Learning Resources
Lesson
This Investigation guides you in developing a way to find a polynomial function from known points on the function. The context used is that of a steamroller going back and forth over a patch while repairing a road. You are given the position of the steamroller at various times and your task is to find its position at 8 seconds. To do this you will need to find the equation of its path. The steps outlined in the procedure will guide you through this process.
You have done extensive work solving systems of equation in Mathematical Modeling, Book 2. You should recall the Investigation in which matrices were used to find the quadratic equation representing the path of a ball thrown in the air.
Do you recall how many points were necessary to determine the quadratic equation? How many points do you think you will need to determine the equation of the cubic function in this Investigation?
Read Investigation 5 on page 82 of your text. Follow the steps outlined in the procedure and answer the related Investigation Questions.
While the system of equations you have created may be solved many ways, the use of matrices is the most efficient. If you choose to solve the system of equation that you have created by matrices, a refresher on how to use the TI-83 calculator is provided.
The equation can also be found using cubic regression on the graphing calculator. You can use this method to verify your solutions. It is necessary, however, that you practice the algebraic and matrix methods in many of the assigned questions.
Please note: To complete investigation 6 in the next section, you will need some materials. Five different sizes of balls will be used. As well, you will need a measuring tape, or string and a metre stick. Be sure to have these supplies gathered by the time you are ready to begin Section 2.3. See your on-site teacher if you are having difficulty finding all the objects.
Activity
C.Y.U. page 83 #'s 58 - 60
When you have completed these questions, ask your on-site teacher to get the solutions for you from the Teacher's Resource Binder and check them against your answers. After you do this, if there is something you had trouble with and still do not understand, contact your on-line teacher for help.
Test Yourself
1. The graph of a cubic function y = f(x) contains the points (-2, 0), (1, -5), (2, -3) and (-1, 4). Find the equation of the function.
SOLUTION
1. You are looking for an equation of the form y = ax3 + bx2 + cx + d, a 
0.
Solution: y = 1.25x3 - 0.33x2 - 5.75x - 0.67