1. 1 Section 5.4 Polynomial Data Modeling

2. 2 LINEAR MODELS Recall that a first-degree polynomial function is, more specifically, a linear function. That is, a first-degree polynomial is of the form P(t) = a + bt. Given any two year-population pairs, we can solve two linear equations in two unknowns to find the values of a and b.

3. 3 EXAMPLE In 1990 the population of Barnesville was 4,747, and in 2000 the population was 5,972. Find a linear model for this data. Given the pairs (1990, 4747) and (2000, 5972), we know that P(1990) = 4747 and P(2000) = 5972. So, we get

4. 4 QUADRATIC MODELS Recall that a second-degree polynomial function is, more specifically, a quadratic function. That is, a second-degree polynomial is of the form P(t) = a + bt + ct 2. Given any three year-population pairs, we can solve three linear equations in three unknowns to find the values of a, b, and c.

5. 5 EXAMPLE The Gwinnett County population in 1960 was 43.5 thousand, in 1980 it was 166.9 thousand, and in 2000 it was 588.4 thousand. Given the pairs (1960, 43.5), (1980, 166.9), and (2000, 588.4), we have P(1960) = 43.5, P(1980) = 166.9, and P(2000) = 588.4. So, we get

6. 6 HIGHER-ORDER MODELS Just as two data points determine a linear (first- degree) model and three data points determine a quadratic (second-degree) model, four data points determine a cubic (third-degree) model. A cubic model has the form P(t) = a + bt + ct 2 + dt 3. Five data points determine a quartic (fourth- degree) model. A quartic model has the form P(t) = a + bt + ct 2 + dt 3 + et 4.

7. 7 EXAMPLE Gwinnett County population data. YearPop. (thous.) 196043.5 197072.3 1980166.9 1990352.9 2000588.4 a)Find a cubic model using the years 1960, 1970, 1980, and 1990. b)Find a quartic model using all five years.