Question

Can you please show an example of a Polynomial Curve-Fitting with 4 points?

Answer 1

Any 4 points?

Answer 2

Any
As easy as possible please

Answer 3

Given the points {a, b, c, d} the polynomial

P(x) = (x-a)(x-b)(x-c)(x-d)

works.

Answer 4

How about if you're given (-1,3) (0,5) (1,-1) (2,2)

Answer 5

Okie dokie, do you know linear algebra/are you comfortable working with matrices?

Answer 6

Yes, I am. This is our upcoming topic in our class.

Answer 7

Perfect (otherwise this problem would be a lot harder)!

In this case, a degree 3 polynomial will work,

\[P(x) = a_0 +a_1x+a_2x^2+a_3x^3\]

and our job is to find

\[{a_0,a_1,a_2,a_3}\]

Answer 8

Yes.Then substitue the ff points to the polynomial?

Answer 9

If you can show the one already at the row echelon form, itcould be great help

Answer 10

Yup,yup. Since P(x) must pass through each of our points so we obtain the system

P( -1 ) = 3
P( 0 ) = 5
P( 1 ) = -1
P( 2 ) = 2

Answer 11

I see. Please continue, sir.

Answer 12

K so our equations look like

\[a_0 + a_1(-1)+a_2(-1)^2+a_3(-1)^3 = 3\]
\[a_0+a_1(0)+a_2(0)^2 + a_3(0)^3 = 5\]
\[a_0+a_1(1)+a_2(1)^2+a_3(1)^3 = -1\]
\[a_0+a_1(2)+a_2(2)^2+a_3(2)^3 = 2\]

Answer 13

The matrix will be.
1 -1 1 -1 3
1 0 0 0 5
1 1 1 1 -1
1 2 4 8 2

Answer 14

In Av=b form, matrix for that would be

A v = b

|1 -1 -1 -1| |a_0| | 5 |
|1 0 0 0| |a_1| = | 3 |
|1 1 1 1| |a_2| |-1 |
|1 2 2 2| |a_3| | 2 |

So we just need to the inverse of A

Answer 15

Your way was much easier to write it lol.

Answer 16

i guess so. lol

Answer 17

I don't know how to work with inverse yet. I'm sorry

Answer 18

Ahh, well it's the same thing as putting the matrix A into row reduced echelon form

Answer 19

shouldn't the first row be 1 -1 1 -1 5?

Answer 20

Oh, I see.

Answer 21

Whoops, you are correct, I forgot to expand any of the powers, was too focused on making it look like a matrix haha

Answer 22

Now, we just row reduce this and we're particularly done : D

|1 -1 1 -1| 5 |
|1 0 0 0 | 3 |
|1 1 1 1 | -1 |
|1 2 4 8 | 2 |

Answer 23

a0 = 3.000
a1 = -4.500
a 2 = -1.000
a3 = 1.500

Answer 24

Is it? :D Thank you for your time sir :D

Answer 25

Yup yup, just sub those values back in and write your answer as

\[P(x) = a_0 +a_1x+a_2x^2+a_3x^3\]

and we're done :) Good job!

Answer 26

Thank you for your help sir :D