A parabola (/pəˈræbələ/; plural parabolas or parabolae, adjective parabolic, from Greek: παραβολή) is a two-dimensional, mirror-symmetrical curve, which is approximately U-shaped when oriented as shown in the diagram, but which can be in any orientation in its plane. It fits any of several superficially different mathematical descriptions which can all be proved to define curves of exactly the same shape.
One description of a parabola involves a point (the focus) and a line (the directrix). The focus does not lie on the directrix. The locus of points in that plane that are equidistant from both the directrix and the focus is the parabola. Another description of a parabola is as a conic section, created from the intersection of a right circular conical surface and a plane which is parallel to another plane which is tangential to the conical surface.[a] A third description is algebraic. A parabola is a graph of a quadratic function, such as
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 on the axis of symmetry that intersects the parabola is called the "vertex", and it is the point where the curvature is greatest. 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 which 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 light, then light which enters a parabola travelling parallel to its axis of symmetry 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 ("collimated") into a parallel beam, leaving the parabola parallel to the axis of symmetry. The same effects occur with sound and other forms of energy. 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 to the design of ballistic missiles. They are frequently used in physics, engineering, and many other areas.
Strictly, the adjective parabolic should be applied only to things that are shaped as a parabola, which is a two-dimensional shape. However, as shown in the last paragraph, the same adjective is commonly used for three-dimensional objects, such as parabolic reflectors, which are really paraboloids. Sometimes, the noun parabola is also used to refer to these objects. Though not perfectly correct, this usage is generally understood.
(http://en.wikipedia.org/wiki/Parabola)
One description of a parabola involves a point (the focus) and a line (the directrix). The focus does not lie on the directrix. The locus of points in that plane that are equidistant from both the directrix and the focus is the parabola. Another description of a parabola is as a conic section, created from the intersection of a right circular conical surface and a plane which is parallel to another plane which is tangential to the conical surface.[a] A third description is algebraic. A parabola is a graph of a quadratic function, such as
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 on the axis of symmetry that intersects the parabola is called the "vertex", and it is the point where the curvature is greatest. 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 which 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 light, then light which enters a parabola travelling parallel to its axis of symmetry 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 ("collimated") into a parallel beam, leaving the parabola parallel to the axis of symmetry. The same effects occur with sound and other forms of energy. 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 to the design of ballistic missiles. They are frequently used in physics, engineering, and many other areas.
Strictly, the adjective parabolic should be applied only to things that are shaped as a parabola, which is a two-dimensional shape. However, as shown in the last paragraph, the same adjective is commonly used for three-dimensional objects, such as parabolic reflectors, which are really paraboloids. Sometimes, the noun parabola is also used to refer to these objects. Though not perfectly correct, this usage is generally understood.
(http://en.wikipedia.org/wiki/Parabola)
As you can see the St. Louis Arch is a REAL LIFE example that can be roughly modelled by a parabola (it's actually a catenary*). To model it we would use a negative parabola because it is facing down. Parabolas are used very often but people don’t tend to notice. For instance they are used to reflect the light beams in your car's headlights and the McDonald’s golden arches. Yum Yum!
But every parabola is different. So how do we know the equation of the parabola we want. The demonstration below courtesy of Wolfram shows us just how. Slide a, b and c below to see how we can use equations to shape a parabola the way we like.
https://mathspace.com.au/blog/wolfram-demo/)
But every parabola is different. So how do we know the equation of the parabola we want. The demonstration below courtesy of Wolfram shows us just how. Slide a, b and c below to see how we can use equations to shape a parabola the way we like.
https://mathspace.com.au/blog/wolfram-demo/)
This is the other example of a parabola
It was constructed between July 28, 1923 through January 19, 1932. It is located in Sydney, Australia. It was designed and built by Dorman Long and Company.
It was constructed between July 28, 1923 through January 19, 1932. It is located in Sydney, Australia. It was designed and built by Dorman Long and Company.
This is an illustration of a parabola that forms in different directions..
General Form: y ² + Dx +Ey + F = 0
Standard Form: ( y - k ) ² = 4p ( x - h )
The vertex of the parabola is at (h,k). The distance (p) from the focus to the vertex
is the same as the the distance from the vertex to the directrix. The focus and the
directix are equidistant from any point on the curve. Try different values of h, k and
p to see their effect.
(http://emathlab.com/Algebra/Conics/parabolaH.php)
Standard Form: ( y - k ) ² = 4p ( x - h )
The vertex of the parabola is at (h,k). The distance (p) from the focus to the vertex
is the same as the the distance from the vertex to the directrix. The focus and the
directix are equidistant from any point on the curve. Try different values of h, k and
p to see their effect.
(http://emathlab.com/Algebra/Conics/parabolaH.php)
PARABOLA TO THE LEFT
Please note that when the parabola opens to the left, then the vertex of the parabola is the rightmost point on the parabola. Please note that this is also the maximum point of this parabola because the value of x at this point is greater than the value of x at any other point on the graph of this parabola.
Graph of x = -y^2 + 10y - 16 is shown below.
A left open parabola y = -2(x - 4) is shown here.
The axis of symmetry is the x axis (y = 0) which
cuts the parabola at the vertex (4, 0)
(http://www.algebra.com/algebra/homework/equations/EQ.lesson)
Please note that when the parabola opens to the left, then the vertex of the parabola is the rightmost point on the parabola. Please note that this is also the maximum point of this parabola because the value of x at this point is greater than the value of x at any other point on the graph of this parabola.
Graph of x = -y^2 + 10y - 16 is shown below.
A left open parabola y = -2(x - 4) is shown here.
The axis of symmetry is the x axis (y = 0) which
cuts the parabola at the vertex (4, 0)
(http://www.algebra.com/algebra/homework/equations/EQ.lesson)
A downward parabola (x - 2)2 = -(y - 5) is shown
here. The axis of symmetry is the vertical line
x =2 which cuts the parabola at the vertex (2, 5).
Vertex for this graph represents the maximum
function value.
(http://www.mathcaptain.com/geometry/vertex.html)
here. The axis of symmetry is the vertical line
x =2 which cuts the parabola at the vertex (2, 5).
Vertex for this graph represents the maximum
function value.
(http://www.mathcaptain.com/geometry/vertex.html)
n upward parabola x2 = y is shown here.
The axis of symmetry is the y axis (x = 0)
and it cuts the parabola at the vertex (0, 0).
Vertex here represents the minimum function value.
(http://www.mathcaptain.com/geometry/vertex.html)
The axis of symmetry is the y axis (x = 0)
and it cuts the parabola at the vertex (0, 0).
Vertex here represents the minimum function value.
(http://www.mathcaptain.com/geometry/vertex.html)
VERTEX FORMULA:
The vertex form of equation to a parabola, not only shows the vertex, but also the line of symmetry.
If the equation is of the form
(y - k)2 = 4p(x -h), the parabola is symmetrical about the horizontal axis, y =k
The parabola opens to the right if p is positive and opens to the left if p is negative.
When the equation is of the form
(x - h)2 = 4p(y - k) , the parabola is symmetrical about the vertical axis x = h.
The parabola opens up if p is positive and opens down if p is negative.
If the equation of the parabola is given in the standard form, then the vertex can be found out by observing the equation. The vertex of the parabola is given by (h, k).
The equation of the parabola has to be rewritten in standard form by completing the squares if necessary.
Solved Example Question: Find the vertex of the parabola 2x2 - 8x + y + 6 = 0, rewriting it in standard form.
Solution:
2x2 - 8x + y + 6 = 0
2x2 - 8x = -y - 6 x and y terms separated
2(x2 - 8x) = -(y + 6)
x2 - 8x = -12(y + 6)
x2 - 8x + 16 = -12(y + 6) + 16 Completed the square on the left side
(x - 4)2 = -12(y - 26) Equation in vertex form.
Hence the vertex of the parabola is (4, 26). The axis of symmetry is x = 4.
The parabola opens down as p = -18 is negative.
(http://www.mathcaptain.com/geometry/vertex.html)
The vertex form of equation to a parabola, not only shows the vertex, but also the line of symmetry.
If the equation is of the form
(y - k)2 = 4p(x -h), the parabola is symmetrical about the horizontal axis, y =k
The parabola opens to the right if p is positive and opens to the left if p is negative.
When the equation is of the form
(x - h)2 = 4p(y - k) , the parabola is symmetrical about the vertical axis x = h.
The parabola opens up if p is positive and opens down if p is negative.
If the equation of the parabola is given in the standard form, then the vertex can be found out by observing the equation. The vertex of the parabola is given by (h, k).
The equation of the parabola has to be rewritten in standard form by completing the squares if necessary.
Solved Example Question: Find the vertex of the parabola 2x2 - 8x + y + 6 = 0, rewriting it in standard form.
Solution:
2x2 - 8x + y + 6 = 0
2x2 - 8x = -y - 6 x and y terms separated
2(x2 - 8x) = -(y + 6)
x2 - 8x = -12(y + 6)
x2 - 8x + 16 = -12(y + 6) + 16 Completed the square on the left side
(x - 4)2 = -12(y - 26) Equation in vertex form.
Hence the vertex of the parabola is (4, 26). The axis of symmetry is x = 4.
The parabola opens down as p = -18 is negative.
(http://www.mathcaptain.com/geometry/vertex.html)