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d ( F , P ) = d ( P , Q ) ( 0 x ) 2 + ( p y ) 2 = ( x x ) 2 + ( p y ) 2 .

Squaring both sides and simplifying yields

x 2 + ( p y ) 2 = 0 2 + ( p y ) 2 x 2 + p 2 2 p y + y 2 = p 2 + 2 p y + y 2 x 2 2 p y = 2 p y x 2 = 4 p y .
A parabola is drawn with vertex at the origin and opening up. A focus is drawn as F at (0, p). A point P is marked on the line at coordinates (x, y), and the distance from the focus to P is marked d. A line marked the directrix is drawn, and it is y = − p. The distance from P to the directrix at (x, −p) is marked d.
A typical parabola in which the distance from the focus to the vertex is represented by the variable p .

Now suppose we want to relocate the vertex. We use the variables ( h , k ) to denote the coordinates of the vertex. Then if the focus is directly above the vertex, it has coordinates ( h , k + p ) and the directrix has the equation y = k p . Going through the same derivation yields the formula ( x h ) 2 = 4 p ( y k ) . Solving this equation for y leads to the following theorem.

Equations for parabolas

Given a parabola opening upward with vertex located at ( h , k ) and focus located at ( h , k + p ) , where p is a constant, the equation for the parabola is given by

y = 1 4 p ( x h ) 2 + k .

This is the standard form    of a parabola.

We can also study the cases when the parabola opens down or to the left or the right. The equation for each of these cases can also be written in standard form as shown in the following graphs.

This figure has four figures, each a parabola facing a different way. In the first figure, a parabola is drawn opening up with equation y = (1/(4p))(x − h)2 + k. The vertex is given as (h, k), the focus is drawn at (h, k + p), and the directrix is drawn as y = k − p. In the second figure, a parabola is drawn opening down with equation y = −(1/(4p))(x − h)2 + k. The vertex is given as (h, k), the focus is drawn at (h, k − p), and the directrix is drawn as y = k + p. In the third figure, a parabola is drawn opening to the right with equation x = (1/(4p))(y − k)2 + h. The vertex is given as (h, k), the focus is drawn at (h + p, k), and the directrix is drawn as x = h − p. In the fourth figure, a parabola is drawn opening left with equation x = −(1/(4p))(y − k)2 + h. The vertex is given as (h, k), the focus is drawn at (h – p, k), and the directrix is drawn as x = h + p.
Four parabolas, opening in various directions, along with their equations in standard form.

In addition, the equation of a parabola can be written in the general form    , though in this form the values of h , k , and p are not immediately recognizable. The general form of a parabola is written as

a x 2 + b x + c y + d = 0 or a y 2 + b x + c y + d = 0 .

The first equation represents a parabola that opens either up or down. The second equation represents a parabola that opens either to the left or to the right. To put the equation into standard form, use the method of completing the square.

Converting the equation of a parabola from general into standard form

Put the equation x 2 4 x 8 y + 12 = 0 into standard form and graph the resulting parabola.

Since y is not squared in this equation, we know that the parabola opens either upward or downward. Therefore we need to solve this equation for y, which will put the equation into standard form. To do that, first add 8 y to both sides of the equation:

8 y = x 2 4 x + 12 .

The next step is to complete the square on the right-hand side. Start by grouping the first two terms on the right-hand side using parentheses:

8 y = ( x 2 4 x ) + 12 .

Next determine the constant that, when added inside the parentheses, makes the quantity inside the parentheses a perfect square trinomial. To do this, take half the coefficient of x and square it. This gives ( −4 2 ) 2 = 4 . Add 4 inside the parentheses and subtract 4 outside the parentheses, so the value of the equation is not changed:

8 y = ( x 2 4 x + 4 ) + 12 4 .

Now combine like terms and factor the quantity inside the parentheses:

8 y = ( x 2 ) 2 + 8 .

Finally, divide by 8:

y = 1 8 ( x 2 ) 2 + 1 .

This equation is now in standard form. Comparing this to [link] gives h = 2 , k = 1 , and p = 2 . The parabola opens up, with vertex at ( 2 , 1 ) , focus at ( 2 , 3 ) , and directrix y = −1 . The graph of this parabola appears as follows.

A parabola is drawn with vertex at (2, 1) and opening up with equation x2 – 4x – 8y + 12 = 0. The focus is drawn at (1, 3). The directrix is drawn at y = − 1.
The parabola in [link] .
Got questions? Get instant answers now!
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Put the equation 2 y 2 x + 12 y + 16 = 0 into standard form and graph the resulting parabola.

x = 2 ( y + 3 ) 2 2
A parabola is drawn with vertex at (−2, −3) and opening to the right with equation x = 2(y + 3)2 – 2. The focus is drawn at (0, −3). The directrix is drawn at x = −4.

Got questions? Get instant answers now!

The axis of symmetry of a vertical (opening up or down) parabola is a vertical line passing through the vertex. The parabola has an interesting reflective property. Suppose we have a satellite dish with a parabolic cross section. If a beam of electromagnetic waves, such as light or radio waves, comes into the dish in a straight line from a satellite (parallel to the axis of symmetry), then the waves reflect off the dish and collect at the focus of the parabola as shown.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
can you provide the details of the parametric equations for the lines that defince doubly-ruled surfeces (huperbolids of one sheet and hyperbolic paraboloid). Can you explain each of the variables in the equations?
Radek Reply

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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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