Parabola and its formula. Three point equation: how to find the vertex of a parabola, formula

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 everything possible types this chart.

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 use 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.

It means that classic look 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; (10).

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.

How to build a parabola? There are several ways to graph a quadratic function. Each of them has its pros and cons. Let's consider two ways.

Let's start by plotting a quadratic function like y=x²+bx+c and y= -x²+bx+c.

Example.

Plot the function y=x²+2x-3.

Solution:

y=x²+2x-3 is a quadratic function. The graph is a parabola with branches up. Parabola vertex coordinates

From the vertex (-1;-4) we build a graph of the parabola y=x² (as from the origin. Instead of (0;0) - the vertex (-1;-4). From (-1;-4) we go to the right by 1 unit and up by 1, then left by 1 and up by 1, then: 2 - right, 4 - up, 2 - left, 4 - up, 3 - right, 9 - up, 3 - left, 9 - up. these 7 points are not enough, then - 4 to the right, 16 - up, etc.).

The graph of the quadratic function y= -x²+bx+c is a parabola whose branches are directed downwards. To build a graph, we are looking for the coordinates of the vertex and from it we build a parabola y= -x².

Example.

Plot the function y= -x²+2x+8.

Solution:

y= -x²+2x+8 is a quadratic function. The graph is a parabola with branches down. Parabola vertex coordinates

From the top we build a parabola y = -x² (1 - right, 1 - down; 1 - left, 1 - down; 2 - right, 4 - down; 2 - left, 4 - down, etc.):

This method allows you to build a parabola quickly and does not cause difficulties if you know how to plot the functions y=x² and y= -x². Disadvantage: if the vertex coordinates are fractional numbers, plotting is not very convenient. If you need to know exact values points of intersection of the graph with the Ox axis, you will have to additionally solve the equation x² + bx + c = 0 (or -x² + bx + c = 0), even if these points can be directly determined from the figure.

Another way to build a parabola is by points, that is, you can find several points on the graph and draw a parabola through them (taking into account the fact that the line x=xₒ is its axis of symmetry). Usually, for this, they take the top of the parabola, the intersection points of the graph with the coordinate axes, and 1-2 additional points.

Plot the function y=x²+5x+4.

Solution:

y=x²+5x+4 is a quadratic function. The graph is a parabola with branches up. Parabola vertex coordinates

that is, the top of the parabola is the point (-2.5; -2.25).

Are looking for . At the point of intersection with the Ox axis y=0: x²+5x+4=0. Roots quadratic equation x1=-1, x2=-4, that is, we got two points on the graph (-1; 0) and (-4; 0).

At the intersection point of the graph with the Oy axis x=0: y=0²+5∙0+4=4. Got a point (0; 4).

To refine the graph, you can find an additional point. Let's take x=1, then y=1²+5∙1+4=10, that is, one more point of the graph - (1; 10). We mark these points on the coordinate plane. Taking into account the symmetry of the parabola with respect to the straight line passing through its vertex, we mark two more points: (-5; 6) and (-6; 10) and draw a parabola through them:

Plot the function y= -x²-3x.

Solution:

y= -x²-3x is a quadratic function. The graph is a parabola with branches down. Parabola vertex coordinates

The top (-1.5; 2.25) is the first point of the parabola.

At the points of intersection of the graph with the x-axis y=0, that is, we solve the equation -x²-3x=0. Its roots are x=0 and x=-3, that is, (0; 0) and (-3; 0) are two more points on the graph. The point (o; 0) is also the point of intersection of the parabola with the y-axis.

At x=1 y=-1²-3∙1=-4, i.e. (1; -4) is an additional point for plotting.

Building a parabola from points is a more time-consuming method compared to the first one. If the parabola does not intersect the Ox axis, more additional points will be required.

Before continuing with plotting quadratic functions of the form y=ax²+bx+c, consider plotting function graphs using geometric transformations. Graphs of functions of the form y=x²+c are also most convenient to build using one of these transformations - parallel translation.

Rubric: |

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

The directory property of a parabola

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

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

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

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

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

FM=MM_d,

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

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

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

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

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

Parabola equation in polar coordinates

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

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

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

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

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

The geometric meaning of the parameter in the parabola equation

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

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

Remarks 3.11.

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

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

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

The equation (x-x_0)^2=2p(y-y_0),\,p>0, also defines a parabola with vertex O "(x_0, y_0) , whose axis is parallel to the y-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 are shown given systems coordinates Oxy and canonical coordinates Ox"y" .

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

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

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

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

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

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

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

Comparing 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 directrix equation x=-\frac(p)(2) , i.e. x=-\frac(1)(2) .

General properties of an ellipse, hyperbola, parabola

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

a) an ellipse if 0\leqslant e<1 ;

b) hyperbola, if e>1;

c) parabola if e=1.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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III level

3.1. Hyperbole touches lines 5 x – 6y – 16 = 0, 13x – 10y– – 48 = 0. Write down the equation of the hyperbola, provided that its axes coincide with the coordinate axes.

3.2. Write the equations of the tangents to the hyperbola

1) passing through a point A(4, 1), B(5, 2) and C(5, 6);

2) parallel to a straight line 10 x – 3y + 9 = 0;

3) perpendicular to the straight line 10 x – 3y + 9 = 0.

parabola is the locus of points in the plane whose coordinates satisfy the equation

Parabola parameters:

Dot F(p/2, 0) is called focus parabolas, magnitude p – parameter , dot ABOUT(0, 0) – summit . At the same time, the direct OF, about which the parabola is symmetrical, defines the axis of this curve.

Value where M(x, y) is an arbitrary point of the parabola, is called focal radius , straight D: x = –p/2 – headmistress (it does not intersect the interior of the parabola). Value is called the eccentricity of the parabola.

The main characteristic property of a parabola: all points of the parabola are equidistant from the directrix and focus (Fig. 24).

There are other forms of the canonical equation of the parabola, which determine other directions of its branches in the coordinate system (Fig. 25):

For parametric definition of a parabola as a parameter t the value of the ordinate of the point of the parabola can be taken:

where t is an arbitrary real number.

Example 1 Determine the parameters and shape of the parabola from its canonical equation:

Solution. 1. Equation y 2 = –8x defines a parabola with vertex at a point ABOUT Ox. Its branches are directed to the left. Comparing this equation with the equation y 2 = –2px, we find: 2 p = 8, p = 4, p/2 = 2. Therefore, the focus is at the point F(–2; 0), directrix equation D: x= 2 (Fig. 26).

2. Equation x 2 = –4y defines a parabola with vertex at a point O(0; 0), symmetrical about the axis Oy. Its branches are directed downwards. Comparing this equation with the equation x 2 = –2py, we find: 2 p = 4, p = 2, p/2 = 1. Therefore, the focus is at the point F(0; –1), directrix equation D: y= 1 (Fig. 27).

Example 2 Define parameters and type of curve x 2 + 8x – 16y– 32 = 0. Make a drawing.

Solution. We transform the left side of the equation using the full square method:

x 2 + 8x– 16y – 32 =0;

(x + 4) 2 – 16 – 16y – 32 =0;

(x + 4) 2 – 16y – 48 =0;

(x + 4) 2 – 16(y + 3).

As a result, we get

(x + 4) 2 = 16(y + 3).

This is the canonical equation of a parabola with vertex at the point (–4; –3), the parameter p= 8, branches pointing up (), axis x= -4. Focus is on point F(–4; –3 + p/2), i.e. F(–4; 1) Headmistress D is given by the equation y = –3 – p/2 or y= -7 (Fig. 28).

Example 4 Compose the equation of a parabola with a vertex at a point V(3; –2) and focus at the point F(1; –2).

Solution. The vertex and focus of this parabola lie on a straight line parallel to the axis Ox(the same ordinates), the branches of the parabola are directed to the left (the abscissa of the focus is less than the abscissa of the vertex), the distance from the focus to the vertex is p/2 = 3 – 1 = 2, p= 4. Hence, the desired equation

(y+ 2) 2 = –2 4( x– 3) or ( y + 2) 2 = = –8(x – 3).

Tasks for independent solution

I level

1.1. Determine the parameters of the parabola and construct it:

1) y 2 = 2x; 2) y 2 = –3x;

3) x 2 = 6y; 4) x 2 = –y.

1.2. Write the equation of a parabola with vertex at the origin if you know that:

1) the parabola is located in the left half-plane symmetrically about the axis Ox And p = 4;

2) the parabola is located symmetrically about the axis Oy and passes through the point M(4; –2).

3) directrix is ​​given by equation 3 y + 4 = 0.

1.3. Write an equation for a curve, all points of which are equidistant from the point (2; 0) and the straight line x = –2.

II level

2.1. Define the type and parameters of the curve.

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 is 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, 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.