Essential Standard:
I can write a function to represent data.
I can analyze which model best fits the data.

Warm Up:

Write an equation of that has a stretch of ½, reflection across the x axis, horizontal translation 2 units to the left and vertical translation 5 units down.

Polynomial Models

Polynomial functions can be used to model many real-world situations. You can use a graphing calculator to find functions that model data. For example the data in the table is plotted below in a scatter plot. Which function appears to be the best fit?

The Point Principle

For any set of points that pass the vertical line test, there is a unique polynomial of degree of at most that fits the points perfectly.

This confirms that two points determine a ______. Three points that are not on a line determine a ______. Four points not on a line or a parabola determine a unique ______, etc.

1)What are the possible degrees of the polynomial that best fits the points and ?

Find the Polynomial of Best Fit

  • First,you must turn on diagnostics in your calculator (not every time, just once).

Go to“Catalog”and then“DiagnositcsOn” thenENTER

  • We will use the calculator to help us determine the polynomial of best fit for a set of data

The polynomial function of best fit will have a ______closest to 1, this is found under r2 on the calculator

Using the Calculator (TI-84)

  1. Press the “STAT” button.
  2. Under the first heading“EDIT” selection option “1 Edit” and push enter.
  3. Delete all of the entries in each list (L1, L2, …). Put the curser over an entry and push delete.
    OR
    Put the curser over the list heading at the top like”L1” and push clear and then enter. Do not push delete (the entire L1 list will be deleted)
  4. Now that your lists have been deleted you need to enter your new information. Put the x into L1and the y into L2. Also, if you are entering years like 1990 and 1992, then you would enter 90 and 92.
  5. Now that your list is entered push “STAT” and go to the second heading of “CALC”.
  6. Under “CALC” you will choose options: 4 LinReg(to see if it is a straight line), 5 QuadReg (to see if it is a quadratic), 6 CubicReg(to see if it is cubic).
  7. Choose the model with r2closest to 1.
  8. Use the numbers given for a , b, c and ….etc towrite your equation.

Determine the polynomial function that best models the given data. Then write the polynomial function.

2)

3)

4) The graph shows the quadratic model for the milk production of Wisconsin dairy farms since 1900. The quadratic model fits the data points exactly because of the point principle. Given that both the linear and quadratic models are good fits, which seems more likely to represent the milk production over time? Why?

5) If four data points were given instead of three for the above scenario, would a cubic function be the best model for the data? Explain your answer.