1.6 – Polynomial Functions from Data A difference table can be used to analyze whether a given set of data points fits into a polynomial function model. A graphing calculator can perform various regressions to determine the strength of this model. Example 1: Place the following data into a difference table and come up with an algebraic model for these data. x 1 5 3 2 6 4 f(x) 4 72 30 15 99 49
Data needs to be organized sequentially. To be useful input (x) values also have to have a constant difference 1st difference is calculated by subtracting two consecutive output values (i.e. y = y2 – y1) or lower minus upper value
3
x 1 2 3 4 5 6
f(x) 4 15 30 49 72 99
f(x) 11 15 19 23 27
2
f(x) 4 4 4 4
f(x)
A constant second difference points to a 2nd degree relation.
Data says that when x=1 y=4 or f(1)=4
So f(x) = ax2 + bx + c and then know
f(1) = 4 f(2) = 15 f(3) = 30
so
a(1)2 + b(1) + c = 4 a(2)2 + b(2) + c = 15 a(3)2 + b(3) + c = 30
Label equations so other can follow your work
Solve using elimination

3a + b = 11 5a + b = 15 2a = 4 or a = 2 5(2) + b = 15 or b = 5 2 + 5 + c = 4 or c = 3 f(x) = 2x2 + 5x  3
Sub a = 2 back into yields Sub a = 2 and b = 5 into yields Therefore functions is; Example 2:
The following data tracked the population in a small town over a 6 year period. Place the data into a difference table as well as a graphing calculator to come up with an algebraic model for this data. Then use the model to estimate what the population was in 1979 and 1990. Year (x) 1981 1982 1983 1984 1985 1986 Populations (y) 4031 4008 3937 3824 3675 3496 x 1981 1982 1983 1984 1985 1986 f(x) 4031 4008 3937 3824 3675 3496 f(x) 23 71 113 149 179
2
f(x)
3
f(x) 6 6 6
A constant third difference points to a 3rd degree relation.
48 42 36 30
1.6 – polynomial functions from data
Difference table show a 3rd degree relation which is confirmed by a regression value of 1 on the graphing calculator So we...