Math question: What's so special about 4?

[JustAnAverageGuy]

x² = 4py | y² = 4px

Why is it four?

A = bh/2. Obviously for area of a triangle you can explain it by putting two of them together to form a square, but what's so special about 4 for parabolas?

 

[CaptainGoodnight]

Make a simple sketch of a parabola opening rightward. Let (x,y) be a point on the parabola. Let the vertex be (h,k). Let p be the *directed* distance *from* the vertex *to* the focus. It should be clear from the sketch that the focus has coordinates (h+p,k) and the directrix has equation x=h-p.

Since a parabola is the set of all points (x,y) that are equidistant from a fixed point (focus) and a fixed line (directrix), the distance formula can be used to state this relationship as an equation.

distance from (x,y) to (h+p,k) = distance from (x,y) to (h-p,y)

sqrt[(x-(h+p))^2 + (y-k)^2] = sqrt[(x-(h-p))^2 + (y-y)^2]

Square both sides, and with some more work you should be able to rewrite this as:
(y-k)^2 = 4p(x-h)

 

[Kyteland]

Open up your calculus book to the chapter on conic sections. There should be a derivation of the equation there.

There are two concepts to use when dealing with a parabola, the focus and the directrix. The focus is a fixed point and the directrix is a fixed line. A parabola is the set of points in a plane that are equidistant from the focus and the driectrix.

If you place the vertex of the parabola on the origin and make the directrix parallel to the x axis you get a simple equation for a parabola. Because of the properties of a parabola, the focus (F) is at the point (0,p) and and the directrix (D) is the line y=-p. Taking the focus and a point on the parabola (P) you can calulate that the distance from F to P (|FP|) as sqrt(x^2+(y-p)^2)
|FP|=sqrt(x^2+(y-p)^2)
The distance from D to P is abs(y+p)
|DP|=abs(y+p)
Since |FP|=|DP| (definition of a parabola)
sqrt(x^2+(y-p)^2)=abs(y+p)
You can simplify this by squaring both sides and solving fro x^2
x^2+(y-p)^2=(y+p)^2
x^2+y^2-2py+p^2=y^2+2py+p^2
Remove like terms.
x^2-2py=2py
x^2=4py

The 4 isn't a magic number. It comes from the fact that you expand the equation (a+b)^2-(a-b)^2 and the 2ab terms combine to make 4ab, while the other tems cancel out.

 

[SouthPaW1227]

math is something humans made up. all the "facts" are just things we believe

 

[CRXican]

yikes

nerd alert!!

 

[Drakkon]

4 is the first non-prime (other than 1) to appear in the list of natural numbers thus it has that special property of being divisble by primes before it...since so much in math revovles around the prime numbers 4 is bound to show up...Actually if you think about it 2 is more common...I mean what is 4 but just 2 x 2

QED

 

[CaptainGoodnight]

Originally posted by: SouthPaW1227
math is something humans made up. all the "facts" are just things we believe

Yay for Nominalism!

 

[aplefka]

Fvck conics, seriously... hyperbolas are blowing my mind right now. Everything was peachy until we got to hyperbolas, then the foci started becoming hella weird answers and since the book was cheap and wanted to save paper, the answers in the back are only every 4th odd cuz of the graphs.

If anyone feels like tutoring me over AIM, PM me.

 

[JustAnAverageGuy]

:beer: kyteland

Originally posted by: CRXican
yikes

nerd alert!!

:Q

I was hoping for a simple answer

 

[RedCOMET]

I thought four was special becuase it came after 3 and was before 5.
Gee, i guess i was wrong.

 

[Kyteland]

Originally posted by: JustAnAverageGuy
:beer: kyteland

Originally posted by: CRXican
yikes

nerd alert!!

:Q

I was hoping for a simple answer

*drinks*

Hopfully that was clear. I left out a lot of detail, but if you are studying parabolas you should already know that stuff.



aplefka,

If you need help, send me a PM. I'm a mathematician by trade (I do math proofs for slot machines) so I'm familiar with this stuff. I'd have to relearn part of it, though. You're better off hitting me with probability questions. (obviously)

DrPizza is a high school math teacher. He would be an even better person to direct questions to.

Edit: Also, if you understand elipses, hyperbolas nearly the same thing. An elipse is defined as all points in a plane where the the distances from the foci sum to a constant:
|P F1| + |P F2| = C
Whereas a hyperbola is defined as all points in a plane where the *difference* is a constant:
|P F1| - |P F2| = C

 

[JustAnAverageGuy]

Originally posted by: Kyteland
DrPizza is a high school math teacher. He would be an even better person to direct questions to.

I would have but Pizza was offline at the time

 

[cKGunslinger]

Originally posted by: SouthPaW1227
math is something humans made up. all the "facts" are just things we believe

Wait, if I have 1 apple, and Suzy give me another apple, you're saying I may not really have 2 apples, I just believe I do? :roll:

 

[Kyteland]

Originally posted by: JustAnAverageGuy

Originally posted by: Kyteland
DrPizza is a high school math teacher. He would be an even better person to direct questions to.

I would have but Pizza was offline at the time

Oh, I see how it is.

I'm just your rebound math guy.

*cries*

 

I'm not sure how I ever got through calculus.

 

[Gibsons]

Originally posted by: SampSon
I'm not sure how I ever got through calculus.

 

[Kyteland]

Originally posted by: cKGunslinger

Originally posted by: SouthPaW1227
math is something humans made up. all the "facts" are just things we believe

Wait, if I have 1 apple, and Suzy give me another apple, you're saying I may not really have 2 apples, I just believe I do? :roll:

He is really just another example of where too little information can be dangerous in the wrong hands.

It's true that a lot of mathematics is built upon unprovable axioms that we assume to be true. In euclidean geometry the terms point, line and plane are undefined. The entire system of is built up on these terms. *If* these axioms were somehow proved to be untrue then the entire system built upon it would be garbage.

A lot of pseudomath that you see is done this way. You can prove a lot of things precisely and mathematically if you make some bad assumptions to start with. Fortunately it is generally easy to disprove these assumptions in those cases. A number of creationism arguments and counter-arguments are formulated this way. They look good on paper, but are unfounded because they use bad axioms as a foundation for the "good" math that follows.

Go ahead and try to disprove three axioms above if you want, SouthPaW1227. If you can then yes, math is just "made up." I'll bet that you can't.

 

[Kyteland]

Originally posted by: SampSon
I'm not sure how I ever got through calculus.

Easy:
- Cram material in to head
- Take exam
- Allow material to slowly leak out of head (heavily aided by :beer: )
- Never use calculus again

That works for most people.