with center at a point 
and axle shafts.
Because , we conclude: the given equation determines that part of the hyperbola that lies to the right of the straight line 
. It is better to draw a hyperbola in an auxiliary coordinate system 
, obtained from the coordinate system 
shift 
, and then highlight the desired part of the hyperbola with a bold line
Example 6. Find out the type of curve and draw its graph.
Solution. Let us select a complete square based on the terms with the variable 
:
Let's rewrite the equation of the curve.
This is the equation of a parabola with its vertex at the point 
. Using a shift transformation, the parabola equation is brought to the canonical form 
, from which it is clear that is a parabola parameter. Focus 
parabolas in the system 
has coordinates 
,, and in the system 
(according to shift transformation). The parabola graph is shown in Fig. 7.
Homework.
1. Draw ellipses given by the equations: 
Find their semi-axes, focal length, eccentricity and indicate on the graphs of ellipses the locations of their foci.
2. Draw hyperbolas given by the equations: 
Find their semi-axes, focal length, eccentricity and indicate on the hyperbola graphs the locations of their foci. Write equations for the asymptotes of the given hyperbolas.
3. Draw parabolas given by the equations: 
. Find their parameter, focal length, and indicate on the parabola graphs the location of the focus.
4. Equation 
defines the 2nd order part of the curve. Find the canonical equation of this curve, write down its name, plot its graph and highlight on it that part of the curve that corresponds to the original equation.
A parabola is the locus of points in the plane that are equidistant from a given point F and a given straight line d that does not pass through the 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 defined by the canonical equation of a parabola:
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)- orthogonal projection of the point M(x,y) onto the directrix. We write this equation in coordinate form:
\sqrt((\left(x-\frac(p)(2)\right)\^2+y^2}=x+\frac{p}{2}. !}
We square both sides of the equation: (\left(x-\frac(p)(2)\right)\^2+y^2=x^2+px+\frac{p^2}{4} !}. Bringing similar terms, we get canonical parabola equation
y^2=2\cdot p\cdot x, those. the chosen coordinate system is canonical.
Carrying out the reasoning in reverse order, we can show 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 of the 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.
- (Greek parabole, from parabollo bringing closer together). 1) allegory, parable. 2) a curved line originating from a section of a cone by a plane parallel to some of its generating planes. 3) a curved line formed during the flight of a bomb, cannonball, etc. Dictionary... ... Dictionary of foreign words of the Russian language
Allegory, parable (Dahl) See example... Synonym dictionary
- (Greek parabole) flat curve (2nd order). A parabola is a set of points M whose distances to a given point F (focus) and to a given straight line D1D2 (directrix) are equal. In the proper coordinate system, the equation of the parabola has the form: y2=2px, where p=2OF.… … Big Encyclopedic Dictionary
PARABOLA, mathematical curve, CONIC SECTION formed by a point moving in such a way that its distance to a fixed point, the focus, is equal to its distance to a fixed straight line, the directrix. A parabola is formed when cutting a cone... ... Scientific and technical encyclopedic dictionary
Female, Greek allegory, parable. | mat. curved line, from among conic sections; cut the sugar loaf obliquely, parallel to the opposite side. Parabolic calculations. Parabolic speech, heterogeneity, foreign speech, figurative... ... Dahl's Explanatory Dictionary
parabola- y, w. parabol f. gr. parabol. 1. outdated Parable, allegory. BAS 1. The Frenchman, wanting to laugh at the Russian coming to Paris, asked: What does parabol, faribol and obol mean? But he soon answered him: Parabolus, there is something that you do not understand;... ... Historical Dictionary of Gallicisms of the Russian Language
PARABOLA- (1) an open curved line of the 2nd order on the plane, which is a graph of the function y2 = 2px, where p is the parameter. A parabola is obtained when a circular plane (see) intersects a plane that does not pass through its vertex and is parallel to one of its generators.... ... Big Polytechnic Encyclopedia
- (from the Greek parabole), a flat curve, the distances of any point M of which to a given point F (focus) and to a given straight line D 1D1 (directrix) are equal (MD=MF) ... Modern encyclopedia
PARABOLA, parabolas, women. (Greek: parabole). 1. A second-order curve representing a conical section of a right circular cone by a plane parallel to one of the generatrices (mat.). || The path described by a heavy body (for example, a bullet) thrown under... ... Ushakov's Explanatory Dictionary
PARABOLA, s, female. In mathematics: an open curve consisting of one branch that is formed when a plane intersects a conical surface. | adj. parabolic, oh, oh. Ozhegov's explanatory dictionary. S.I. Ozhegov, N.Yu. Shvedova. 1949 1992 … Ozhegov's Explanatory Dictionary
- “PARABOLA”, Russia, 1992, color, 30 min. Documentary essay. An attempt to understand the mystical essence of the tales of the Udmurts, a small people in the Volga region. Director: Svetlana Stasenko (see Svetlana STASENKO). Scriptwriter: Svetlana Stasenko (see STASENKO... ... Encyclopedia of Cinema
Books
- Parabola of the dream job search plan. Archetypes of HR managers..., Marina Zorina. Marina Zorina’s book “The Parabola of the Dream Job Search Plan” is based on the author’s real experience and is filled with useful information regarding the patterns of the internal recruitment process.…
- Parabola of my life, Titta Ruffo. The author of the book is the most famous Italian singer, soloist of the world's leading opera houses. Titta Ruffo's memoirs, written vividly and directly, contain sketches of the theatrical life of the first...
Probably everyone knows what a parabola is. But we’ll look at how to use it correctly and competently when solving various practical problems below.
First, let us outline the basic concepts that algebra and geometry give to this term. Let's consider all possible types of this graph.
Let's find out all the main characteristics of this function. Let's understand the basics of curve construction (geometry). Let's learn how to find the top and other basic values of a graph of this type.
Let's find out: how to correctly construct the desired curve using the equation, what you need to pay attention to. Let's look at 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 branches of the function drawing along the abscissa axis.
The canonical equation is:
y 2 = 2 * p * x,
where coefficient p is the focal parameter of the parabola (AF).
In algebra it will be 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 abscissa axis.
The range of values of the function – (-∞, M) or (M, +∞) depends on the direction of the branches of the curve. 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 a curve of this type from an expression, you need to determine the sign before the first parameter of the algebraic expression. If a ˃ 0, then they are directed upward. If it's the other way around, 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 determine it? There is a special formula. When b is not equal to 0, we need to look for the coordinates of this point.
Formulas for finding the vertex:
- x 0 = -b / (2 * a);
- y 0 = y (x 0).
Example.
There is a function y = 4 * x 2 + 16 * x – 25. Let’s find the vertices of this function.
For a line like this:
- x = -16 / (2 * 4) = -2;
- y = 4 * 4 - 16 * 2 - 25 = 16 - 32 - 25 = -41.
We get the coordinates of the vertex (-2, -41).
Parabola displacement
The classic case is when in a quadratic function y = a x 2 + b x + c, the second and third parameters are equal to 0, and = 1 - the vertex is at the point (0; 0).
Movement along the abscissa or ordinate axes is due to changes in the parameters b and c, respectively. The line on the plane will be shifted by exactly the number of units equal to the value of the parameter.
Example.
We have: b = 2, c = 3.
This means that the classic form 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 using given parameters.
By analyzing the expressions and equations, you can see the following:
- The point of intersection of the desired line with the ordinate vector will have a value equal to c.
- All points of the graph (along the x-axis) will be symmetrical with respect to the main extremum of the function.
In addition, the intersection points with OX can be found by knowing the discriminant (D) of such a function:
D = (b 2 - 4 * a * c).
To do this, you need to equate the expression to zero.
The presence of roots of a parabola depends on the result:
- D ˃ 0, then x 1, 2 = (-b ± D 0.5) / (2 * a);
- D = 0, then x 1, 2 = -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 ordinate axis;
- find the intersection with the x-axis.
Example 1.
Given the function y = x 2 - 5 * x + 4. It is necessary to construct a parabola. We follow the algorithm:
- a = 1, therefore, the branches are directed upward;
- extremum coordinates: x = - (-5) / 2 = 5/2; y = (5/2) 2 - 5 * (5/2) + 4 = -15/4;
- intersects with the ordinate axis at the value y = 4;
- let's find the discriminant: D = 25 - 16 = 9;
- looking for roots:
- X 1 = (5 + 3) / 2 = 4; (4, 0);
- X 2 = (5 - 3) / 2 = 1; (10).
Example 2.
For the function y = 3 * x 2 - 2 * x - 1 you need to construct a parabola. We act according to the given algorithm:
- a = 3, therefore, the branches are directed upward;
- extremum coordinates: x = - (-2) / 2 * 3 = 1/3; y = 3 * (1/3) 2 - 2 * (1/3) - 1 = -4/3;
- will intersect with the y-axis at the value y = -1;
- let's find the discriminant: D = 4 + 12 = 16. So the roots are:
- X 1 = (2 + 4) / 6 = 1; (1;0);
- X 2 = (2 - 4) / 6 = -1/3; (-1/3; 0).
Using the obtained points, you can construct a parabola.
Directrix, eccentricity, focus of a parabola
Based on the canonical equation, the focus of F has coordinates (p/2, 0).
Straight line AB is a directrix (a kind of chord of a parabola of a certain length). Its equation is x = -p/2.
Eccentricity (constant) = 1.
Conclusion
We looked at a 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 a displacement along the axes, and, having a construction algorithm, you can draw its graph.