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

Perhaps everyone knows what a parabola is. But how to correctly, competently use it in solving various practical problems, we will figure it out below.

First, we outline the basic concepts that algebra and geometry give to this term. Consider all possible types this graph.

Let's find out all the main characteristics of this function. Let's understand the basics of curve building (geometry). Let's learn how to find the top and other main values ​​of a chart of this type.

We will find out: how to construct the desired curve correctly 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 branches of the drawing of a function along the abscissa axis.

The canonical equation is:

y 2 = 2 * p * x,

where the 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).

Quadratic Function Properties and Plot

The function has an axis of symmetry and a center (extremum). Domain of definition - 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 M parameter 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 in front of the first parameter of the algebraic expression. If a ˃ 0, then they are directed upward. If on the contrary - down.

How to find the vertex of a parabola using the formula

Finding an 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 can you define it? There is a special formula. When b is not equal to 0, you need to look for the coordinates of this point.

Vertex finding formulas:

• 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 such a line:

• 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 offset

The classical case, when in the 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).

The 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 by exactly 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 the curve will shift by 2 unit segments along the abscissa axis and by 3 - along the ordinate axis.

How to build a parabola using a quadratic equation

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

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

1. The point of intersection of the sought line with the ordinate vector will have a value equal to c.
2. All points of the graph (along the abscissa) will be symmetrical about the main extremum of the function.

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

D = (b 2 - 4 * a * c).

To do this, set 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 = 0, then x 1, 2 = -b / (2 * a);
• D ˂ 0, then there are no intersection points 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 abscissa.

Example 1.

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

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

Example 2.

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

1. a = 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. will intersect with the y-axis at the value y = -1;
4. find the discriminant: D = 4 + 12 = 16. So the roots:

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

From the points obtained, you can build a parabola.

Headmistress, eccentricity, parabola focus

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

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

Eccentricity (constant) = 1.

Conclusion

We looked at a topic that students learn 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 plotting algorithm, you can draw its graph.

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

We start by plotting a quadratic function of the form 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) - vertex (-1; -4). From (-1; -4) go to the right by 1 unit and up by 1 unit, 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 downward. 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 is not difficult if you know how to plot the functions y = x² and y = -x². Disadvantage: if the coordinates of the vertex are fractional numbers, it is not very convenient to plot the graph. 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 picture.

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 that the line x = xₒ is its axis of symmetry). Usually for this they take the vertex of the parabola, the points of intersection 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 vertex 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 in the graph (-1; 0) and (-4; 0).

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

An additional point can be found to refine the graph. Take x = 1, then y = 1² + 5 ∙ 1 + 4 = 10, that is, one more point on 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

Vertex (-1.5; 2.25) - the first point of the parabola.

At the points of intersection of the graph with the abscissa 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.

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

The construction of a parabola by points is more time consuming, in comparison with the first, method. If the parabola does not intersect the Ox axis, more additional points are required.

Before continuing with charting quadratic functions of the form y = ax² + bx + c, consider the construction of graphs of functions using geometric transformations. It is also most convenient to construct graphs of functions of the form y = x² + c using one of such transformations - parallel transfer.

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

The directory property of a parabola

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

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

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

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

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

FM = MM_d,

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

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

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

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

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

Equation of a parabola in a polar coordinate system

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

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

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

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

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

The geometric meaning of the parameter in the parabola equation

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

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

Remarks 3.11.

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

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

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

The equation (x-x_0) ^ 2 = 2p (y-y_0), \, p> 0, also defines a parabola with a vertex O "(x_0, y_0), the axis of which is parallel to the ordinate axis (Fig. 3.47, c). This equation is reduced to the canonical one using parallel translation (3.36) and renaming the coordinate axes (3.38). 3.47, b, c are shown preset systems coordinates Oxy and canonical coordinate systems 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), the axis of which is parallel to the ordinate axis, the branches of the parabola are directed upward (for a> 0) or down (for a<0 ). Действительно, выделяя полный квадрат, получаем уравнение

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

which is reduced to the canonical form (y ") ^ 2 = 2px", where p = \ left | \ frac (1) (2a) \ right |, using the 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 composition: parallel transport (3.36) with x_0 = - \ frac (b) (2a) and y_0 = - \ frac (b ^ 2-4ac) (4a), renaming the coordinate axes (3.38), and in the case of a<0 еще и изменения направления координатной оси (3.37). На рис.3.48,а,б изображены заданные системы координат Oxy и канонические системы координат O"x"y" для случаев a>0 and a<0 соответственно.

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

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

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

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

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

b) hyperbola if e> 1;

c) a parabola if e = 1.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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III level

3.1. Hyperbola touches straight 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. Make equations of tangents to hyperbola

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

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

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

Parabola is called the locus of points of the plane, the coordinates of which satisfy the equation

Parabola parameters:

Point F(p/ 2, 0) is called focus parabolas, magnitude p – parameter , point O(0, 0) – apex ... Moreover, the straight OF, with respect to which the parabola is symmetric, defines the axis of this curve.

The magnitude where M(x, y) Is an arbitrary point of the parabola, called focal radius , straight D: x = –p/2 – headmistress (it does not cross the inner region of the parabola). The magnitude called the eccentricity of the parabola.

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

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

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

where t- an arbitrary real number.

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

Solution. 1. Equation y 2 = –8x defines a parabola with apex at the point O 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 apex at the point O(0; 0) symmetric about the axis Oy... Its branches are directed downward. 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 curve type x 2 + 8x – 16y- 32 = 0. Make a drawing.

Solution. Let's transform the left side of the equation using the full square selection 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 apex at the point (–4; –3), the parameter p= 8, branches directed upwards (), axis x= –4. The focus is on the point F(–4; –3 + p/ 2), i.e. F(–4; 1) Headmistress D given by the equation y = –3 – p/ 2 or y= –7 (Fig. 28).

Example 4. Equate a parabola with apex 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 required equation

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

Self-help assignments

Level I

1.1. Define the parameters of the parabola and plot 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 apex 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 goes through the point M(4; –2).

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

1.3. Equate a curve where all points are equidistant from point (2; 0) and straight line x = –2.

II level

2.1. Determine the type and parameters of the curve.

Consider a line on the plane and a point not lying on this line. AND ellipse, and hyperbola can be defined in a unified manner 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

of 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 locus of points equidistant from a given point and from a given straight line.

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

The fixed point is called focus parabola, and the straight line - the headmistress of the parabola... Moreover, it is believed that parabola eccentricity is equal to one.

From geometric considerations it follows that the parabola is symmetric with respect to the straight line perpendicular to the directrix and passing through the focus of the parabola. This line is called the axis of symmetry of a parabola or simply parabola axis... The parabola intersects its axis of symmetry at a single point. This point is called apex of a 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, choose on the plane origin at the apex of the parabola, as abscissa axis- the axis of the parabola, the positive direction on which is set by the focus position (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 by p. He's 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 point F and from line d is given by the equation

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

which is called the 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.

Parabola view. If the parabola y 2 = x, the form of which is assumed to be known, is compressed with the coefficient 1 / (2p) along the abscissa axis, then a general parabola is obtained, which is described by equation (8.3).

Example 8.2. Let us find the focus coordinates and the directrix equation of a 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 in 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 line makes equal angles with the focal radius MF and the abscissa axis.