Presentation on how to draw a parabola. Presentation on mathematics on the topic "Parabola. Relatives of the parabola, near and far." Parabolic orbit and satellite motion along it

The goal of the project: to study one of the second-order curves (parabola) and the scope of its application. Project objectives: 1. Give a strict mathematical definition of a parabola. 2. Study the properties of a parabola. 3. Find out why a parabola is called a conic section. 4. Identify areas of application of the parabola.

Parabola (Greek παραβολή appendix) is a curve whose points are equally distant from some point called the focus and from some straight line called the directrix of the parabola. Along with the ellipse and hyperbola, a parabola is a conic section. An image of a conic section that is a parabola. Construction of a parabola as a conic section.

Constructing a parabola First method. A parabola can be constructed “point by point” using a compass and ruler, without knowing the equation and having only the focus and directrix. The vertex is the midpoint of the segment between the focus and the directrix. An arbitrary reference system with the required single segment. Each subsequent point is the intersection of the perpendicular bisector of the segment between the focus and the directrix point, located at a distance from the origin that is a multiple of the unit segment, and a straight line passing through this point and parallel to the axis of the parabola

Constructing a parabola The second method. In order to draw a parabola, you will need a ruler, a square, a thread with a length equal to the larger leg of the square, and buttons. Attach one end of the thread to the focus, and the other to the top of the smaller corner of the square. Let's apply a ruler to the directrix and place a square on it with the smaller leg. Use a pencil to pull the thread so that its point touches the paper and presses against the larger leg. We will move the square and press the pencil against its side so that the thread remains taut. In this case, the pencil will draw a parabola on the paper.

Properties of a parabola 1. A parabola is a second-order curve. 2. It has an axis of symmetry called the axis of the parabola. The axis passes through the focus and the vertex perpendicular to the directrix. 3. Optical property. A beam of rays parallel to the axis of the parabola, reflected in the parabola, is collected at its focus. And vice versa, light from a source located in focus is reflected by a parabola into a beam of rays parallel to its axis. 4. For a parabola, the focus is at the point (0; 0.25). For a parabola, the focus is at the point (0; f). 5. All parabolas are similar. The distance between the focus and the directrix determines the scale. 6. When a parabola rotates around the axis of symmetry, an elliptical paraboloid is obtained.

Properties of a parabola The distance from Pn to the focus F is the same as from Pn to Qn. Illustration of the proof of Pascal's theorem using the 9-point theorem. The lengths of the F-Pn-Qn lines are the same. We can say that, unlike the ellipse, the second focus of the parabola is at infinity (see also Dandelin Balls).

Using paraboloids in the technique A paraboloid of rotation focuses a beam of rays parallel to the main axis into one point. The property of a paraboloid of revolution is often used to collect a beam of rays parallel to the main axis into one focal point, or, conversely, to form a parallel beam of radiation from a source located at the focus. Parabolic antennas, reflector telescopes, searchlights, and car headlights are based on this principle. Radio telescope antenna.

Solar lighter An original way to use the sun's energy. The solar lighter is a stainless steel parabolic mirror, much like the one used to light the Olympic flame in Athens. A parabolic mirror makes it possible to collect all the energy at one focal point and ignite a fire. The temperature at this point can reach 537 degrees Celsius. Such a device will be indispensable on a hike and in other field conditions.

Parabolas in physical space Trajectories of some cosmic bodies (comets, asteroids and others) passing near a star or other massive object (star, black hole or simply planets) at a sufficiently high speed have the shape of a parabola (or hyperbola). Due to their high speed and low mass, these bodies are not captured gravitational field the stars continue their free flight. This phenomenon is used for gravity maneuvers of spacecraft.

Application of parabolas in ballistics Ballistics (from the Greek βάλλειν to throw) is the science of the movement of bodies thrown in space, based on mathematics and physics. She primarily studies the motion of projectiles fired from firearms, rockets and ballistic missiles. A distinction is made between internal ballistics, which studies the movement of a projectile in the gun channel, as opposed to external ballistics, which studies the movement of a projectile as it exits the gun. External ballistics, as a rule, is understood as the science of the movement of bodies in air and airless space under the influence of only external forces.

Suspension bridge structure structure. The main stresses in a suspension bridge are tensile stresses in the main cables and compressive stresses in the supports; the stresses in the span itself are small. Almost all the forces in the supports are directed vertically downwards and are stabilized by cables, so the supports can be very thin. The relatively simple distribution of loads across different structural elements simplifies the calculation of suspension bridges. Under the influence of their own weight and the weight of the bridge span, the cables sag and form an arc close to a parabola. An unloaded cable suspended between two supports takes the form of a so-called. a "catenary line" that is close to a parabola in an almost horizontal section. If the weight of the cables can be neglected, and the weight of the span is uniformly distributed along the length of the bridge, the cables take the shape of a parabola. If the weight of the cable is comparable to the weight of the road surface, then its shape will be intermediate between a catenary line and a parabola.

Results During the work on this project: 1. A strict mathematical definition of a parabola was formulated. 2. A method for constructing a parabola is considered. 3. Some properties of the parabola were studied. 4. The connection between the concepts of “parabola” and “conic sections” has been revealed. 5. The areas of application of the parabola are determined (physics, technology, ballistics, astronomy, architecture, bridge construction). 6. The importance of mathematics in the world around us has been confirmed.

Internet resources Parabola Conic section Antenna Reflector _ (telescope) Spotlight Focus _ (physics) Suspension bridge Elliptical paraboloid

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Lesson objectives: reproduction and correction of necessary knowledge and skills on this topic;

analysis of tasks and methods of their implementation;

develop logical thinking;

consolidate the ability to build and “read” graphs;

instill an interest in the history of mathematics.

Lesson type: lesson on consolidating and testing students' knowledge, skills and abilities.

Equipment:

• PowerPoint presentation;
• drawing tools.

I. Historical background. (Slide 2)

Apollonius of Perga (Perge, 262 BC - 190 BC) - ancient Greek mathematician, one of the three (along with Euclid and Archimedes) great geometers of antiquity who lived in the 3rd century BC.

Apollonius became famous primarily for his monograph “Conic sections”(8 books), in which he gave meaningful general theory ellipse, parabola and hyperbola. It was Apollonius who proposed the common names for these curves; before him they were simply called “cone sections.” He introduced other mathematical terms, the Latin analogues of which have forever entered science, in particular: asymptote, abscissa, ordinate, applicate.

“Parable” means an application or parable. For a long time, this was the name given to the cutting line of a cone, until the quadratic function appeared.

Application of the properties of a parabola in life.

It turns out that the parabola graph quadratic function- has this interesting property: there is such a point and such a line that each point of the parabola is equally distant from this point and from this line (the point is called the focus of the parabola, and the line is its directrix). This property of a parabola was already known to the mathematicians of ancient Greece.

A stone thrown at an angle to the horizon, or a projectile fired from a cannon, fly along a trajectory shaped like a parabola.

If you rotate a parabola around its axis of symmetry, you get a surface called a paraboloid of revolution. If you vigorously stir water in a glass with a spoon and then remove the spoon, the surface of the water will take the shape of such a paraboloid.

And here is another interesting property: if a paraboloid of revolution is rotated around its axis at a suitable speed, then the resultant centrifugal force and gravity at each point of the paraboloid will be directed perpendicular to its surface.

A funny attraction is based on this property: if you rotate a large paraboloid, then each of the people located inside it seems that he himself is firmly standing on the floor, and all the other people are somehow miraculously holding on to the walls.

II. Generalization of knowledge about the location of the graph of a parabola. (Slide 3-5)

Looking at a parabola...

In this section we will show how you can get a lot of information about the coefficients of a quadratic trinomial y = ax 2 + bx + c, looking at his graph - a parabola.

First, let us recall well-known facts.

1) Coefficient sign A(at x 2) shows the direction of the branches of the parabola:

a > O - branches up;

A< 0 - ветви вниз.

Coefficient modulus, A responsible for “coolness”

parabolas: the more the “steeper” the parabola.

Decide exercise 1. (Slide 6, 7)

For each of the quadratic trinomials:

2) Coefficient b(together with A) determines the abscissa of the vertex of the parabola:

In particular, when A= 1 abscissa of vertex quadratic trinomial y = x 2 + bx + c equal to .

At b> 0 vertex is located to the left of the axis Oh, at b< 0 - to the right, at b = 0- on the axis Oh.

Decide exercise 2. (Slide 8, 9)

For each of their quadratic trinomials:

find its graph on the drawing.

3) Keeping the odds a and b and changing With, we will “raise” and “lower” the parabola. How to “read” the value in a drawing With?

It's clear that c = y (0)-ordinate of the point of intersection of the parabola with the axis Oh.

Decide exercise 3. (Slide 11, 12)

a) Where is which graph?

b) What is more: With or 1 ?

c) Determine the sign b.

Decide exercise 4. (Slide 13, 14)

The drawing shows the graphs of the functions:

and the axis Oh, going, as always, “from bottom to top” perpendicular to the axis Oh, erased.

a) Which function has graph 1 and which has graph 2?

b) Determine the signs of c and d.

c) Determine the sign of b.

Decide exercise 5. (Slide 15, 16)

The drawing shows the graphs of the functions:

y = x 2 + 4x + c,

y = x 2 + bx + d and y = x 2 + 1,

and the axis Oh, going, as always, “from left to right” perpendicular to the axis Oh, erased.

a) Which function has graph 1, which has graph 2, and which has graph 3?

b) Determine the sign b.

c) What's more: With or d?

d) Identify the signs With And d.

Decide exercise 6. (Slide 17–19)

The drawing shows the graphs of the functions:

y = ax 2 + x + c,

y = –x 2 + bx + 2

and the axes Oh And Oh, located in a standard manner (parallel to the edges of the sheet, Oh- horizontally “from left to right”, Oh- vertically (“bottom up”), erased.

a) Determine the sign b.

b) Determine the sign With.

c) Prove that:

• the solution to the exercises is based on the facts that we know about the coefficients of the quadratic trinomial;
• The properties of a parabola are extremely rich and varied, use them to solve the problem.

Task (slide 20, 21).

It is known that a parabola, which is the graph of a quadratic trinomial y = ax 2 + 10x + c, has no points in the third quarter.

Which of the following statements may not be true?

(A) a>0

(B) The vertex of the parabola lies in the second quadrant.

(C) with > 0

(E) 1OO – 4 ac < 0.

Since the parabola has no points in the third quarter, it cannot be negative. So, a> 0, therefore, the abscissa of the vertex x 0< 0. То есть вершина не может лежать ни в I, ни в IV четвертях. В III четверти ее нет по условию, значит, она лежит во II четверти. Итак, парабола обязана иметь такой вид, как показано на рисунке, поэтому условия А, В и С обязательно выполняются. Неравенство в Е означает, что дискриминант неположителен, то есть у квадратного трехчлена не более одного корня, - это условие тоже обязательно выполняется. Условие With> 0.1 does not follow from anything.

Indeed, it can be violated, for example, for a parabola at= x 2 + 10x + 0.01, satisfying the conditions of the problem.

Answer: (D).

This term has other meanings . (Literature)

Parabola – “comparison, juxtaposition, similarity, approximation.”

A short story of an allegorical nature, having an instructive meaning and a special form of narration, which moves as if along a curve (parabola): starting with abstract subjects, the story gradually approaches the main topic, and then returns again.

PARABOLA.

RELATIVES OF THE PARABOLA -

NEAR AND FAR

Silchenko Olga, Izotova Anna

9th grade students of MBOU Strashevichi Secondary School

teacher: Samolysova Tatyana Vasilievna

Project goal:

study one of the second-order curves (parabola) and the scope of its application.

Project objectives:

1.Give a mathematical definition of a parabola.

2. Study the properties of a parabola.

3. Find out why a parabola is called a conic section.

4. Find information about the “relatives” of the parabola

5. Identify areas of application of the parabola

We are all familiar with the quadratic trinomial, about which it would seem that we all know: how to find roots, and how to construct a graph, and how to solve quadratic inequalities... But this is a hasty judgment - our old friend has many secrets and surprises!

Parabola (Greek παραβολή - appendix) - a curve whose points are equally distant from some point called the focus and from some straight line called the directrix of the parabola.

Parabola- this is a section cone plane parallel to its generatrix.

Another way to build

It turns out that a parabola - the graph of a quadratic function - has an interesting property: there is such a point and such a line that each point of the parabola is equally distant from this point and from this line (the point is called the focus of the parabola, and the line is called the directrix). This property of a parabola was known to the mathematicians of ancient Greece. For the graph of the function y = x 2, the focus is the point with coordinates (0;0.25), and the directrix is ​​the straight line y = -0.25.

Try to figure out how you can build a parabola using this property.

Properties of a parabola

1. Parabola is a second order curve.

2. It has an axis of symmetry called the axis of the parabola. The axis passes through the focus and the vertex perpendicular to the directrix.

3.Optical property. A beam of rays parallel to the axis of the parabola, reflected in the parabola, is collected at its focus. And vice versa, light from a source located in focus is reflected by a parabola into a beam of rays parallel to its axis.

4. For a parabola, the focus is at the point (0; 0.25).

For a parabola, the focus is at the point (0; f).

5. All parabolas are similar. The distance between the focus and the directrix determines the scale.

The closest relatives of the parabola- This circle , hyperbola And ellipse.

And what all these curves have in common is an ordinary cone:

draw a plane that is parallel to the axis of the cone,

then the line of intersection will be a hyperbola

• if the plane is perpendicular to the axis, then the intersection is a circle ,

• if the plane is placed between the last two,

then the intersection will result in an ellipse.

if the plane is parallel to the generatrix of the cone, then the intersection will result in a parabola ,

Therefore, all these curves together are called conic sections.

Already in 340 BC, the Greek mathematician Menaechmus knew about this property of these curves, and in the second century BC Apollonius of Perga wrote a similar treatise “Conic Sections”.

Cycloid.

Another famous relative of the parabola is the cycloid. This is the trajectory of a point on the rim of a wheel that rolls in a straight line without slipping. This name was given to the curve by Galileo. If you go down on a sled from a hill built in the form of a cycloid, then the time of descent does not depend on the place from which the sled began to roll. But descending from the same height on a slide of any other shape will take longer. Because of this property, the cycloid is also called “brachistochrone.” (from Greek words meaning "shortest" and "time").

Paraboloid of rotation.

If you rotate a parabola around its axis of rotation, you get a surface called a paraboloid of revolution.

If you vigorously stir water in a glass with a spoon and then remove the spoon, the surface of the water will take the shape of such a paraboloid.

The use of paraboloids in technology

A paraboloid of rotation focuses a beam of rays parallel to the main axis into one point.

The property of a paraboloid of rotation is often used to collect a beam of rays parallel to the main axis into one point - the focus, or, conversely, to form a parallel beam of radiation from a source located at the focus.

Parabolic antennas, reflecting telescopes, searchlights, and car headlights are based on this principle.

Use of paraboloids in technology

Reflecting telescopes

Spotlight

Car headlights

Solar lighter

An original way to use solar energy. The solar lighter is a stainless steel parabolic mirror, much like the one used to light the Olympic flame in Athens.

A parabolic mirror makes it possible to collect all the energy at one focal point and ignite a fire. The temperature at this point can reach 537 degrees Celsius. Such a device will be indispensable on a hike and in other field conditions.

Parabolas in physical space

Parabolic orbit and satellite motion along it

Fall basketball ball

Parabolic solar power plant in California, USA.

Parabola in nature

Parabola. Its shape is incredible, as is its height. Some people

They still don’t believe in the existence of this strange rock. This is what they say:

“There is neither God nor Parabola. And what they show is photoshop.”

Parabola in nature

Anyone who believes that a parabola can only be found on the pages of a textbook is undoubtedly mistaken. Look carefully at the pictures and find the parabolas in them.

Make some drawings of leaves, flowers, animals yourself and find the parabolas in them.

Parabolas in the animal world

Animal jumping trajectories are close to a parabola

Results

While working on this project :

1. A strict mathematical definition of a parabola is formulated.

2. A method for constructing a parabola is considered.

3. Some properties of the parabola were studied.

4. The connection between the concepts of “parabola” and “conic sections” was revealed, and relatives of the parabola were found.

5. The areas of application of the parabola have been determined (physics, technology, astronomy, architecture, etc.).

6. The importance of mathematics in the world around us has been confirmed.

List of sources used:

1. Encyclopedic Dictionary young mathematician. Compiled by A.P. Savin, M, Pedagogy, 1982.

2. Encyclopedia for children, volume 11, “Mathematics”, M, “Avanta+”, 1998.

3. Mathematical club "Kangaroo", "Around the square trinomial" St. Petersburg, 2002.

4. Website http://www/uvlekat- matem.narod.ru/

5.Website www.bigpi.biysk.ru

6.Website en.wikipedia.org › Conical section