5-8 Polynomial Models in the Real World

Apply polynomial models to real-world situations by fitting data to

linear, quadratic, cubic, or quartic models.

Regression You can use a graphing calculator to

find a polynomial to model a particular data set.

Ex: find the polynomial function that models the data.

Enter into L1 and L2 Adjust window and make sure your STAT PLOT is on. Use 2nd STAT then CALC

• Try graphing the LinReg, QuadReg, and CubicReg until you find the best fit.

The (n+1) Point Principle For any set of n+1 points in the coordinate

plane that pass the vertical line test, there is a unique polynomial of degree at most n that fits the points perfectly.• Basically, if you have 2 points, you can graph a line. 3

points that are not on a line can determine a parabola. 4 points not on a line can graph a cubic, etc.

Practice What polynomial function has a graph

that passes through (0, -3), (1, -1), (2, 5), and (-1, -7)?

Using the (n+1) Point Principle, there are 4 points that are not on a line so we can find a cubic polynomial.

Substitute each point into standard form for x and y.

Solve the system using a 45 matrix in your calculator.

Modeling Data What linear model best fits the data?

Use the model to estimate the milk production in 2000.

Enter the data into lists (use years since 1955) and find the LinReg.

Substitute 45 years to find the production in 2000.

23.16 billion pounds of milk

Comparing Models Using the same data (milk production),

compare the linear model and a quadratic model for the data.

The quadratic appears to be a better fit and applies to the (n+1) Point Principle.• Look at values

Which makes more sense given the context?• The linear model continues to rise as time goes on.• The quadratic has down and down end behavior so it

will decrease to zero and into negative values.• Linear model is more likely.

Interpolation and Extrapolation Model appears to be cubic so try the CubicReg Notice the value is close to 1

• Fits the data well. Use the model to estimate consumption in 1980, 2000, and 2012 Use interpolation for 1980 and 2000

because it fits with the pattern in the table. You would be least confident in 2012

estimate because it’s extrapolated and the cubic function increases rapidly after 2001.