5.9 Write Polynomial Functions and Models

 

Finite differences

The differences of consecutive y-values when the x-values in a data set are equally spaced

 

Example 1

Write a cubic function

 

Write the cubic function whose graph is shown.

 

 


Solution

1.  Use the three given x-intercepts to write the function in intercept form.

f(x) = a __(x+ 2)(x+ 1)(x - 2)__

2.  Find a by substituting the coordinates of the fourth point.

__-8__ = a(0 + 2) (0 + 1) (0 - 2)

__-8__ = __-4__ a

a = __2__

 

The function is f(x) = __2(x + 2)(x + 1)(x - 2)__.

CHECK Check the end behavior for f. The degree of f is __odd__ and a _>_0. So,
f(x) ® __- ¥__ as x ® - ¥ and f(x) ® __+ ¥__ as x ® + ¥ , which matches the graph.

 

 

Example 2

Finding finite differences

 

An equation for a polynomial function is f(n) = n3 + 2n2 - 4n + 3. Show that this function has constant third-order differences.

 

Solution

Write the first several function values. Find the first-order differences by subtracting consecutive function values. Then find the second-order differences by subtracting consecutive __first-order__ differences. Finally, find the third-order differences by subtracting consecutive __second-order__ differences.

 

 

 

 

 

 

f(l)        f(2)      f(3)      f(4)      f(5)      f(6)

Function values for

_0_     _7_     _30_     _75_   _148_    _255_

equally-spaced n-values

_7_     _23_    _45_    _73_    _107_

First-order differences

           _16_     _22_    _28_    _34_

Second-order difference

                 _6_       _6_       _6_

Thrid-order differences

 

 

 

Each third-order difference is _6_, so the third-order differences are constant.

 

PROPERTIES OF FINITE DIFFERENCES

1.      If a polynomial function f(x) has degree n, then the nth-order differences of function values for equally-spaced x-values are __nonzero and constant__.

2.      Conversely, if the nth-order differences of equally spaced data are __nonzero and constant__, then the data can be represented by a polynomial function of degree n.

Example 3

 

The values of a polynomial function for five consecutive whole numbers are given below. Write a polynomial function for f(n).

f(l) = 3     f(2) = 8     f(3) = 14        f(4) = 21    f(5) = 29

 

Solution

Begin by finding the finite differences.

f(l)        f(2)      f(3)      f(4)      f(5)

Function values for equally

_3_      _8_      _14_    _21_    _29_

spaced n-values

     _5_      _6_       _7_      _8_

First-order differences

            _1_      _1_      _1_

Second-order differences

 

Because the __second-order__ differences are constant, you know that the numbers can be represented by a __quadratic__ function which has the form f(n) = __an2 + bn + c__.

Substitute the first three values into the function to obtain a system of linear equations in __three__ variables.

 

__a(1)2 + b(1) + c= 3 ____             __a + b + c= 3__

__a(2)2 + b(2) + c= 8____              __4a + 2b + c= 8__

__a(3)2 + b(3) + c= 14 ___             __9a+3b + c= 14__

 

The solution to the linear system is

a =       b =       c = __-1__.

 


So, the polynomial function is f(n) =