12.5 Parabolas: from definition to equation

 Page 1 / 1
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 $\frac{1}{4a}$ .

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. $\sqrt{{x}^{2}+{y}^{2}}$ . 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. $\sqrt{{x}^{2}+{y}^{2}}=y+4$ . This is the key step! By asserting that $d1=d2$ , 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 ${x}^{2}=8y+16$ . Solve for $y$ , and you end up with $y=\frac{1}{8}{x}^{2}-2$ . Some of them will have difficulty seeing that this is the final form—point out that we can rewrite it as $y=\frac{1}{8}\left(x-0{\right)}^{2}-2$ , 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.

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
what is the stm
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?
what is a peer
What is meant by 'nano scale'?
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
what is Nano technology ?
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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
hi
Loga
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
Got questions? Join the online conversation and get instant answers!