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!

2,3, 5 2 3 5f x x x x

2 6 5x x x 3 24 11 30x x x

1,1 2,1 2 1 1 2 1 2f x x x x

21 1 2x x 3 23 1x x x

Modeling with Higher-Order PolynomialsSec. 2.3b…

Homework: p. 204-205 49-59 odd, 63, 67

Practice ProblemsPractice ProblemsDixie Packaging Company has contracted to make boxes witha volume of approximately 484 cubic inches. Squares are tobe 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 sizesquares should be cut from the cardboard?

20

25

x

x

Volume = (height)(length)(width)

Height = x Length = 25 – 2x

Width = 20 – 2x

25 2 20 2V x x x

Practice ProblemsPractice ProblemsDixie Packaging Company has contracted to make boxes witha volume of approximately 484 cubic inches. Squares are tobe 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 sizesquares should be cut from the cardboard?

20

25

x

x

Solve the equation:

25 2 20 2 484x x x Solve graphically!!!Solve graphically!!!

Window: [0, 10] by [0, 1000]

Squares with length 1.222 in. or 6.871 in. should be cut from the cardboard

Practice ProblemsPractice ProblemsHow many points do we need to determine a slant line? 22

How many points do we need to determine a quadratic function?

3, noncollinear3, noncollinear

In general, we need (n + 1) points positioned with sufficientgenerality 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 ProblemsPractice ProblemsUse cubic regression to fit a curve through the four points givenin the following table.

x

y

–2

2

1

0.5

3

–0.2

8

1.253 20.0061 0.0177 0.5007 0.9769y x x x

Use quartic regression to fit a curve through the five points givenin the following table.

x

y

3

–2

4

–4

5

–1

6

84 3 20.375 6.917 44.125 116.583 111y x x x x

8

3

How good are the fits of these curves??? PERFECT!!!PERFECT!!!

Practice ProblemsPractice ProblemsU.S. Farm Exports

Year Amount (billions)1980 41.21985 29.01990 39.51992 43.21993 42.91994 46.31995 56.31996 60.41997 57.21998 51.81999 48.52000 51.6

Try various polynomial regressionmodels (degrees n = 1, 2, 3, 4) forthe given data, and decide whichis the best model (let x = 0 be 1980).

Practice ProblemsPractice ProblemsTry various polynomial regression models (degrees n = 1, 2, 3, 4)for the given data, and decide which is the best model.

3 20.0276 0.9036 6.6274 41.6526y x x x Cubic model:

1.070 33.154y x Linear model:

20.0410 0.223 35.864y x x Quadratic model:

4 3 20.00109 0.0173 0.324 4.290 41.179y x x x x Quartic model:

Which is the best???Which is the best???(think (think graphicallygraphicallyand and algebraically)algebraically)

Practice ProblemsPractice ProblemsEconomists for Smith Brothers, Inc., find the company profit Pby using the formula P = R – C, where R is the total revenuegenerated by the business and C is the total cost of operatingthe business.

(a) Using data from past years, the economists determined therevenue and cost functions below, where x is the number ofcustomers patronizing the business. How many customers mustSmith Bros. have to be profitable each year?

20.0125 412R x x x 312,225 0.00135C x x Graph P x R x C x

Window: [–10, 600] by [–1000, 100000]

Where is thisfunction positive?29.732 541.743x

They need between 30 and 541 customersThey need between 30 and 541 customers

Practice ProblemsPractice ProblemsEconomists for Smith Brothers, Inc., find the company profit Pby using the formula P = R – C, where R is the total revenuegenerated by the business and C is the total cost of operatingthe business.

(b) How many customers must there be for Smith Bros. to realizean annual profit of $60,000?

20.0125 412R x x x 312,225 0.00135C x x

Graph P(x) with the line y = 60,000 Where is the profit functionequal to the line?

200.491 429.729x Either 201 or 429 gives a profit slightly overEither 201 or 429 gives a profit slightly over$60,000; 200 or 430 customers both yield slightly$60,000; 200 or 430 customers both yield slightlyless than $60,000less than $60,000