Cubic vs. Quadratic Modeling

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Cubic vs. Quadratic Modeling Powered By Docstoc
					Professor Al Jabra and his assistant Cody Braker have been working on a secret
excavation of an ancient lost city, “Polynomialville.” They have come across a
tarnished silver box which they feel may contain clues about this lost civilization.
The problem is that the box is locked and can only be opened using a combination.
The following list of numbers is inscribed on the box:

-5, 120, -4, 76, -3, 45, -2, 22, -1, 7, 0, 0, 1, -3, 2, -2 as well as some other numbers
that are not visible because the box is so tarnished.

Cody Braker believes that these numbers correspond to x- and y-coordinates and
tells the Professor that they should plot the points on a grid and determine an
equation to obtain the missing x- and y-values!

      x                 y                                     y
     -5               120
     -4                76
     -3                45
     -2                22
     -1                7
     0                 0
      1                -3                                      0                       x
     2                 -2

Your expertise in mathematics tells you that this relationship appears to be:


MCR 3U0 – Polynomial Functions                                                   Page 1 of 3
You perform a regression analysis to determine the curve of best fit for this data:

    1. Enter the data into two lists (STAT – EDIT)
    2. 2ND – Y= (STATPLOT). Turn the PLOT ON and make sure variables are L1
       and L2.
    3. Set WINDOW appropriately and GRAPH.
    4. Select STAT – CALC and select appropriate model.
    5. Select 2ND – 1, 2ND – 2, and ENTER. You will obtain values of the parameters
       that give you a model as well as a value of R2 (goodness of fit).

Equation: _________________________________________

Cody Braker has just discovered another set numbers!

6, -7, 10, -108

Add these to your list.

How does the graph change?

Does your model from above still work?

Perform another regression analysis to determine another model that could fit this
data. You may need multiple attempts – decide on the best one.

Equation: _________________________________

MCR 3U0 – Polynomial Functions                                              Page 2 of 3

    1. How did the domain and range of the original data affect your predictions?

    2. The model that you should have obtained has general form
       y = ax 3 + bx 2 + cx + d . This is a CUBIC EQUATION. Determine the zeros
           of your model. How many do you have?

    3. Experiment with your calculator – come up with different cubic equations
       and complete the following table. What is the maximum number of real
       zeros for a cubic equation?

            Equation                     Zeros                   # of Zeros

    4. A CUBIC EQUATION can also be expressed in the form y = a(x-r)(x-s)(x-t)
       where r, s, and t are the zeros. Can you come up with a cubic equation with:
    i)        Three unique zeros?

    ii)       One repeated zero and one unique zero?

    iii)      One repeated zero

MCR 3U0 – Polynomial Functions                                             Page 3 of 3

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