EntranceParabola is a common occurrence in everyday life. Presentation on mathematics on the topic "Parabola
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Lesson objectives: reproduction and correction of necessary knowledge and skills on this topic;
Lesson type: lesson on consolidating and testing students' knowledge, skills and abilities.
Equipment:
- PowerPoint presentation;
- drawing tools.
I. Historical reference. (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 cone cutting line, until it appeared quadratic function.
Application of the properties of a parabola in life.
It turns out that a parabola, a graph of a 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 to each of the people located inside it, it seems that he himself is standing firmly 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 the vertex of a quadratic trinomial y = x 2 + bx + c equal to .
At b> 0 vertex is located to the left of the axis OU, at b< 0 - to the right, at b = 0- on the axis OU.
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 OU.
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 OU, 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 OU, 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 OU And Oh, located in a standard manner (parallel to the edges of the sheet, Oh- horizontally “from left to right”, OU- 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.
Presentation on the topic: Parabola and its properties Completed by: Student of 10th grade Grechkin Yaroslav Teacher Shamsutdinova R.R. School
Parabola. Focus. The directrix of a parabola is the geometric locus of points on a plane, for each of which the distance to a fixed point of this plane, called the focus, is equal to the distance to a fixed straight line lying in the same plane and called the directrix of the parabola. A parabola is a second order curve. Focus is an arbitrary point of the parabola. A directrix is a straight line lying in the plane of a parabola and having the property that the ratio of the distance from any point on the curve to the focus of the curve to the distance from the same point to this straight line is a constant value equal to the eccentricity. Eccentricity – numerical characteristic conical section.
Historical background The discoverer of conic sections is supposedly considered to be Menaechmus (4th century BC), a student of Plato and teacher of Alexander the Great. Menaechmus used a parabola and an equilateral hyperbola to solve the problem of doubling a cube. Treatises on conic sections written by Aristaeus and Euclid at the end of the 4th century. BC, were lost, but materials from them were included in the famous Conic Sections of Apollonius of Perga (c. 260–170 BC), which have survived to this day. Apollonius abandoned the requirement that the secant plane of the cone's generatrix be perpendicular and, by varying the angle of its inclination, obtained all conic sections from one circular cone, straight or inclined. We are indebted to Apollo and modern names curves - ellipse, parabola and hyperbola. In his constructions, Apollonius used a two-cavity circular cone. Since the time of Apollonius, conic sections have been divided into three types depending on the inclination of the cutting plane to the generatrix of the cone. A parabola is formed when the cutting plane is parallel to one of the tangent planes of the cone. The foci of the ellipse and hyperbola were known to Apollonius, but the focus of the parabola was apparently first established by Pappus (2nd half of the 3rd century), who defined this curve as the locus of points equidistant from a given point (focus) and a given straight line, which is called the director. The construction of a parabola using a stretched thread, based on the definition of Pappus, was proposed by Isidore of Miletus (6th century).
Derivation of the equation of a parabola To obtain the equation of a curve corresponding to this definition, we introduce a suitable coordinate system. To do this, from the focus F we lower the perpendicular FD to the directrix l. The origin of coordinates O will be located in the middle of the segment FD, the axis will be directed along the segment FD so that its direction coincides with the direction of the vector. Let's draw the axis perpendicular to the axis. Let the distance between the focus and the directrix of the parabola be equal to p. Then in the chosen coordinate system the parabola has the equation
Derivation of the equation of a parabola In the chosen coordinate system, the focus of the parabola is the point, and the directrix has the equation. Let the current point be a parabola. Then according to the formula for flat case we find The distance from point M to directrix l is the length of the perpendicular MK dropped onto the directrix from point M. From the figure it is obvious that Then, by the definition of a parabola, MK = FM, that is: Canonical equation parabolas
Properties of a parabola A parabola has an axis of symmetry. Proof: The variable y appears in the equation only to the second power. Therefore, if the coordinates of the point M (x ; y) satisfy the parabola equation, then the coordinates of the point N (x ; – y) will satisfy it. Point N is symmetrical to point M relative to the Ox axis. Therefore, the Ox axis is the axis of symmetry of the parabola in the canonical coordinate system. The axis of symmetry is called the axis of the parabola. The point where the parabola intersects the axis is called the vertex of the parabola. The vertex of a parabola in the canonical coordinate system is at the origin.
Properties of a parabola Let F be the focus of the parabola, M an arbitrary point of the parabola, l a ray with its origin at a point parallel to the axis of the parabola. Then the normal to the parabola at point M divides the angle formed by the segment FM and the ray l in half. This property means that a ray of light leaving the focus, reflected from a parabola, will then go parallel to the axis of this parabola. And vice versa, all rays coming from infinity and parallel to the axis of the parabola will converge at its focus. This property is widely used in technology. Spotlights usually have a mirror, the surface of which is obtained by rotating a parabola around its axis of symmetry (parabolic mirror). The light source in spotlights is placed at the focus of a parabola. As a result, the spotlight produces a beam of almost parallel rays of light. The same property is used in receiving antennas for space communications and in telescope mirrors, which collect a stream of parallel rays of radio waves or a stream of parallel rays of light and concentrate it at the focus of the mirror.
Constructing a parabola 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.
Constructing a parabola If you make a mirror surface in the shape of a paraboloid and place a light source at its focus, then the light rays, reflected from the mirror surface, will go in one direction, perpendicular to the directrix of the parabola. Therefore, the reflective surfaces of spotlights, car headlights, flashlights, telescopes, parabolic antennas, etc. made in the shape of a paraboloid.
Dialogues about the parabola MBOU Igrimskaya Secondary School No. 2, Saliy Tatyana Anatolyevna, mathematics teacher
Goals and objectives of the lesson: Review the properties of a quadratic function. Show the connection between a quadratic function and its graph and the real world. Systematize knowledge on the application of the properties of a parabola.
Definition. A function of the form y = ax 2 + b x + c, where a, b, c are given numbers, a≠0, x is a real variable, is called a quadratic function. Examples: 1) y = 5x+1 4) y =x 3 +7x-1 2) y=3x 2 -1 5) y=4x 2 3) y=-2x 2 +x+3 6) y=-3x 2 +2x
Determine the coordinates of the vertex of the parabola. Equation of the axis of symmetry of a parabola. Function zeros. The intervals in which the function increases and decreases. Intervals in which the function takes positive values, negative values. What is the sign of the coefficient a? How does the position of the parabola branches depend on the coefficient a?
Top of a parabola: Assignment. Find the coordinates of the vertex of the parabola: 1) y = x 2 -4x-5 2) y = -5x 2 +3 Answer: (2;-9) Answer: (0;3) Equation of the axis of symmetry: x = x 0
Coordinates of the points of intersection of the parabola with the coordinate axes. C Ox: y=0 ax 2 + b x+c=0 C Oy: x=0 y=c Task. Find the coordinates of the points of intersection of the parabola with the coordinate axes: 1) y = x 2 - x; 2)y=x 2 +3; 3)y=5x 2 -3x-2 (0;0);(1;0) (0;3) (1;0);(-0.4;0);(0;2)
Test. (-1;1) (- ∞ ;0) (1; ∞) (-∞;∞) (-1;0) x≠-1 No values x y 0 y > 0 y
Construct a graph of the function and use the graph to find out its properties. Y = -x 2 -6x-8 Properties of the function: y > 0 on the interval y
Graph of a quadratic function - Parabola Parabola (Greek παραβολή - appendix) is the locus of points equidistant from a given line (called the directrix of the parabola) and a given point (called the focus of the parabola).
Properties Parabola is a second order curve. It has an axis of symmetry called the axis of the parabola. The axis passes through the focus and is perpendicular to the directrix. If the focus of the parabola is reflected relative to the tangent, then its image will lie on the directrix. A parabola is the antipode of a line. All parabolas are similar. The distance between the focus and the directrix determines the scale. When a parabola rotates around the axis of symmetry, an elliptical paraboloid is obtained. y > 0
Archimedes' Focus This day in 212 BC. the surviving Romans remembered it for the rest of their lives. Almost half a thousand small suns suddenly lit up on the fortress wall. At first they simply blinded, but after a while something fantastic happened: the leading Roman ships that approached Syracuse, one after another, suddenly began to flare up like torches. The Romans fled in panic...
According to legend, Archimedes of Syracuse burned the Roman fleet while defending his city with parabolic mirrors. The properties of such mirrors are used in the construction of solar ovens, telescopes, etc.
Wonderful parabola I love to sing and have fun, to spin around in a merry dance. When I rotate around an axis, I turn into an important figure. And the gentlemen run up and escort you to the car. And everyone wants to invite you to stay on the roof of the house. Mystery
A body thrown upward moves in a parabola. Let a ball be thrown vertically upward from a height of 1.5 m, giving it an initial speed of 10 m/s². Then the height h (in m) at which the ball is located is a quadratic function of the flight time t (in s). If we assume that g =10 m/s, then the function h= f(t) can be described by the formula h= 1.5+10t-5 t². The graph of this function is part of a parabola.
Application of the properties of a parabola in solving problems of increased complexity. 1. How many roots does the equation have: (x -100)(x -101)+(x - 101)(x -102)+(x -102)(x -100)=0?