Formula of a curve?


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If I have three points (normal x,y) on a graph, how do I calculate the formula of the curve that would run through these three points?

I've sure I learnt about this stuff back at school but damned if I remember. That's what comes of not listening... :blush:

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If I have three points (normal x,y) on a graph, how do I calculate the formula of the curve that would run through these three points?

I've sure I learnt about this stuff back at school but damned if I remember. That's what comes of not listening... :blush:

What type of curve? You need to be more specific. I assume you mean a quadratic curve (i.e. x^2+2*x+1), but please specify before I go to the trouble explaining how to solve for one.

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Yes, quadratic, sorry.

I'm pretty sure it's simply (x^z) - 1, where z is presently unknown, but I'm not certain.

Well, there is a simple way to solve for an equation after you have three points. You see, the formula for any quadratic equation is ax^2+bx+c=y; you want to find a,b, and c, then. To do this, plug in your points for x and y to get three simultaneous equations, for example, using the points (1,2), (4,6), (-1,5), one would get

2=a+b+c

6=16a+4b+c

5=a-b+c

Then just solve those equations using the normal methods for solving simultaneous systems of equations (I won't explain it here, as they are pretty easy and tedious; look them up if you don't know them), and then you'll have your quadratic equation.

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OK, I think I get it.

So for (x,y), one of the equations is:

y = (x^2)a + (x^1)b + (x^0)c

or

y = (x^2)a + xb + c

Right?

Once I get the forumulas, it's no problem (but tedious, yes) to solve.

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Exactly. You could always use a calculator program to do it for you (I think some such programs are made for the TI-83 Plus), but they use the method I have described to get the equation.

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Hehe ok I'll give you a cool example. Let?s say I have the costs of a 10", 12" and a 14" pizza, which would be $5.95, $6.95, and $8.50 respectively. The length of the pizza would be the (x) variable and the price would be the y variable.i>---------------------b>

(10", $5.95)

(12", $6.95)

(14", $8.50)---------------------b>I would subtract the $6.95 from the $5.95 to get a difference of $1.00, then subtract $8.50 from $6.95 and get a difference of $1.55. Then you would subtract $1.55 from $1.00 and get a difference of $.55. Therefore we now know it's a 2nd degree polynomial so the form of the equation would be y = ax^2 + bx + c.i>---------------------b>Now let's say I wanted to find what the price of a 16" pizza would be. I would take those three points and plug them into the y = ax^2 + bx + c form of the equation so that the prices of the pizza are going from greatest to least.i>---------------------b>1.) $8.50 = 196a + 14b + c

2.) $6.95 = 144a + 12b + c

3.) $5.95 = 100a +10b + ci>---------------------b>

Now, the successive subtraction begins. Subtract equation numbers 1 and 2.

$1.55 = 52a + 2b (the c term cancels out)

Subtract equation numbers 2 and 3.

$1.00 = 44a + 2b (the c term cancels out)

Now subtract these two equations . . .

$.55 = 8a

Divide both sides by 8 and (a) is equal to .06875.

Plug (a) back into one of the equations with (a) and (b) in it to solve for (b).

$1.00 = 44(.06875) + 2b

$1.00 = 3.025 + 2b

Subtract 3.025 from both sides

-2.025 = 2b

Divided by 2 on both sides

b is equal to -1.0125

Now take an equation with (a), (b), and ? in it and solve for ?.

$5.95 = 100(.06875) + 10(-1.0125) + c

$5.95 = 6.875 + -10.125 + c

$5.95 = -3.25 + c

Add 3.25 on both sides

c is equal to 9.2---------------------

Now once you have (a), (b), and ?, plug them into the formula of a quadratic: y = ax^2 + bx + c.

y = .06875(x)^2 -1.0125(x) + 9.2

Now since I want to find how much a 16" pizza would cost me I would plug in 16" for x, and I'll show it using the Euler notation.

f(16") = .06875(16")^2 -1.0125(16") + 9.2

= 17.6 - 16.2 + 9.2

= $10.60 for a 16" p:Dza :D

Hmmm I am hungry now . . . .<To check this using your TI-83 if you have one, go to the STAT menu, 1: (Where it says EDIT), enter in the X variables for L1, and the Y variables for L2. Then go to STAT -- CALC, 5: (QuadReg), and type 2nd key L1, 2nd key, L2, and go to VARS -- Y-VARS, 1: (Function) and hit enter. Once you have hit enter again your calculator should display the equation and all the answers to the (a), (b) and ? variables.s.

Edited by Louisville Slugger
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