Question

Scientific experiments often produce a set of ordered pairs from measurements taken. Sometimes you can fit polynomial functions through these points as a way to model the data and make predictions.

Answer 1

Example

Consider the following four ordered pairs: (1, 6) (2, 11) (4, 39) (7, -54). Each of these ordered pairs represents an (x, y) pair. Let y = f(x). We can also think of each pair as an (x, f(x)) pair.

Let’s say we wanted to find a third-degree polynomial that fits these ordered pairs. You could substitute the values of the ordered pairs into this equation:

f(x) = A + B(x – xo) + C(x – xo)(x – x1) + D(x – xo)(x – x1)(x – x2), where A, B, C, and D are constants.

In the equation above, xo, x1, and x2 represent the x-coordinates of three of the ordered pairs that are given. In this case we will let xo=1, x1 =2, and x2=4.

By substituting the known values from the ordered pairs into the equation, we can find the constants A, B, C, and D.

Substituting (1, 6) in the equation by letting x=1 and f(x)=6, after doing the math and simplifying, we obtain 6 = A.
Substituting (2,11) in the equation, 11 = A + (2 – 1)B; so 11 = A + B.
Substituting (4, 39) in the equation, 39 = A + (4 – 1)B + (4 -1)(4 – 2)C; so 39 = A + 3B + 6C .
Substituting (7, -54) in the equation, we get -54 = A+ (7 – 1)B + (7 – 1)(7 – 2)C + (7 -1)(7 -2)(7 -4)D; so -54 =A + 6B + 30C + 90D.

Answer 2

Instructions

In this project we will find the polynomial fits though the four ordered pairs given. Some of the work has been done already in the example. With the equations above, we now have enough information to help determine the values of A, B, C, and D. After A, B, C, and D are found, we can expand the polynomial to find the reduced form of the function.

Answer 3

What is the value of A, B, C, D?