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A teacher's guide on teaching the connection between the definition and equation of a parabola, and how to get from one to the other.

OK, where are we? We started with the geometric definition of a parabola. Then we jumped straight to the machinery, and we never attempted to connect the two. But that is what we’re going to do now.

Remember what we did with circles? We started with our geometric definition. We picked an arbitrary point on the circle, called it ( x , y ), and wrote an equation that said “you, Mr. ( x , y ), are exactly 5 units away from the origin.” That equation became the equation for the circle.

Now we’re going to do the same thing with a parabola. We’re going to write an equation that says “you, Mr. ( x , y ), are the same distance from the focus that you are from the directrix.” In doing so, we will write the equation for a parabola, based on the geometric definition. And we will discover, along the way, that the distance from the focus to the vertex really is 1 4a size 12{ { {1} over {4a} } } {} .

That’s really all the setup this assignment needs. But they will need a lot of help doing it. Let them go at it, in groups, and walk around and give hints when necessary. The answers we are looking for are:

  1. x 2 + y 2 size 12{ sqrt {x rSup { size 8{2} } +y rSup { size 8{2} } } } {} . As always, hint at this by pushing them toward the Pythagorean triangle.
  2. y + 4 . The way I always hint at this is by saying “Try numbers. Suppose instead of ( x , y ) this were (3,10). Now, how about (10,3)?” and so on, until they see that they are just adding 4 to the y-coordinate. Then remind them of the rule, from day one of this unit—to find distances, subtract. In this case, subtract -4.
  3. x 2 + y 2 size 12{ sqrt {x rSup { size 8{2} } +y rSup { size 8{2} } } } {} = y + 4 . This is the key step! By asserting that d 1 = d 2 , we are writing the definition equation for the parabola.
  4. Good algebra exercise! Square both sides, the y 2 terms cancel, and you’re left with x 2 = 8 y + 16 . Solve for y , and you end up with y = 1 8 x 2 - 2 . Some of them will have difficulty seeing that this is the final form—point out that we can rewrite it as y = 1 8 ( x - 0 ) 2 - 2 , if that helps. So the vertex is (0,-2) and the distance from focus to vertex is 2, just as they should be.

Warn them that this will be on the test!

Which brings me to…

Time for another test!

Our first test on conics. I go back and forth as to whether I should give them a bunch of free information at the top of the test—but it’s probably a good idea to give them a chart, sort of like the one on top of my sample.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Advanced algebra ii: teacher's guide. OpenStax CNX. Aug 13, 2009 Download for free at http://cnx.org/content/col10687/1.3
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