Lets start with some whiteboard todayUsing only algebra

Let’s start with some whiteboard 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. 3 b…

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 – 2 x Width = 20 – 2 x 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!!! Window: [0, 10] by [0, 1000] 25 x x 20 Squares with length 1. 222 in. or 6. 871 in. should be cut from the cardboard

Practice Problems How many points do we need to determine a slant line? 2 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 Use cubic regression to fit a curve through the four points given in the following table. x y – 2 2 1 0. 5 3 – 0. 2 8 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 y 3 – 2 4 – 4 5 – 1 6 8 8 3

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

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. (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? Graph Window: [– 10, 600] by [– 1000, 100000] Where is this function positive? They need between 30 and 541 customers

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? Graph P(x) with the line y = 60, 000 Where is the profit function equal to the line? Either 201 or 429 gives a profit slightly over $60, 000; 200 or 430 customers both yield slightly less than $60, 000