A simple story illustrating a moral or religious lesson. Switch to new thesaurus. Aesop's fables - a collection of fables believed to have been written by the Greek storyteller Aesop.
New Testament - the collection of books of the Gospels, Acts of the Apostles, the Pauline and other epistles, and Revelation; composed soon after Christ's death; the second half of the Christian Bible. Related words adjectives parabolicparabolical.
Parabel Gleichnis. Jesus told parables. Mentioned in? Aesop's fables Aggadah allegory apologue assistance broad interpretation bysen comparable worth Dives Dodd explanation fable forbysen form criticism good Samaritan gospel interpret interpretation interpreting. References in classic literature? View in context. And also this parable give I unto you: Not a few who meant to cast out their devil, went thereby into the swine themselves.
He remembered the parable of the unjust judge, and though he had previously felt sure that he ought to refuse, he now began to hesitate and, having hesitated, took to prayer and prayed until a decision formed itself in his soul.
After exhausting life in his efforts for mankind's spiritual good, he had made the manner of his death a parablein order to impress on his admirers the mighty and mournful lesson, that, in the view of Infinite Purity, we are sinners all alike. He thought of Cronshaw's parable of the Persian carpet. It seems an easier and shorter way to dignity, to observe that-- since there never was a true story which could not be told in parableswhere you might put a monkey for a margrave, and vice versa-- whatever has been or is to be narrated by me about low people, may be ennobled by being considered a parable ; so that if any bad habits and ugly consequences are brought into view, the reader may have the relief of regarding them as not more than figuratively ungenteel, and may feel himself virtually in company with persons of some style.
The parable of Pythagoras is dark, but true; Cor ne edito; Eat not the heart. One of the greatest of poets, Coleridge was one of the wisest of men, and it was not for nothing that he read us this parable. Let us have a little less of "hands across the sea," and a little more of that elemental distrust that is the security of nations. A real element of Socratic teaching, which is more prominent in the Republic than in any of the other Dialogues of Plato, is the use of example and illustration 'taphorhtika auto prhospherhontez' : "Let us apply the test of common instances.
It was in the Umpqua Valley that they heard the parable of the white sparrow. The simple pathos, and the apparent indirectness of such a tale as that of 'Poticoushka,' the peasant conscript, is of vastly more value to the world at large than all his parables ; and 'The Death of Ivan Ilyitch,' the Philistine worldling, will turn the hearts of many more from the love of the world than such pale fables of the early Christian life as "Work while ye have the Light.
Dictionary browser? Full browser?A parabola is set of all points in a plane which are an equal distance away from a given point and given line. The point is called the focus of the parabola and the line is called the directrix. The focus lies on the axis of symmetry of the parabola.
Notice that here we are working with a parabola with a vertical axis of symmetry, so the x -coordinate of the focus is the same as the x -coordinate of the vertex. The focus is at 02. Names of standardized tests are owned by the trademark holders and are not affiliated with Varsity Tutors LLC. Media outlet trademarks are owned by the respective media outlets and are not affiliated with Varsity Tutors. Varsity Tutors connects learners with experts. Instructors are independent contractors who tailor their services to each client, using their own style, methods and materials.
Focus of a Parabola A parabola is set of all points in a plane which are an equal distance away from a given point and given line. Subjects Near Me. Download our free learning tools apps and test prep books. Varsity Tutors does not have affiliation with universities mentioned on its website.In mathematicsa parabola is a plane curve which is mirror-symmetrical and is approximately U- shaped.
It fits several other superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One description of a parabola involves a point the focus and a line the directrix. The focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from both the directrix and the focus. Another description of a parabola is as a conic sectioncreated from the intersection of a right circular conical surface and a plane parallel to another plane that is tangential to the conical surface.
The line perpendicular to the directrix and passing through the focus that is, the line that splits the parabola through the middle is called the " axis of symmetry ". The point where the parabola intersects its axis of symmetry is called the " vertex " and is the point where the parabola is most sharply curved. The distance between the vertex and the focus, measured along the axis of symmetry, is the "focal length".
The " latus rectum " is the chord of the parabola that is parallel to the directrix and passes through the focus. Parabolas can open up, down, left, right, or in some other arbitrary direction. Any parabola can be repositioned and rescaled to fit exactly on any other parabola—that is, all parabolas are geometrically similar.
Parabolas have the property that, if they are made of material that reflects lightthen light that travels parallel to the axis of symmetry of a parabola and strikes its concave side is reflected to its focus, regardless of where on the parabola the reflection occurs.
Conversely, light that originates from a point source at the focus is reflected into a parallel " collimated " beam, leaving the parabola parallel to the axis of symmetry. The same effects occur with sound and other waves. This reflective property is the basis of many practical uses of parabolas.
The parabola has many important applications, from a parabolic antenna or parabolic microphone to automobile headlight reflectors and the design of ballistic missiles. They are frequently used in physicsengineeringand many other areas. The earliest known work on conic sections was by Menaechmus in the 4th century BC.
He discovered a way to solve the problem of doubling the cube using parabolas. The solution, however, does not meet the requirements of compass-and-straightedge construction.
The area enclosed by a parabola and a line segment, the so-called "parabola segment", was computed by Archimedes by the method of exhaustion in the 3rd century BC, in his The Quadrature of the Parabola.
The name "parabola" is due to Apolloniuswho discovered many properties of conic sections. It means "application", referring to "application of areas" concept, that has a connection with this curve, as Apollonius had proved.
Galileo showed that the path of a projectile follows a parabola, a consequence of uniform acceleration due to gravity. The idea that a parabolic reflector could produce an image was already well known before the invention of the reflecting telescope.
Parabolic mirrors are used in most modern reflecting telescopes and in satellite dishes and radar receivers. A parabola can be defined geometrically as a set of points locus of points in the Euclidean plane:. The horizontal chord through the focus see picture in opening section is called the latus rectum ; one half of it is the semi-latus rectum. The latus rectum is parallel to the directrix. From the picture one obtains.
The latus rectum is the line drawn through a focus of a conic section parallel to the directrix and terminated both ways by the curve. The implicit equation of a parabola is defined by an irreducible polynomial of degree two:.
The previous section shows that any parabola with the origin as vertex and the y axis as axis of symmetry can be considered as the graph of a function. From the section above one obtains:. Completing the square yields. Two objects in the Euclidean plane are similar if one can be transformed to the other by a similaritythat is, an arbitrary composition of rigid motions translations and rotations and uniform scalings.
A suitable rotation around the origin can then transform the parabola to one that has the y axis as axis of symmetry.Conics: Parabolas: Introduction page 1 of 4. In the context of conics, however, there are some additional considerations. To form a parabola according to ancient Greek definitions, you would start with a line and a point off to one side. The line is called the "directrix"; the point is called the "focus".
The parabola is the curve formed from all the points xy that are equidistant from the directrix and the focus. The line perpendicular to the directrix and passing through the focus that is, the line that splits the parabola up the middle is called the " axis of symmetry ".costruzione parabola
The point on this axis which is exactly midway between the focus and the directrix is the " vertex "; the vertex is the point where the parabola changes direction. The name "parabola" is derived from a New Latin term that means something similar to "compare" or "balance", and refers to the fact that the distance from the parabola to the focus is always equal to that is, is always in balance with the distance from the parabola to the directrix.
In practical terms, you'll probably only need to know that the vertex is exactly midway between the directrix and the focus. In previous contextsyour parabolas have either been "right side up" or "upside down" graph, depending on whether the leading coefficient was positive or negative, respectively.
In the context of conics, however, you will also be working with "sideways" parabolas, parabolas whose axes of symmetry parallel the x -axis and which open to the right or to the left.
A basic property of parabolas "in real life" is that any light or sound ray entering the parabola parallel to the axis of symmetry and hitting the inner surface of the parabolic "bowl" will be reflected back to the focus. The focus of a parabola is always inside the parabola; the vertex is always on the parabola; the directrix is always outside the parabola. The "vertex" form of a parabola with its vertex at hk is:.
The conics form of the parabola equation the one you'll find in advanced or older texts is:. Why " hk " for the vertex? Why " p " instead of " a " in the old-time conics formula? The relationship between the "vertex" form of the equation and the "conics" form of the equation is nothing more than a rearrangement:.
Since the y part is squared and p is positive, then this is a sideways parabola that opens to the right. The focus is inside the parabola, so it has to be two units to the right of the vertex:.This form is called the standard form of a quadratic function. The graph of the quadratic function is a U-shaped curve is called a parabola. In the graph, the highest or lowest point of a parabola is the vertex.
In this case the vertex is the minimum, or lowest point, of the parabola. A large positive value of a makes a narrow parabola; a positive value of a which is close to 0 makes the parabola wide. In this case the vertex is the maximum, or highest point, of the parabola. Again, a large negative value of a makes the parabola narrow; a value close to zero makes it wide.
For an equation in standard form, the value of c gives the y -intercept of the graph. The line that passes through the vertex and divides the parabola into two symmetric parts is called the axis of symmetry. In the graphs below, the axis of symmetry is different marked in red. Note that c still gives the y -intercept. Note that in this case, c is the x -intercept.
If a is positive, the graph opens to the right; if a is negative, the graph opens to the left. Axis of Symmetry of a Parabola. Graphing quadratic equations using the axis of symmetry. Names of standardized tests are owned by the trademark holders and are not affiliated with Varsity Tutors LLC.
Media outlet trademarks are owned by the respective media outlets and are not affiliated with Varsity Tutors. Varsity Tutors connects learners with experts. Instructors are independent contractors who tailor their services to each client, using their own style, methods and materials. Subjects Near Me.The graph of a quadratic function is a parabola, and its parts provide valuable information about the function.
The graph of a quadratic function is a U-shaped curve called a parabola. This shape is shown below. This is shown below.
Parabolas have several recognizable features that characterize their shape and placement on the Cartesian plane. One important feature of the parabola is that it has an extreme point, called the vertex. If the parabola opens up, the vertex represents the lowest point on the graph, or the minimum value of the quadratic function.
If the parabola opens down, the vertex represents the highest point on the graph, or the maximum value. In either case, the vertex is a turning point on the graph. Parabolas also have an axis of symmetry, which is parallel to the y-axis. The axis of symmetry is a vertical line drawn through the vertex. The y -intercept is the point at which the parabola crosses the y -axis.
There cannot be more than one such point, for the graph of a quadratic function. The x -intercepts are the points at which the parabola crosses the x -axis. Recall that if the quadratic function is set equal to zero, then the result is a quadratic equation. The solutions to the equation are called the roots of the function. Describe the solutions to a quadratic equation as the points where the parabola crosses the x-axis. The roots of a quadratic function can also be found graphically by making observations about its graph.
These are two different methods that can be used to reach the same values, and we will now see how they are related. Consider the quadratic function that is graphed below. Notice that these are the same values that when found when we solved for roots graphically. Solve graphically and algebraically. Therefore, it has no real roots.A superparabola is a geometric curve defined in the Cartesian coordinate system as a set of points x, y with.
The curve can be described parametrically on the complex plane as. Thus, the centroid can be expressed as one-half the ratio of the area of the square of the curve to the area of the curve.
Two instances of exact solutions have been found. A natural generalization for the superparabola is to relax the constraint on the power of x. For example. The curve can be described parametrically on the complex plane as well. Now, it is apparent that the generalized superparabola contains within it the superellipse, i.
Here, however, we have the analytic solution for the area under the curve. Waldman and Gray  used the superparabola in their analyses of the Archimedean hoof. The superparabola and its generalization have been applied to the Archimedean hoof. In the first image, the portion on the right is called the hoofand is taken from the remaining half-cylinder leaving the complement.
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Please introduce links to this page from related articles ; try the Find link tool for suggestions. August Waldman and S. Gray, Superparabola and Superellipse in the Method of Archimedes. Gray, D. Yang, G.