Let’s start with a problem today…Using only algebra, find a cubic function with the given zeros. Answer should be in standard form. Support by graphing your answer…labeling your graph well!

Modeling with Higher-Order Polynomials Sec. 2.3b… Homework: p. 204-205 49-59 odd, 63, 67

Practice Problems Dixie Packaging Company has contracted to make boxes with a volume of approximately 484 cubic inches. Squares are to be cut from the corners of a 20-in. by 25-in. piece of cardboard, and the flaps folded up to make an open box. What size squares should be cut from the cardboard? Volume = (height)(length)(width) Height = x Length = 25 – 2x Width = 20 – 2x 25 x x 20

Practice Problems Dixie Packaging Company has contracted to make boxes with a volume of approximately 484 cubic inches. Squares are to be cut from the corners of a 20-in. by 25-in. piece of cardboard, and the flaps folded up to make an open box. What size squares should be cut from the cardboard? Solve the equation: Solve graphically!!! 25 Window: [0, 10] by [0, 1000] x  Squares with length 1.222 in. or 6.871 in. should be cut from the cardboard x 20

Practice Problems 2 3, noncollinear How many points do we need to determine a slant line? How many points do we need to determine a quadratic function? 3, noncollinear In general, we need (n + 1) points positioned with sufficient generality to determine a polynomial function of degree n. The process of fitting a polynomial of degree n to (n + 1) points is called polynomial interpolation.

Practice Problems PERFECT!!! Use cubic regression to fit a curve through the four points given in the following table. x –2 1 3 8 y 2 0.5 –0.2 1.25 How good are the fits of these curves??? PERFECT!!! Use quartic regression to fit a curve through the five points given in the following table. x 3 4 5 6 8 y –2 –4 –1 8 3

Practice Problems U.S. Farm Exports Year Amount (billions) 1980 41.2 1980 41.2 1985 29.0 1990 39.5 1992 43.2 1993 42.9 1994 46.3 1995 56.3 1996 60.4 1997 57.2 1998 51.8 1999 48.5 2000 51.6 Try various polynomial regression models (degrees n = 1, 2, 3, 4) for the given data, and decide which is the best model (let x = 0 be 1980).

Practice Problems Which is the best??? Try various polynomial regression models (degrees n = 1, 2, 3, 4) for the given data, and decide which is the best model. Linear model: Which is the best??? (think graphically and algebraically) Quadratic model: Cubic model: Quartic model:

Practice Problems They need between 30 and 541 customers Economists for Smith Brothers, Inc., find the company profit P by using the formula P = R – C, where R is the total revenue generated by the business and C is the total cost of operating the business. (a) Using data from past years, the economists determined the revenue and cost functions below, where x is the number of customers patronizing the business. How many customers must Smith Bros. have to be profitable each year? Where is this function positive? Graph Window: [–10, 600] by [–1000, 100000] They need between 30 and 541 customers

Where is the profit function Practice Problems Economists for Smith Brothers, Inc., find the company profit P by using the formula P = R – C, where R is the total revenue generated by the business and C is the total cost of operating the business. (b) How many customers must there be for Smith Bros. to realize an annual profit of $60,000? Where is the profit function equal to the line? Graph P(x) with the line y = 60,000 Either 201 or 429 gives a profit slightly over $60,000; 200 or 430 customers both yield slightly less than $60,000